Prasanna

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Big Ideas Math Book 7th Grade Advanced Answer Key | Big Ideas Math Answers 7th Grade Advanced Solutions Pdf

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• Chapter 1 Equations
• Chapter 2: Transformations
• Chapter 3: Angles and Triangles
• Chapter 4: Graphing and Writing Linear Equations
• Chapter 5: Systems of Linear Equations
• Chapter 6: Data Analysis and Displays
• Chapter 7: Functions
• Chapter 8 Exponents and Scientific Notation
• Chapter 9 Real Numbers and the Pythagorean Theorem
• Chapter 10 Volume and Similar Solids
• Chapter A: Equations and Inequalities
• Chapter B: Probability
• Chapter C: Statistics
• Chapter D: Geometric Shapes and Angles
• Chapter E: Surface Area and Volume

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Big Ideas Math Book Algebra 1 Answer Key Chapter 11 Data Analysis and Displays

By referring to the ultimate material of Big Ideas Math Algebra 1 Answers Chapter 11 Data Analysis and Displays, every students and learner get deep knowledge about the topic and become top in any examinations. Because each and every solution involved in this BIM Algebra 1 Ch 11 Data Analysis and Displays Solution Key is explained by the subject experts based on the Common Core Curriculum. Practice more and improve your math proficiency by simply tapping on the below link of respective Big Ideas Math Algebra 1 Answers of Chapter 11 & attempt all Data Analysis and Displays problems effectively.

• Data Analysis and Displays Maintaining Mathematical Proficiency – Page 583
• Data Analysis and Displays Mathematical Practices – Page 584
• Lesson 11.1 Measures of Center and Variation – Page(586-592)
• Measures of Center and Variation 11.1 Exercises – Page(590-592)
• Lesson 11.2 Box-and-Whisker Plots – Page(594-598)
• Box-and-Whisker Plots 11.2 Exercises – Page(597-598)
• Lesson 11.3 Shapes of Distributions – Page(600-606)
• Shapes of Distributions 11.3 Exercises – Page(604-606)
• Data Analysis and Displays Study Skills: Studying for Finals – Page 607
• Data Analysis and Displays 11.1–11.3 Quiz – Page 608
• Lesson 11.4 Two-Way Tables – Page(610-616)
• Two-Way Tables 11.4 Exercises – Page(614-616)
• Lesson 11.5 Choosing a Data Display – Page(618-622)
• Choosing a Data Display 11.5 Exercises – Page(621-622)
• Data Analysis and Displays Performance Task: College Students Study Time – Page – 623
• Data Analysis and Displays Chapter Review – Page(624-626)
• Data Analysis and Displays Chapter Test – Page 627
• Data Analysis and Displays Cumulative Assessment – Page (628-630)

Data Analysis and Displays Maintaining Mathematical Proficiency

The table shows the results of a survey. Display the data in a histogram.
Question 1.

Answer:

Question 2.

Answer:

The table shows the results of a survey. Display the data in a circle graph.
Question 3.

Answer:

Question 4.
ABSTRACT REASONING
Twenty people respond “yes” or “no” to a survey question. Let a and b represent the frequencies of the responses. What must be true about the sum of a and b? What must be true about the sum when “maybe” is an option for the response?
Answer:

Data Analysis and Displays Mathematical Practices

Mathematically proficient students use diagrams and graphs to show relationships between data. They also analyze data to draw conclusions.
Using Data Displays

Monitoring Progress

Question 1.
The table shows the estimated populations of males and females by age in the United States in 2012. Use a spreadsheet, graphing calculator, or some other form of technology to make two different displays for the data.
Answer:

Question 2.
Explain why you chose each type of data display in Monitoring Progress Question 1. What conclusions can you draw from your data displays?
Answer:

Lesson 11.1 Measures of Center and Variation

Essential Question
How can you describe the variation of a data set?

EXPLORATION 1
Describing the Variation of Data
Work with a partner. The graphs show the weights of the players on a professional football team and a professional baseball team.

a. Describe the data in each graph in terms of how much the weights vary from the mean. Explain your reasoning.
b. Compare how much the weights of the players on the football team vary from the mean to how much the weights of the players on the baseball team vary from the mean.
c. Does there appear to be a correlation between the body weights and the positions of players in professional football? in professional baseball? Explain.
Answer:

EXPLORATION 2

Describing the Variation of Data
Work with a partner. The weights (in pounds) of the players on a professional basketball team by position are as follows.
Power forwards: 235, 255, 295, 245; small forwards: 235, 235;
centers: 255, 245, 325; point guards: 205, 185, 205; shooting guards: 205, 215, 185
Make a graph that represents the weights and positions of the players. Does there appear to be a correlation between the body weights and the positions of players in professional basketball? Explain your reasoning.
Answer:

Communicate Your Answer
Question 3.
How can you describe the variation of a data set?
Answer:

Monitoring Progress

Question 1.
WHAT IF?
The park hires another student at an hourly wage of $8.45. (a) How does this additional value affect the mean, median, and mode? Explain. (b) Which measure of center best represents the data? Explain.

Answer:
a. Mean = 9.51, median = 8.65, mode = 8.25, 8.45
b. The median best represents the data. The mode is less than most of the data and the mean is greater than most of the data.

Explanation:
a. Mean = \(\frac { 8.45 + 16.5 + 8.75 + 8.65 + 9.10 + 8.25 + 8.45 + 8.25 + 9.25 }{ 9 } \)
= 9.51
Order the data
8.25, 8.25, 8.45, 8.45, 8.65, 8.75 9.10, 9.25, 16.50
Median = 8.65
Mode = 8.25, 8.45

Question 2.
The table shows the annual salaries of the employees of an auto repair service. (a) Identify the outlier. How does the outlier affect the mean, median, and mode? (b) Describe one possible explanation for the outlier.

Answer:
a. When you remove the outlier, the mean decreases 49000 – 38714 = 10286, the median decreases 39500 – 38000 = 1500 and the mode remains the same.

Explanation:
The outlier is 72,000
Find mean, median and mode without outlier
Mean = \(\frac { 32000 + 41000 + 38000 + 45000 + 42000 + 38000 + 35000 }{ 7 } \)
= 38714
arrange the data 32000, 35000, 38000, 38000, 41000, 42000, 45000
Median = 38000
Mode = 38000
When you remove the outlier, mean decreases 49000 – 38714 = 10286, the median decreases 39500 – 38000 = 1500 and mode remains the same.

Question 3.
After the first week, the 25-year-old is voted off Show A and the 48-year-old is voted off Show B. How does this affect the range of the ages of the remaining contestants on each show in Example 3? Explain.

Answer:
It does not show any effect on the ages of the remaining contestants on each show in Example 3

Explanation
Order the data
Show A: 19, 20, 20, 21, 22, 25, 25, 27, 27, 29, 29, 30, 31
So, the range 31 – 19 = 12 years
Show B: 19, 20, 21, 22, 22, 24, 25, 25, 27, 27, 32, 48, 48
So, the range 48 – 19 = 29 years

Question 4.
Find the standard deviation of the ages for Show B in Example 3. Interpret your result.

Answer:
The standard deviation is about 7.47. This means that the typical age of a contestant on show b differs from the mean about 7.47 years.

Explanation:

x	μ	x – μ	(x – μ)²
25	26	-1	1
20	26	-6	36
22	26	-4	16
27	26	1	1
48	26	22	484
32	26	6	36
19	26	-7	49
27	26	1	1
25	26	-1	1
22	26	-4	16
21	26	-5	25
24	26	-2	4

Variance = \(\frac { 1 + 36 + 16 + 1 + 484 + 36 + 49 + 1 + 1 + 16 + 25 + 4 }{ 12 } \)
= 55.833
Standard deviation = √55.833 = 7.47
The standard deviation is about 7.47. This means that the typical age of a contestant on show b differs from the mean of about 7.47 years.

Question 5.
Compare the standard deviations for Show A and Show B. What can you conclude?

Answer:
Show A standard deviation is 4.2 years and show b standard deviation is 7.47

Question 6.
Find the mean, median, mode, range, and standard deviation of the altitudes of the airplanes when each altitude increases by \(\frac{1}{2}\) miles.

Answer:
Mean = 3.35, Median = 2.95, Threre is no mode
Range = 3.3
Standard deviation = 1.85

Explanation:
The airplanes are 19/10, 7/5, 21/5, 3/2, 6/5, 9/10
Mean = 1.85 + 3/2 = 3.35
Arrange the data 9/10, 6/5, 7/5, 3/2, 19/10, 21/5
Median = 1.45 + 3/2 = 2.95
Threre is no mode
Range = 21/5 – 9/10 = 3.3

x	μ	x – μ	(x – μ)²
19/10	3.35	-1.45	2.102
7/5	3.35	-1.95	3.802
21/5	3.35	0.85	0.7225
3/2	3.35	-1.85	3.422
6/5	3.35	-2.15	4.622
9/10	3.35	-2.45	6

Variance = \(\frac { 2.102 + 3.802 + 0.7225 + 3.422 + 4.622 + 6 }{ 6 } \)
= 3.44
standard deviation = 1.85

Measures of Center and Variation 11.1 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
In a data set, what does a measure of center represent? What does a measure of variation describe?
Answer:

Question 2.
WRITING
Describe how removing an outlier from a data set affects the mean of the data set.
Answer:

Question 3.
OPEN-ENDED
Create a data set that has more than one mode.
Answer:

Question 4.
REASONING
What is an advantage of using the range to describe a data set? Why do you think the standard deviation is considered a more reliable measure of variation than the range?

Answer:
An advantage of using the range is that it is wasy to calculate.
Range = Difference between highest and lowest observed values
The disadvantage of using the range is that it uses only two values of the data set.
The standard deviation is more reliable measure of variation because it uses all the values of data set.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–8, (a) Find the mean, median, and mode of the data set and (b) determine which measure of center best represents the data. Explain. (See Example 1.)
Question 5.
3, 5, 1, 5, 1, 1, 2, 3, 15
Answer:

Question 6.
12, 9, 17, 15, 10

Answer:
Mean = 12.6, median = 12, no mode
b. Median represents the best data.

Explanation:
Mean = \(\frac { 12 + 9 + 17 + 15 + 10 }{ 5 } \)
= 12.6
Arrange the data 9, 10, 12, 15, 17
Median = 12
There is no mode.
b. Median represents the best data.

Question 7.
13, 30, 16, 19, 20, 22, 25, 31

Answer:
Mean = 22, median = 21, no mode
b. Median represents the best data.

Explanation:
Mean = \(\frac { 13 + 30 + 16 + 19 + 20 + 22 + 25 + 31 }{ 8 } \)
= 22
Arrange the data 13, 16, 19, 20, 22, 25, 30, 31
Median = 20 + 22/2 = 21
There is no mode.
b. Median represents the best data.

Question 8.
14, 15, 3, 15, 14, 14, 18, 15, 8, 16

Answer:
Mean = 13.2, median = 14.5, mode = 14 and 15

Explanation:
Mean = \(\frac { 14 + 15 + 3 + 15 + 14 + 14 + 18 + 15 + 8 + 16 }{ 10 } \)
= 13.2
Arrange the data 3, 8, 14, 14, 14, 15, 15, 15, 16, 18
Median = 14 + 15/2 = 14.5
Mode = 14, 15
b. Median represents the best data.

Question 9.
ANALYZING DATA
The table shows the lengths of nine movies.

a. Find the mean, median, and mode of the lengths.
b. Which measure of center best represents the data? Explain.
Answer:

Question 10.
ANALYZING DATA
The table shows the daily changes in the value of a stock over 12 days.

a. Find the mean, median, and mode of the changes in stock value.
b. Which measure of center best represents the data? Explain.
c. On the 13th day, the value of the stock increases by $4.28. How does this additional value affect the mean, median, and mode? Explain.

Answer:
a. Mean = -0.4016, median = 0.86 and no mode
b. median best represents the data.
c. The mean increases -0.4016 + 0.0414 = -0.3602, median increases by 0.19.

Explanation:
a. Mean = \(\frac { 1.05 + 2.64 + 0.66 + 2.03 + 0.67 – 0.28 – 13.78 + 4.02 – 3.01 – 2.41 + 1.39 + 2.20 }{ 12 } \)
= -0.4016
Arrange the data -13.78, -3.01, -2.41, -0.28, 0.66, 0.67, 1.05, 1.39, 2.03, 2.20, 2.64, 4.02
Median = 0.67 + 1.05/2 = 0.86
There is no mode
b. median best represents the data.
c. Mean = \(\frac { 1.05 + 2.64 + 0.66 + 2.03 + 0.67 – 0.28 – 13.78 + 4.02 – 3.01 – 2.41 + 1.39 + 2.20 + 4.28 }{ 13 } \)
= -0.0414
Arrange the data -13.78, -3.01, -2.41, -0.28, 0.66, 0.67, 1.05, 1.39, 2.03, 2.20, 2.64, 4.02, 4.28
Median = 1.05
No mode

In Exercises 11–14, find the value of x.
Question 11.
2, 8, 9, 7, 6, x; The mean is 6.
Answer:

Question 12.
12.5, -10, -7.5, x; The mean is 11.5.

Answer:
x = 51

Explanation:
Mean = \(\frac { 12.5 – 10 – 7.5 + x }{ 4 } \)
4 x 11.5 = -5 + x
46 + 5 = x
x = 51

Question 13.
9, 10, 12, x, 20, 25; The median is 14.

Answer:
x = 16

Explanation:
Arrange the data
9, 10, 12, x, 20, 25
Median = 14 = \(\frac { 12 + x }{ 2 } \)
28 = 12 + x
x = 16

Question 14.
30, 45, x, 100; The median is 51.

Answer:
x = 57

Explanation:
Median = 51 = \(\frac { 45 + x }{ 2 } \)
102 = 45 + x
x = 57

Question 15.
ANALYZING DATA
The table shows the masses of eight polar bears. (See Example 2.)

a. Identify the outlier. How does the outlier affect the mean, median, and mode?
b. Describe one possible explanation for the outlier.
Answer:

Question 16.
ANALYZING DATA
The sizes of emails (in kilobytes) in your inbox are 2, 3, 5, 2, 1, 46, 3, 7, 2, and 1.
a. Identify the outlier. How does the outlier affect the mean, median, and mode?
b. Describe one possible explanation for the outlier.

Answer:
a. Outlier increases mean and median but has no effect on the mode
b. One possible explanation for having 46 kilobyte email is it may have an attachments such as photos, videos or large documents.

Explanation:
The outlier is 46
Mean with outlier = \(\frac { 2 + 3 + 5 + 2 + 1 + 46 + 3 + 7 + 2 + 1 }{ 10 } \)
= 7.2
Mean without outlier = \(\frac { 2 + 3 + 5 + 2 + 1 + 3 + 7 + 2 + 1 }{ 9 } \) = 2.89
The difference of two means = 2.89 – 7.2 = -4.31
Hence the outlier increases the mean because by removing it, mean decreases by -4.31 kilobytes
Order the data
1, 1, 2, 2, 2, 3, 3, 5, 7, 46
Median with outlier = 2 + 3/2 = 2.5
Median without outlier = 2
The difference of medians = 2 – 2.5 = -0.5
Therefore, the outlier increases median because by removing it, the median decreases by -0.5 kilobytes.
Mode = 2

Question 17.
ANALYZING DATA
The scores of two golfers are shown. Find the range of the scores for each golfer. Compare your results. (See Example 3.)

Answer:

Question 18.
ANALYZING DATA
The graph shows a player’s monthly home run totals in two seasons. Find the range of the number of home runs for each season. Compare your results.

Answer:
The ranges of months in an order are june, april, may, july, august and september.

Explanation:
Range of april = 4 – 1 = 3
Range of May = 6 – 0 = 6
Range of June = 6 – 4 = 2
Range of July = 8 – 2 = 6
Range of August = 7 – 0 = 7
Range of September = 13 – 3 = 10

In Exercises 19–22, find (a) the range and (b) the standard deviation of the data set.
Question 19.
40, 35, 45, 55, 60
Answer:

Question 20.
141, 116, 117, 135, 126, 121

Answer:
Range = 19
Standard deviation = 9.237

Explanation:
Range = 135 – 116 = 19
Mean = \(\frac { 141 + 116 + 117 + 135 + 126 + 121 }{ 6 } \)
= 126

x	μ	x – μ	(x – μ)²
141	126	15	225
116	126	-10	100
117	126	-9	81
135	126	9	81
126	126	0	0
121	126	-5	25

Variance = \(\frac { 225 + 100 + 81 + 81 + 0 + 25 }{ 6 } \) = 85.33
Standard deviation = 9.237

Question 21.
0.5, 2.0, 2.5, 1.5, 1.0, 1.5
Answer:

Question 22.
8.2, 10.1, 2.6, 4.8, 2.4, 5.6, 7.0, 3.3

Answer:
Range = 7.7
Standard deviation = 9.58

Explanation:
Range = 10.1 – 2.4 = 7.7
Mean = \(\frac { 8.2 + 10.1 + 2.6 + 4.8 + 2.4 + 5.6 + 7.0 + 3.3 }{ 8 } \)
= 14.725

x	μ	x – μ	(x – μ)²
8.2	14.725	6.525	42.575
10.1	14.725	-4.625	21.39
2.6	14.725	-12.125	147.015
4.8	14.725	-9.925	98.505
2.4	14.725	-12.325	151.905
5.6	14.725	-9.125	83.265
7	14.725	-7.725	59.675
3.3	14.725	-11.425	130.53

Variance = \(\frac { 42.575 + 21.39 + 147.015 + 98.505 + 151.905 + 83.265 + 59.675 + 130.53 }{ 8 } \) = 91.8575
Standard deviation = 9.58

Question 23.
ANALYZING DATA
Consider the data in Exercise 17.
a. Find the standard deviation of the scores of Golfer A. Interpret your result.
b. Find the standard deviation of the scores of Golfer B. Interpret your result.
c. Compare the standard deviations for Golfer A and Golfer B. What can you conclude?
Answer:

Question 24.
ANALYZING DATA
Consider the data in Exercise 18.
a. Find the standard deviation of the monthly home run totals in the player’s rookie season. Interpret your result.
b. Find the standard deviation of the monthly home run totals in this season. Interpret your result.
c. Compare the standard deviations for the rookie season and this season. What can you conclude?

Answer:
a. SD = 1.87
b. SD = 3.05
c. This season has more standard deviation when compared with player’s rookie season.

Explanation:
a. Mean = \(\frac { 1 + 6 + 2 + 3 }{ 4 } \) = 3

x	μ	x – μ	(x – μ)²
1	3	-2	4
6	3	3	9
2	3	-1	1
3	3	0	0

Variance = \(\frac { 4 + 9 + 1 + 0 }{ 4 } \) = 3.5
Standard deviation = 1.87
b. Mean = \(\frac { 4 + 6 + 4 + 8 + 7 + 13 }{ 6 } \) = 7

x	μ	x – μ	(x – μ)²
4	7	-3	9
6	7	-1	1
4	7	-3	9
8	7	1	1
7	7	0	0
13	7	6	36

Variance = \(\frac { 9 + 1 + 9 + 1 + 0 + 36 }{ 6 } \) = 9.33
Standard deviation = 3.05

In Exercises 25 and 26, find the mean, median, and mode of the data set after the given transformation.
Question 25.
In Exercise 5, each data value increases by 4.
Answer:

Question 26.
In Exercise 6, each data value increases by 20%.

Answer:
Mean = 15.12, medan = 14.4 and no mode

Explanation:
The data values are 12, 9, 17, 15, 10
The new data values are 14.4, 10.8, 20.4, 18, 12
Mean = \(\frac { 14.4 + 10.8 + 20.4 + 18 + 12 }{ 5 } \)
= 15.12
10.8, 12, 14.4, 18, 20.4
Median = 14.4
There is no mode

Question 27.
TRANSFORMING DATA
Find the values of the measures shown when each value in the data set increases by 14.
Mean: 62
Median: 55
Mode: 49
Range: 46
Standard deviation: 15.5
Answer:

Question 28.
TRANSFORMING DATA
Find the values of the measures shown when each value in the data set is multiplied by 0.5.
Mean: 320
Median: 300
Mode: none
Range: 210
Standard deviation: 70.6

Answer:
Mean = 160
Median = 150
Mode = none
Range = 210
Standard deviation = 70.6

Explanation:
Mean = 320 x 0.5 = 160
Median = 300 x 0.5 = 150
Mode = none
Range = 210
Standard deviation = 70.6

Question 29.
ERROR ANALYSIS
Describe and correct the error in finding the median of the data set.

Answer:

Question 30.
ERROR ANALYSIS
Describe and correct the error in finding the range of the data set after the given transformation.

Answer:
Range = 23 + 23 = 46

Question 31.
PROBLEM SOLVING
In a bowling match, the team with the greater mean score wins. The scores of the members of two bowling teams are shown.
Team A: 172, 130, 173, 212
Team B: 136, 184, 168, 192

a. Which team wins the match? If the team with the greater median score wins, is the result the same? Explain.
b. Which team is more consistent? Explain.
c. In another match between the two teams, all the members of Team A increase their scores by 15 and all the members of Team B increase their scores by 12.5%. Which team wins this match? Explain.
Answer:

Question 32.
MAKING AN ARGUMENT
Your friend says that when two data sets have the same range, you can assume the data sets have the same standard deviation, because both range and standard deviation are measures of variation. Is your friend correct? Explain.

Answer:
No, you cannot assume that two sets that have the same range have also the same standard deviation.

Explanation:
Let two data samples are
A = 1, 2, 3, 4, 5
B = 1, 3, 3, 4, 5
Range of A = 5 – 1 = 4
Range of B = 5 – 1 = 4
Mean of A = 3
Mean of B = 3.2
Standard deviation of A = 1.83
Standard deviation of B = 1.68

Question 33.
ANALYZING DATA
The table shows the results of a survey that asked 12 students about their favorite meal. Which measure of center (mean, median, or mode) can be used to describe the data? Explain.

Answer:

Question 34.
HOW DO YOU SEE IT?
The dot plots show the ages of the members of three different adventure clubs. Without performing calculations, which data set has the greatest standard deviation? Which has the least standard deviation? Explain your reasoning.

Answer:

Question 35.
REASONING
A data set is described by the measures shown.
Mean: 27
Median: 32
Mode: 18
Range: 41
Standard deviation: 9
Find the mean, median, mode, range, and standard deviation of the data set when each data value is multiplied by 3 and then increased by 8.
Answer:

Question 36.
CRITICAL THINKING
Can the standard deviation of a data set be 0? Can it be negative? Explain.

Answer:
The standard deviation of a data set can never be negative.

Question 37.
USING TOOLS
Measure the heights (in inches) of the students in your class.
a. Find the mean, median, mode, range, and standard deviation of the heights.
b. A new student who is 7 feet tall joins your class. How would you expect this student’s height to affect the measures in part (a)? Verify your answer.
Answer:

Question 38.
THOUGHT PROVOKING
To find the arithmetic mean of n numbers, divide the sum of the numbers by n. To find the geometric mean of n numbers a₁, a₂, a₃, . . . , a_n, take the nth root of the product of the numbers.
geometric mean = \(\sqrt[n]{a_{1} \cdot a_{2} \cdot a_{3} \cdot \ldots \cdot a_{n}}\)
Compare the arithmetic mean to the geometric mean of n numbers.

Answer:
Let us take the data values 3, 8, 14, 14, 14, 15, 15, 15, 16, 18
Remove 10% of the lowest and highest value
Therefore, the trimmed data set is 8, 14, 14, 14, 15, 15, 15, 16
Trimmed mean = 13.875
Difference of trimmed and arithmetic mean = 13.875 – 13.2 = 0.675
The trimmed mean is greater than the arithmetic means by 0.675. the trimmed mean is useful because it can estimate the effects of outliers.

Question 39.
PROBLEM SOLVING
The circle graph shows the distribution of the ages of 200 students in a college Psychology I class.

a. Find the mean, median, and mode of the students’ ages.
b. Identify the outliers. How do the outliers affect the mean, median, and mode?
c. Suppose all 200 students take the same Psychology II class exactly 1 year later. Draw a new circle graph that shows the distribution of the ages of this class and find the mean, median, and mode of the students’ ages.
Answer:

Maintaining Mathematical Proficiency

Solve the inequality.(Section 2.4)
Question 40.
6x + 1 ≤ 4x – 9

Answer:
x ≤ -5

Explanation:
6x – 4x ≤ -9 – 1
2x ≤ -10
x ≤ -5

Question 41.
-3(3y – 2) < 1 – 9y
Answer:

Question 42.
2(5c – 4) ≥ 5(2c + 8)

Answer:
The inequality has no solution.

Explanation:
2(5c – 4) ≥ 5(2c + 8)
10c – 8 ≥ 10c + 40
10c – 10 ≥ 40 + 8
0 ≥ 48

Question 43.
4(3 – w) > 3(4w – 4)
Answer:

Evaluate the function for the given value of x.(Section 6.3)
Question 44.
f(x) = 4^x; x = 3

Answer:
f(3) = 64

Explanation:
f(x) = 4^x
f(3) = 4³
f(3) = 64

Question 45.
f(x) = 7^x; x = -2
Answer:

Question 46.
f(x) = 5(2)^x; x = 6

Answer:
f(6) = 320

Explanation:
f(x) = 5(2)^x
f(6) = 5(2)⁶
f(6) = 5 x 64 = 320

Question 47.
f(x) = -2(3)^x; x = 4
Answer:

Lesson 11.2 Box-and-Whisker Plots

Essential Question How can you use a box-and-whisker plot to describe a data set?

EXPLORATION 1

Drawing a Box-and-Whisker Plot
Work with a partner. The numbers of first cousins of the students in a ninth-grade class are shown. A box-and-whisker plot is one way to represent the data visually.
a. Order the data on a strip of grid paper with 24 equally spaced boxes. Fold the paper in half to find the median.

b. Fold the paper in half again to divide the data into four groups. Because there are 24 numbers in the data set, each group should have 6 numbers. Find the least value, the greatest value, the first quartile, and the third quartile.

c. Explain how the box-and-whisker plot shown represents the data set.

Answer:

CommunicateYour Answer

Question 2.
How can you use a box-and-whisker plot to describe a data set?
Answer:

Question 3.
Interpret each box-and-whisker plot.

a. body mass indices (BMI) of students in a ninth-grade class

b. heights of roller coasters at an amusement park

Answer:

Monitoring Progress

Question 1.
A basketball player scores 14, 16, 20, 5, 22, 30, 16, and 28 points during a tournament. Make a box-and-whisker plot that represents the data.

Answer:

Explanation:
Order the data
5, 14, 16, 16, 20, 22, 28, 30
Median = 16 + 20/2 = 18
First quartile = 14.5
Third quartile = 26.5
Interquartile range = 12
outliers = none

Use the box-and-whisker plot in Example 1.
Question 2.
Find and interpret the range and interquartile range of the data.

Answer:
Range = 30 – 5 = 25
Interquartile range = 26.5 – 14.5 = 12

Question 3.
Describe the distribution of the data.

Answer:

Question 4.
The double box-and-whisker plot represents the surfboard prices at Shop A and Shop B. Identify the shape of each distribution. Which shop’s prices are more spread out? Explain.

Answer:
a. For shop A, the median is middle
For shop B, the right whisker is longer than the left whisker, and most of the data is on the left side of the plot.
b. Shop A prices are more spread out.

Box-and-Whisker Plots 11.2 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
Describe how to find the first quartile of a data set.
Answer:

Question 2.
DIFFERENT WORDS, SAME QUESTION
Consider the box-and-whisker plot shown. Which is different? Find “both” answers.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, use the box-and-whisker plot to find the given measure.

Question 3.
least value
Answer:

Question 4.
greatest value

Answer:
The greatest value is 14

Question 5.
third quartile
Answer:

Question 6.
first quartile

Answer:
The first quartile is 6

Question 7.
median
Answer:

Question 8.
range

Answer:
Range = 14 – 3 = 11

In Exercises 9–12, make a box-and-whisker plot that represents the data.
Question 9.
Hours of television watched: 0, 3, 4, 5, 2, 4, 6, 5
Answer:

Question 10.
Cat lengths (in inches): 16, 18, 20, 25, 17, 22, 23, 21

Answer:

Median = 20.5
First quartile = 17.25
Third quartile = 22.75
ICR = 5.5

Question 11.
Elevations (in feet): -2, 0, 5, -4, 1, -3, 2, 0, 2, -3, 6
Answer:

Question 12.
MP3 player prices (in dollars): 124, 95, 105, 110, 95, 124, 300, 190, 114

Answer:

Median: 114
Minimum: 95
Maximum: 300
First quartile: 100
Third quartile: 157
Interquartile Range: 57

Question 13.
ANALYZING DATA
The dot plot represents the numbers of hours students spent studying for an exam. Make a box-and-whisker plot that represents the data.

Answer:

Question 14.
ANALYZING DATA
The stem-and-leaf plot represents the lengths (in inches) of the fish caught on a fishing trip. Make a box-and-whisker plot that represents the data.

Answer:

Question 15.
ANALYZING DATA
The box-and-whisker plot represents the prices (in dollars) of the entrées at a restaurant.

a. Find and interpret the range of the data.
b. Describe the distribution of the data.
c. Find and interpret the interquartile range of the data.
d. Are the data more spread out below Q1 or above or Q3? Explain.
Answer:

Question 16.
ANALYZING DATA
A baseball player scores 101 runs in a season. The box-and-whisker plot represents the numbers of runs the player scores against different opposing teams.

a. Find and interpret the range and interquartile range of the data.
b. Describe the distribution of the data. c. Are the data more spread out between Q1 and Q2 or between Q2 and Q3? Explain.
Answer:
a. Range = 17 – 0 = 17
ICR = 9 – 2 = 7

Question 17.
ANALYZING DATA
The double box-and-whisker plot represents the monthly car sales for a year for two sales representatives.

a. Identify the shape of each distribution.
b. Which representative’s sales are more spread out? Explain.
c. Which representative had the single worst sales month during the year? Explain.
Answer:

Question 18.
ERROR ANALYSIS
Describe and correct the error in describing the box-and-whisker plot.

Answer:
The distribution is skewed left. So most of the data are on the right side of the plot.

Question 19.
WRITING
Given the numbers 36 and 12, identify which number is the range and which number is the interquartile range of a data set. Explain.
Answer:

Question 20.
HOW DO YOU SEE IT?
The box-and-whisker plot represents a data set. Determine whether each statement is always true. Explain your reasoning.

a. The data set contains the value 11.
b. The data set contains the value 6.
c. The distribution is skewed right.
d. The mean of the data is 5.

Answer:
a. Always true
b. Always true
c. Never true
d. Never true

Question 21.
ANALYZING DATA
The double box-and-whisker plot represents the battery lives (in hours) of two brands of cell phones.

a. Identify the shape of each distribution.
b. What is the range of the upper 75% of each brand?
c. Compare the interquartile ranges of the two data sets.
d. Which brand do you think has a greater standard deviation? Explain.
e. You need a cell phone that has a battery life of more than 3.5 hours most of the time. Which brand should you buy? Explain.
Answer:

Question 22.
THOUGHT PROVOKING
Create a data set that can be represented by the box-and-whisker plot shown. Justify your answer.

Answer:
Median = 10.5
Minimum = 3
Maximum = 18
Range = 18 – 3 = 15
Q1 = 2
Q3 = 15
ICR = 15 – 2 = 13

Question 23.
CRITICAL THINKING
Two data sets have the same median, the same interquartile range, and the same range. Is it possible for the box-and-whisker plots of the data sets to be different? Justify your answer.
Answer:

Maintaining Mathematical Proficiency

Use zeros to graph the function. (Section 8.5)
Question 24.
f(x) = -2(x + 9)(x – 3)

Answer:
The zeros are p = -9, q = 3
The x coordinate = \(\frac { -9 + 3 }{ 2 } \) = -3
y = -2(x + 9)(x – 3)
= -2(-3 + 9)(-3 – 3)
= -2(6)(-6) = 72
So, plot the x-intercepts (3, 0) and (-9, 0) as well as the vertex (0, 72). The parabola opens down.

Question 25.
y = 3(x – 5)(x + 5)
Answer:

Question 26.
y = 4x² – 16x – 48

Answer:
y = 4x² – 16x – 48
y = 4(x + 2)(x − 6)
The zeros are p = -2, q = 6
The x coordinate of the vertex is \(\frac { -2 + 6 }{ 2 } \) = 2
y = 4(2)² – 16(2) – 48
y = -64
So, plot the x intercepts (-2, 0), (6, 0) and vertex (2, -64)

Question 27.
h(x) = -x² + 5x + 14
Answer:

Lesson 11.3 Shapes of Distributions

Essential Question How can you use a histogram to characterize the basic shape of a distribution?

EXPLORATION 1

Analyzing a Famous Symmetric Distribution
Work with a partner. A famous data set was collected in Scotland in the mid-1800s. It contains the chest sizes, measured in inches, of 5738 men in the Scottish Militia. Estimate the percent of the chest sizes that lie within (a) 1 standard deviation of the mean, (b) 2 standard deviations of the mean, and (c) 3 standard deviations of the mean. Explain your reasoning.

Answer:

EXPLORATION 2

Comparing Two Symmetric Distributions
Work with a partner. The graphs show the distributions of the heights of 250 adult American males and 250 adult American females.

Answer:
a. Which data set has a smaller standard deviation? Explain what this means in the context of the problem.
b. Estimate the percent of male heights between 67 inches and 73 inches.

Answer:

CommunicateYour Answer
Question 3.
How can you use a histogram to characterize the basic shape of a distribution?
Answer:

Question 4.
All three distributions in Explorations 1 and 2 are roughly symmetric. The histograms are called “bell-shaped.”
a. What are the characteristics of a symmetric distribution?
b.Why is a symmetric distribution called “bell-shaped?”
c. Give two other real-life examples of symmetric distributions.Shapes of Distributions.
Answer:

Monitoring Progress

Question 1.
The frequency table shows the numbers of pounds of aluminium cans collected by classes for a fundraiser. Display the data in a histogram. Describe the shape of the distribution.

Answer:

The distribution is skewed left. Most of the data is on the right.

Question 2.
You record the numbers of email attachments sent by 30 employees of a company in 1 week. Your results are shown in the table. (a) Display the data in a histogram using six intervals beginning with 1–20. (b) Which measures of center and variation best represent the data? Explain.

Answer:

Email Attachments Sent	Frequency
1 – 20	2
21 – 40	3
41 – 60	9
61 – 80	10
81 – 100	4
101 – 120	2

Most of the data is on the right.

Question 3.
Compare the distributions using their shapes and appropriate measures of center and variation.

Answer:
Most of the data is on the right side for the professional football players and most of the data is on the right side for company employees.
The mean of professional football players is 3-5 and the median is 6-8. The mean for company employees is 9-11 and the median is 12-14.

Question 4.
Why is the mean greater than the median for the men?
Answer:

Question 5.
If 50 more women are surveyed, about how many more would you expect to own between 10 and 18 pairs of shoes?
Answer:

Shapes of Distributions 11.3 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Describe how data are distributed in a symmetric distribution, a distribution that is skewed left, and a distribution that is skewed right.
Answer:

Question 2.
WRITING
How does the shape of a distribution help you decide which measures of center and variation best describe the data?

Answer:
The shape of the distribution describes the values of the data. In a symmetric distribution, the left and right side of the graph are mirrored images. Hence, the case of outliers are rare.
We know that mean best measures the center of data without outliers because it will work smothly with even or odd number of values.
Hence, it is best to use mean and standard deviation as measures of center and variation when the shape of the distribution is ymmetric
In a skewed distribution, the mean of the data will be in the direction in which the distribution is skewed.
hence, it would be best to use the median and the five number summery as a measure of center and variation because those would be less affected by the skewness since it only considers the ordered data.

Monitoring Progress and Modeling with Mathematics

Question 3.
DESCRIBING DISTRIBUTIONS
The frequency table shows the numbers of hours that students volunteer per month. Display the data in a histogram. Describe the shape of the distribution.

Answer:

Question 4.
DESCRIBING DISTRIBUTIONS
The frequency table shows the results of a survey that asked people how many hours they spend online per week. Display the data in a histogram. Describe the shape of the distribution.

Answer:

The distribution is skewed right. So most of the data is towards the right.

In Exercises 5 and 6, describe the shape of the distribution of the data. Explain your reasoning.
Question 5.

Answer:

In Exercises 7 and 8, determine which measures of center and variation best represent the data. Explain your reasoning.
Question 7.

Answer:

Question 8.

Answer:
Most of the data is on the right side.

Question 9.
ANALYZING DATA
The table shows the last 24 ATM withdrawals at a bank.

a. Display the data in a histogram using seven intervals beginning with 26–50.
b. Which measures of center and variation best represent the data? Explain.
c. The bank charges a fee for any ATM withdrawal less than $150. How would you interpret the data?
Answer:

Question 10.
ANALYZING DATA
Measuring an IQ is an inexact science. However, IQ scores have been around for years in an attempt to measure human intelligence. The table shows some of the greatest known IQ scores.

a. Display the data in a histogram using five intervals beginning with 151–166.
b. Which measures of center and variation best represent the data? Explain.
c. The distribution of IQ scores for the human population is symmetric. What happens to the shape of the distribution in part (a) as you include more and more IQ scores from the human population in the data set?

Answer:
a.
b. Mean and standard deviation best represents the data.

ERROR ANALYSIS In Exercises 11 and 12, describe and correct the error in the statements about the data displayed in the histogram.

Question 11.

Answer:

Question 12.

Answer:

Question 13.
USING TOOLS
For a large data set, would you use a stem-and-leaf plot or a histogram to show the distribution of the data? Explain.
Answer:

Question 14.
REASONING
For a symmetric distribution, why is the mean used to describe the center and the standard deviation used to describe the variation? For a skewed distribution, why is the median used to describe the center and the  ve-number summary used to describe the variation?

Answer:
For Symmetry distribution:
Because the mean corresponds to half of the data, on Q2 the standard deviation is the variation form of the mean.
For Skewed distribution:
Because the median corresponds most likely line where the most data are and the five number summary because the median is closer to either Q1 or Q3 in this case.

Question 15.
COMPARING DATA SETS
The double histogram shows the distributions of daily high temperatures for two towns over a 50-day period. Compare the distributions using their shapes and appropriate measures of center and variation.

Answer:

Question 16.
COMPARING DATA SETS
The frequency tables show the numbers of entrées in certain price ranges (in dollars) at two different restaurants. Display the data in a double histogram. Compare the distributions using their shapes and appropriate measures of center and variation.

Answer:

Question 17.
OPEN-ENDED
Describe a real-life data set that has a distribution that is skewed right.
Answer:

Question 18.
OPEN-ENDED
Describe a real-life data set that has a distribution that is skewed left.
Answer:

Question 19.
COMPARING DATA SETS
The table shows the results of a survey that asked freshmen and sophomores how many songs they have downloaded on their MP3 players.

a. Make a double box-and-whisker plot that represents the data. Describe the shape of each distribution.
b. Compare the number of songs downloaded by freshmen to the number of songs downloaded by sophomores.
c. About how many of the freshmen surveyed would you expect to have between 730 and 1570 songs downloaded on their MP3 players?
d. If you survey100 more freshmen, about how many would you expect to have downloaded between 310 and 1990 songs on their MP3 players?
Answer:

Question 20.
COMPARING DATA SETS
You conduct the same survey as in Exercise 19 but use a different group of freshmen. The results are as follows.Survey size: 60; minimum: 200; maximum: 2400; 1st quartile: 640; median: 1670; 3rd quartile: 2150; mean: 1480; standard deviation: 500
a. Compare the number of songs downloaded by this group of freshmen to the number of songs downloaded by sophomores.
b. Why is the median greater than the mean for this group of freshmen?
Answer:

Question 21.
REASONING
A data set has a symmetric distribution. Every value in the data set is doubled. Describe the shape of the new distribution. Are the measures of center and variation affected? Explain.
Answer:

Question 22.
HOW DO YOU SEE IT?
Match the distribution with the corresponding box-and-whisker plot.

Answer:

Question 23.
REASONING
You record the following waiting times at a restaurant.

a. Display the data in a histogram using five intervals beginning with 0–9. Describe the shape of the distribution.
b. Display the data in a histogram using 10 intervals beginning with 0–4. What happens when the number of intervals is increased?
c. Which histogram best represents the data? Explain your reasoning.
Answer:

Question 24.
THOUGHT PROVOKING
The shape of a bimodal distribution is shown. Describe a real-life example of a bimodal distribution.

Answer:

Maintaining Mathematical Proficiency

Find the domain of the function.(Section 10.1)
Question 25.
f(x) = \(\sqrt{x+6}\)
Answer:

Question 26.
f(x) = \(\sqrt{2x}\)

Answer:
2x ≥ 0
x ≥ 0
So, the domain of the function is x ≥ 0.

Question 27.
f(x) = \(\frac{1}{4} \sqrt{x-7}\)
Answer:

Data Analysis and Displays Study Skills: Studying for Finals

11.1–11.3What Did YouLearn?
Core Vocabulary

Core Concepts

Mathematical Practices
Question 1.
Exercises 15 and 16 on page 590 are similar. For each data set, is the outlier much greater than or much less than the rest of the data values? Compare how the outliers affect the means. Explain why this makes sense.
Answer:

Question 2.
In Exercise 18 on page 605, provide a possible reason for why the distribution is skewed left.
Answer:

Data Analysis and Displays 1.1–11.3 Quiz

Find the mean, median, and mode of the data set. Which measure of center best represents the data? Explain.(Section 11.1)
Question 1.

Answer:
Mean = 3
Median = 3.25
Mode = 3.5
Median best represents the data.

Explanation:
Mean = \(\frac { 3.5 + 5 + 2.5 + 3 + 3.5 + 0.5 }{ 6 } \) = 3
Arrange data 0.5, 2.5, 3, 3.5, 3.5, 5
Median = 3 + 3.5/2 = 3.25
Mode = 3.5

Question 2.

Answer:
Mean = 1314.8
Median = 1191
Mode = 1000
Median best represents the data.

Explanation:
Mean = \(\frac { 1000 + 1200 + 2568 + 1267 + 1180 + 1191 + 1328 + 1000 + 1100 }{ 9 } \) = 1314.8
Arrange data 1000, 1000, 1100, 1180, 1191, 1200, 1267, 1328, 2568
Median = 1191
Mode = 1000

Find the range and standard deviation of each data set. Then compare your results.(Section 11.1)
Question 3.
Absent students during a week of school
Female: 6, 2, 4, 3, 4
Male: 5, 3, 6, 6, 9

Answer:
The range of male is greater than the range of female.
The standard deviation of male is also greater than the female.

Explanation:
Range of female = 6 – 2 = 4
Mean of female = \(\frac { 6 + 2 + 4 + 3 + 4 }{ 5 } \) = 3.8

x	μ	x – μ	(x – μ)²
6	3.8	2.2	4.84
2	3.8	-1.8	3.24
4	3.8	0.2	0.04
3	3.8	-0.8	.64
4	3.8	0.2	0.04

Variance = \(\frac { 4.84 + 3.24 + 0.04+ 0.64 + 0.04 }{ 5 } \) = 1.76
Standard deviation = 1.32
Range of male = 9 – 3 = 6
Mean of male = \(\frac { 5 + 3 + 6 + 6 + 9 }{ 5 } \) = 5.8

x	μ	x – μ	(x – μ)²
5	5.8	-0.8	0.64
3	5.8	-2.8	7.84
6	5.8	0.2	0.04
6	5.8	0.2	0.04
9	5.8	3.2	10.24

Variance = \(\frac { 0.64 + 7.84 + 0.04 + 0.04 + 10.24 }{ 5 } \) = 3.76
Standard deviation = 1.93

Question 4.
Numbers of points scored
Juniors: 19, 15, 20, 10, 14, 21, 18, 15
Seniors: 22, 19, 29, 32, 15, 26, 30, 19

Answer:
Seniors have the highest range, mean and standard deviation when compared with juniors.

Explanation:
Range of juniors = 21 – 10 = 11
Mean of juniors = \(\frac { 19 + 15 + 20 + 10 + 14 + 21 + 18 + 15 }{ 8 } \) = 16.5

x	μ	x – μ	(x – μ)²
19	16.5	2.5	6.25
15	16.5	-1.5	2.25
20	16.5	3.5	12.25
10	16.5	-6.5	45.25
14	16.5	-2.5	6.25
21	16.5	4.5	20.25
18	16.5	1.5	2.25
15	16.5	-1.5	2.25

Variance = \(\frac { 6.25 + 2.25 + 12.25 + 45.25 + 6.25 + 20.25 + 2.25 + 2.25 }{ 8 } \) = 12.125
Standard deviation = 3.48
Range of seniors = 32 – 15 = 17
Mean of seniors = \(\frac { 22 + 19 + 29 + 32 + 15 + 26 + 30 + 19 }{ 8 } \) = 24

x	μ	x – μ	(x – μ)²
22	24	-2	4
19	24	-5	25
29	24	5	25
32	24	8	64
15	24	-9	81
26	24	2	4
30	24	6	36
19	24	-5	25

Variance = \(\frac { 4 + 25 + 25 + 64 + 81 + 4 + 36 + 25 }{ 8 } \) = 33
Standard deviation = 5.7

Make a box-and-whisker plot that represents the data.(Section 11.2)
Question 5.
Ages of family members:
60, 15, 25, 20, 55, 70, 40, 30

Answer:

Median: 35
Minimum: 15
Maximum: 70
First quartile: 21.25
Third quartile: 58.75
Interquartile Range: 37.5

Question 6.
Minutes of violin practice:
20, 50, 60, 40, 40, 30, 60, 40, 50, 20, 20, 35

Answer:

Median: 40
Minimum: 20
Maximum: 60
First quartile: 22.5
Third quartile: 50
Interquartile Range: 27.5

Question 7.
Display the data in a histogram. Describe the shape of the distribution. (Section 11.3)

Answer:

The shape is skewed right. As the most of the data is on the right.

Question 8.
The table shows the prices of eight mountain bikes in a sporting goods store. (Section 11.1 and Section 11.2)

a. Find the mean, median, mode, range, and standard deviation of the prices.
b. Identify the outlier. How does the outlier affect the mean, median, and mode?
c. Make a box-and-whisker plot that represents the data. Find and interpret the interquartile range of the data. Identify the shape of the distribution.
d. Find the mean, median, mode, range, and standard deviation of the prices when the store offers a 5% discount on all mountain bikes.

Answer:
a. Mean = 122, Median = 149.5, Mode = 98, Range = 116, Standard deviation = 36
b. Outlier = 211, mean and median decreases and mode remains same
c.
Interquartile Range: 30.75
d. Mean = 115.9, Median = 142.025, Mode = 93.1, Range = 116, Standard deviation = 36

Explanation:
a. Mean = \(\frac { 98 + 119 + 95 + 211 + 130 + 98 + 100 + 125 }{ 8 } \) = 122
Arrange the data 95, 98, 98, 100, 119, 125, 130, 211
Median = 149.5
Mode = 98
Range = 211 – 95 = 116

x	μ	x – μ	(x – μ)²
98	122	-24	576
119	122	-3	9
95	122	-27	729
211	122	89	7921
130	122	8	64
98	122	-24	576
100	122	-22	484
125	122	3	9

Variance = \(\frac { 576 + 9 + 729 + 7921 + 64 + 576 + 484 + 9 }{ 8 } \) = 1296
Standard deviation = 36
b. Mean without outlier = \(\frac { 98 + 119 + 95 + 130 + 98 + 100 + 125 }{ 7 } \) = 109.28
Median without outlier = 100
Mode without outlier = 98
d. Apply 5% discount on mean, median and mode. It does not affect change on standard deviation and range.
Mean = 115.9, Median = 142.025, Mode = 93.1, Range = 116, Standard deviation = 36

Question 9.
The table shows the times of 20 presentations. (Section 11.3)

a. Display the data in a histogram using five intervals beginning with 3–5.
b. Which measures of center and variation best represent the data? Explain.
c. The presentations are supposed to be 10 minutes long. How would you interpret these results?

Answer:
a.
The data is symmetrical
b. Mean and standard deviation best represents the data.

Lesson 11.4 Two-Way Tables

Essential Question How can you read and make a two-way table?

EXPLORATION 1

Reading a Two-Way Table
Work with a partner. You are the manager of a sports shop. The two-way tables show the numbers of soccer T-shirts in stock at your shop at the beginning and end of the selling season. (a) Complete the totals for the rows and columns in each table. (b) How would you alter the number of T-shirts you order for next season? Explain your reasoning.

Answer:

EXPLORATION 2

Making a Two-Way Table
Work with a partner. The three-dimensional bar graph shows the numbers of hours students work at part-time jobs.
a. Make a two-way table showing the data. Use estimation to find the entries in your table.

b. Write two observations that summarize the data in your table.
Answer:

Communicate Your Answer
Question 3.
How can you read and make a two-way table?
Answer:

Monitoring Progress

Question 1.
You conduct a technology survey to publish on your school’s website. You survey students in the school cafeteria about the technological devices they own. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Answer:

158 cell phones are responded
85 Cell Phones are not responded
52 table computers are responded
191 table computer are not responded.

Question 2.
You survey students about whether they are getting a summer job. Seventy-five males respond, with 18 of them responding “no.” Fifty-seven females respond, with 45 of them responding “yes.” Organize the results in a two-way table. Include the marginal frequencies.

Answer:

Question 3.
Use the survey results in Monitoring Progress Question 2 to make a two-way table that shows the joint and marginal relative frequencies. What percent of students are not getting a summer job?

Answer:
Total number of students = 75 + 57 = 132
Number of students not getting summer job = 30
Percent of students are not getting a summer job = 30/132 = 22.72%

Question 4.
Use the survey results in Example 3 to make a two-way table that shows the conditional relative frequencies based on the row totals. Given that a student is a senior, what is the conditional relative frequency that he or she is planning to major in a medical field?

Answer:

Question 5.
Using the results of the survey in Monitoring Progress Question 1, is there an association between owning a tablet computer and owning a cell phone? Explain your reasoning.

Answer:

Two-Way Tables 11.4 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Each entry in a two-way table is called a(n) __________.
Answer:

Question 2.
WRITING
When is it appropriate to use a two-way table to organize data?
Answer:

Question 3.
VOCABULARY
Explain the relationship between joint relative frequencies, marginal relative frequencies, and conditional relative frequencies.
Answer:

Question 4.
WRITING
Describe two ways you can find conditional relative frequencies.

Answer:
You can find a conditional relative frequency using a row total OR a column total of a two way table.

Monitoring Progress and Modeling with Mathematics

You conduct a survey that asks 346 students whether they buy lunch at school. In Exercises 5–8, use the results of the survey shown in the two-way table.

Question 5.
How many freshmen were surveyed?
Answer:

Question 6.
How many sophomores were surveyed?

Answer:
168 sophomores are surveyed.

Question 7.
How many students buy lunch at school?
Answer:

Question 8.
How many students do not buy lunch at school?

Answer:
138 students do not buy luch at school.

In Exercises 9 and 10, find and interpret the marginal frequencies.
Question 9.

Answer:

Question 10.

Answer:

Question 11.
USING TWO-WAY TABLES
You conduct a survey that asks students whether they plan to participate in school spirit week. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Answer:

Question 12.
USING TWO-WAY TABLES
You conduct a survey that asks college-bound high school seniors about the type of degree they plan to receive. The results are shown in the two-way table. Find and interpret the marginal frequencies.

Answer:

So, 120 associates plan to recieve, 244 bachelor’s plan to recieve and 90 master’s plan to recieve. 226 male plan to recieve and 228 female plan to recieve.

USING STRUCTURE In Exercises 13 and 14, complete the two-way table..
Question 13.

Answer:

Question 14.

Answer:

Question 15.
MAKING TWO-WAY TABLES
You conduct a survey that asks 245 students in your school whether they have taken a Spanish or a French class. One hundred nine of the students have taken a Spanish class, and 45 of those students have taken a French class. Eighty-two of the students have not taken a Spanish or a French class. Organize the results in a two-way table. Include the marginal frequencies.
Answer:

Question 16.
MAKING TWO-WAY TABLES
A car dealership has 98 cars on its lot. Fifty-five of the cars are new. Of the new cars, 36 are domestic cars. There are 15 used foreign cars on the lot. Organize this information in a two-way table. Include the marginal frequencies. In Exercises 17 and 18, make a two-way table that shows the joint and marginal relative frequencies.

Answer:

In Exercises 17 and 18, make a two-way table that shows the joint and marginal relative frequencies.
Question 17.

Answer:

Question 18.

Answer:

Question 19.
USING TWO-WAY TABLES
Refer to Exercise 17. What percent of students prefer aerobic exercise? What percent of students are males who prefer anaerobic exercise?
Answer:

Question 20.
USING TWO-WAY TABLES
Refer to Exercise 18. What percent of the sandwiches are on wheat bread? What percent of the sandwiches are turkey on white bread?

Answer:
Percentage of sandwiches on wheat bread is 55%
Percentage of sandwiches are turkey on white bread is 24%

Explanation:

ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in using the two-way table.

Question 21.

Answer:

Question 22.

Answer:

Question 23.
USING TWO-WAY TABLES
A company is hosting an event for its employees to celebrate the end of the year. It asks the employees whether they prefer a lunch event or a dinner event. It also asks whether they prefer a catered event or a potluck. The results are shown in the two-way table. Make a two-way table that shows the conditional relative frequencies based on the row totals. Given that an employee prefers a lunch event, what is the conditional relative frequency that he or she prefers a catered event?

Answer:

Question 24.
USING TWO-WAY TABLES
The two-way table shows the results of a survey that asked students about their preference for a new school mascot. Make a two-way table that shows the conditional relative frequencies based on the column totals. Given that a student prefers a hawk as a mascot, what is the conditional relative frequency that he or she prefers a cartoon mascot?

Answer:

Explanation:
Total number of observations = 67 + 74 + 51 + 58 + 18 + 24 = 292
For instance, the joint relative frequency of mascot of realiastic style of tiger type is 67/292 = 0.23
The correct calculation is

Next, divide the joint relative frequencies by the marginal relative frequencies on the column totals.

The conditional relative frequency that the students prefer a cartoon mascot that is a hawk is 0.22

Question 25.
ANALYZING TWO-WAYTABLES
You survey college-bound seniors and find that 85% plan to live on campus, 35% plan to have a car while at college, and 5% plan to live off campus and not have a car. Is there an association between living on campus and having a car at college? Explain.
Answer:

Question 26.
ANALYZING TWO-WAYTABLES
You survey students and find that 70% watch sports on TV, 48% participate in a sport, and 16% do not watch sports on TV or participate in a sport. Is there an association between participating in a sport and watching sports on TV? Explain.

Answer:
Use the given information to make a two-way table. Use reasoning to find the missing joint & marginal relative frequencies. Note that the total in row & column is 100%

Use conditional relative frequencies based on the column totals to determine whether there is an association:

Of the students who participate in sports, about 71% watch sports. Of the students who do not participate in sports, about 69% watch sports. It appears that students those who participate & those who do not participate in sports are equally likely to watch sports.
therefore, there is no association between participating in sports & watching sports in a tv.

Question 27.
ANALYZING TWO-WAY TABLES
The two-way table shows the results of a survey that asked adults whether they participate in recreational skiing. Is there an association between age and recreational skiing?

Answer:

Question 28.
ANALYZING TWO-WAY TABLES
Refer to Exercise 12. Is there an association between gender and type of degree? Explain.
Answer:

Question 29.
WRITING
Compare Venn diagrams and two-way tables.
Answer:

Question 30.
HOW DO YOU SEE IT?
The graph shows the results of a survey that asked students about their favorite movie genre.

a. Display the given information in a two-way table.
b. Which of the data displays do you prefer? Explain.

Answer:
a.
B. I prefer a two-way table. Because it is easy to read the information from a two way table.

Question 31.
PROBLEM SOLVING
A box office sells 1809 tickets to a play, 800 of which are for the main floor. The tickets consist of 2x + y adult tickets on the main floor, x – 40 child tickets on the main floor, x + 2y adult tickets in the balcony, and 3x – y – 80 child tickets in the balcony.

a. Organize this information in a two-way table.
b. Find the values of x and y.
c. What percent of tickets are adult tickets?
d. What percent of child tickets are balcony tickets?
Answer:

Question 32.
THOUGHT PROVOKING
Compare “one-way tables” and “two-way tables.” Is it possible to have a “three-way table?” If so, give an example of a three-way table.

Answer:
One way tables represent data for a single variable while two way tables present two different categories.
Yes it is possible to have a three way table if one of the categories have another subcategory.
An example would be:

Maintaining Mathematical Proficiency

Tell whether the table of values represents a linear, an exponential, or a quadratic function. (Section 8.6)
Question 33.

Answer:

Question 34.

Answer:

Lesson 11.5 Choosing a Data Display

Essential Question How can you display data in a way that helps you make decisions?

EXPLORATION 1

Displaying Data
Work with a partner. Analyze the data and then create a display that best represents the data. Explain your choice of data display.
a. A group of schools in New England participated in a 2-month study and reported 3962 animals found dead along roads.
birds: 307
mammals: 2746
amphibiAnswer: 145
reptiles: 75
unknown: 689
b. The data below show the numbers of black bears killed on a state’s roads from 1993 to 2012.

c. A 1-week study along a 4-mile section of road found the following weights (in pounds) of raccoons that had been killed by vehicles.

d. A yearlong study by volunteers in California reported the following numbers of animals killed by motor vehicles.

Answer:

Communicate Your Answer
Question 2.
How can you display data in a way that helps you make decisions?
Answer:

Question 3.
Use the Internet or some other reference to find examples of the following types of data displays.

bar graph
circle graph
scatter plot
stem-and-leaf plot
pictograph
line graph
box-and-whisker plot
histogram
dot plot

Monitoring Progress

Tell whether the data are qualitative or quantitative. Explain your reasoning.
Question 1.
telephone numbers in a directory

Answer:
Telephone numbers are numerical. So the data are quantitative.

Question 2.
ages of patients at a hospital

Answer:
Ages are numerical entries. So, the data are quantitative.

Question 3.
lengths of videos on a website

Answer:
The lengths of videos are numerical. So the data are quantitative.

Question 4.
types of flowers at a florist

Answer:
Types of flowers are nonnumerical entries. So the data are qualitative.

Question 5.
Display the data in Example 2(a) in another way.

Answer:

Question 6.
Display the data in Example 2(b) in another way.

Answer:

Question 7.
Redraw the graphs in Example 3 so they are not misleading.

Answer:

Choosing a Data Display 11.5 Exercises

Vocabulary and Core Concept Check

Question 1.
OPEN-ENDED
Describe two ways that a line graph can be misleading.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG?
Which data set does not belong with the other three? Explain your reasoning.

Answer:
breeds of dogs at a pet store

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, tell whether the data are qualitative or quantitative. Explain your reasoning.
Question 3.
brands of cars in a parking lot
Answer:

Question 4.
weights of bears at a zoo

Answer:
Weights are numerical entries. So the data are quantitative.

Question 5.
budgets of feature films
Answer:

Question 6.
file formats of documents on a computer

Answer:
file formats are non-numeric entries. So the data are qualitative.

Question 7.
shoe sizes of students in your class
Answer:

Question 8.
street addresses in a phone book

Answer:
Street address are non-numeric entries. So the data are qualitative.

In Exercises 9–12, choose an appropriate data display for the situation. Explain your reasoning.
Question 9.
the number of students in a marching band each year
Answer:

Question 10.
a comparison of students’ grades (out of 100) in two different classes

Answer:
An appropriate data display the comparison of student grades. It can be displayed on a pie chart.

Question 11.
the favorite sports of students in your class
Answer:

Question 12.
the distribution of teachers by age

Answer:
An appropriate data display fo the distribution of teachers age by a bar graph.

In Exercises 13–16, analyze the data and then create a display that best represents the data. Explain your reasoning.
Question 13.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

Question 17.
DISPLAYING DATA
Display the data in Exercise 13 in another way.
Answer:

Question 18.
DISPLAYING DATA
Display the data in Exercise 14 in another way.

Answer:

Question 19.
DISPLAYING DATA
Display the data in Exercise 15 in another way.
Answer:

Question 20.
DISPLAYING DATA
Display the data in Exercise 16 in another way.

Answer:

In Exercises 21–24, describe how the graph is misleading. Then explain how someone might misinterpret the graph.
Question 21.

Answer:

Question 22.

Answer:
The frequency on the vertical axis has very small increment. So, someone might believe that minutes are doubled from 0 – 29 to 60 – 119.

Question 23.

Answer:

Question 24.

Answer:
The increments on the y-axis are not equal.

Question 25.
DISPLAYING DATA
Redraw the graph in Exercise 21 so it is not misleading.
Answer:

Question 26.
DISPLAYING DATA
Redraw the graph in Exercise 22 so it is not misleading.
Answer:

Question 27.
MAKING AN ARGUMENT
A data set gives the ages of voters for a city election. Classmate A says the data should be displayed in a bar graph, while Classmate B says the data would be better displayed in a histogram. Who is correct? Explain.
Answer:

Question 28.
HOW DO YOU SEE IT?
The manager of a company sees the graph shown and concludes that the company is experiencing a decline. What is missing from the graph? Explain why the manager may be mistaken.

Answer:
The y-axis label is missing.

Question 29.
REASONING
A survey asked 100 students about the sports they play. The results are shown in the circle graph.

a. Explain why the graph is misleading.
b. What type of data display would be more appropriate for the data? Explain.
Answer:

Question 30.
THOUGHT PROVOKING
Use a spreadsheet program to create a type of data display that is not used in this section.
Answer:

Question 31.
REASONING
What type of data display shows the mode of a data set?
Answer:

Maintaining Mathematical Proficiency

Determine whether the relation is a function. Explain.(Section 3.1)
Question 32.
(-5, -1), (-6, 0), (-5, 1), (-2, 2), (3 , 3)

Answer:
Each input has exactly one output. So, the relation is a function.

Question 33.
(0, 1), (4, 0), (8, 1), (12, 2), (16, 3)
Answer:

Data Analysis and Displays Performance Task: College Students Study Time

11.4–11.5 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices
Question 1.
Consider the data given in the two-way table for Exercises 5–8 on page 614. Your sophomore friend responded to the survey. Is your friend more likely to have responded “yes” or “no” to buying a lunch? Explain.
Answer:

Question 2.
Use your answer to Exercise 28 on page 622 to explain why it is important for a company manager to see accurate graphs.
Answer:

Data Analysis and Displays Chapter Review

11.1 Measures of Center and Variation (pp. 585–592)

Question 1.
Use the data in the example above. You run 4.0 miles on Day 11. How does this additional value affect the mean, median, and mode? Explain.

Answer:
Mean remains the same, median decreases and mode remain the same.

Explanation:
Mean is 4 miles. Median is 4.2 miles and mode is 4.3 miles.

Question 2.
Use the data in the example above. You run 10.0 miles on Day 11. How does this additional value affect the mean, median, and mode? Explain.

Answer:
Mean, median increases and mode remains the same.

Explanation:
Mean = 4.5
Median = 4.3
Mode = 4.3

Find the mean, median, and mode of the data.
Question 3.

Answer:
Mean = 4, Median = 3, mode = 10

Explanation:
Mean = \(\frac { 11 + 0 + 10 + 3 – 9 + 10 + 3 – 2 + 10 }{ 9 } \) = 4
Arrange the data
-9, -2, 0, 3, 3, 10, 10, 10, 11
Median = 3
Mode = 10

Question 4.

Answer:
Mean = 1.7
Median = 1
Mode = 1

Explanation:
Arrange the data
0, 0, 1, 1, 1, 1, 2, 2, 4, 5
Mean = 1.7
Median = 1
Mode = 1

Find the range and standard deviation of each data set. Then compare your results.
Question 5.

Answer:
Range of part A = 56
The standard deviation of part A = 17.97
Range of part B = 86
The standard deviation of part B = 31.63

Explanation:
Range of part A = 230 – 174 = 56
Mean = \(\frac { 205 + 185 + 210 + 174 + 194 + 190 + 200 + 219 + 203 + 230 }{ 10 } \) = 201

x	μ	x – μ	(x – μ)²
205	210	-5	25
185	210	-25	625
210	210	0	0
174	210	-36	1296
194	210	-16	256
190	210	-20	400
200	210	-10	100
219	210	9	81
203	210	-7	49
230	210	20	400

Variance = \(\frac { 25 + 625 + 1296 + 256 + 400 + 100 + 81 + 49 + 400 }{ 10 } \) = 323.2
Standard deviation = 17.97
Range of part B = 240 – 154 = 86
Mean = \(\frac { 228 + 172 + 154 + 235 + 168 + 205 + 181 + 240 + 235 + 192 }{ 10 } \) = 201

x	μ	x – μ	(x – μ)²
228	210	18	324
172	210	-38	1444
154	210	-56	3136
235	210	25	625
168	210	-42	1764
205	210	-5	25
181	210	-29	841
240	210	30	900
235	210	25	625
192	210	-18	324

Variance = \(\frac { 324+1444+3136+625+1764+25+841+900+625+324 }{ 10 } \) = 1000.8
Standard deviation = 31.63

Question 6.

Answer:
Range of store A = 110
Standard deviation of store A = 39.37
Range of store B = 150
Standard deviation of store B = 48.08

Explanation:

Range of store A = 250 – 140 = 110
Mean = 190

x	μ	x – μ	(x – μ)²
140	190	-50	2500
200	190	10	100
150	190	-40	1600
250	190	60	3600
180	190	-10	100
250	190	60	3600
190	190	0	0
160	190	-30	900

Variance = 1550
Standard deviation of store A = 39.37
Range of store B = 310 – 160 = 150
Mean = 232.5

x	μ	x – μ	(x – μ)²
225	232.5	-7.5	56.25
260	232.5	27.5	756.25
190	232.5	-42.5	1806.25
160	232.5	-72.5	5256.25
310	232.5	77.5	6006.25
190	232.5	-42.5	1806.25
285	232.5	52.5	2756.25
240	232.5	7.5	56.25

Variance = 2312.5
The standard deviation of store B = 48.08

Find the values of the measures shown after the given transformation.
Mean: 109 Median: 104 Mode: 96 Range: 45 Standard deviation: 3.6
Question 7.
Each value in the data set increases by 25

Answer:
mean = 109 + 25 = 134
median = 104 + 25 = 129
mode = 96 + 25 = 121
range = 45
standard deviation = 3.6

Question 8.
Each value in the data set is multiplied by 0.6

Answer:
Mean = 109 x 0.6 = 65.4
Median = 104 x 0.6 = 62.4
Mode = 96 x 0.4 = 38.4
range = 45
standard deviation = 3.6

11.2 Box-and-Whisker Plots (pp. 593–598)

Make a box-and-whisker plot that represents the data. Identify the shape of the distribution.
Question 9.
Ages of volunteers at a hospital:
14, 17, 20, 16, 17, 14, 21, 18, 22

Answer:

Median: 17
Minimum: 14
Maximum: 22
First quartile: 15
Third quartile: 20.5
Interquartile Range: 5.5

Question 10.
Masses (in kilograms) of lions:
120, 230, 180, 210, 200, 200, 230, 160

Answer:

Median: 200
Minimum: 120
Maximum: 230
First quartile: 165
Third quartile: 225
Interquartile Range: 60

11.3 Shapes of Distributions (pp. 599–606)

Question 11.
The frequency table shows the amounts (in dollars) of money the students in a class have in their pockets.
a. Display the data in a histogram. Describe the shape of the distribution.
b. Which measures of center and variation best represent the data?
c. Compare this distribution with the distribution shown above using their shapes and appropriate measures of center and variation.

Answer:
a.
The shape of the distribution is skewed right.
b. Mean and median best represents the data.

11.4 Two-Way Tables (pp. 609–616)

Question 12.
The two-way table shows the results of a survey that asked shoppers at a mall about whether they like the new food court.

a. Make a two-way table that shows the joint and marginal relative frequencies.
b. Make a two-way table that shows the conditional relative frequencies based on the column totals.

Answer:

11.5 Choosing a Data Display (pp. 617–622)

Question 13.
Analyze the data in the table at the right and then create a display that best represents the data. Explain your reasoning.

Answer:

Tell whether the data are qualitative or quantitative. Explain.
Question 14.
heights of the members of a basketball team

Answer:
Hights are numerical entries. So heights of the members of a basketball team are quantitative data.

Question 15.
grade level of students in an elementary school

Answer:
Grade levels are non-numerical entries. So grade level of students in an elementary school are qualittaivedata.

Data Analysis and Displays Chapter Test

Describe the shape of the data distribution. Then determine which measures of center and variation best represent the data.
Question 1.

Answer:
The shape is symmetric. Mean best represents the data.

Question 2.

Answer:
The shape is skewed right. Mean and median best represents the data.

Question 3.

Answer:
The shape is skewed left. Mean and median best represents the data.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning.
a. The sum of the marginal relative frequencies in the “total” row and the “total” column of a two-way table should each be equal to 1.
b. In a box-and-whisker plot, the length of the box to the left of the median and the length of the box to the right of the median are equal.
c. Qualitative data are numerical.

Answer:
a. Always true
b. sometimes true
c. Sometimes true

Explanation:
a. Always true
The marginal relative frequency is the sum of the joint relative frequencies. Each joint relative frequency is the ratio a frequency that is not in the total observations.
Because the total row corresponds to the total observations or 100% of the number of observation, and the total row and column are equal. It is always true that the total row and column are equal
b. sometimes true
The length of each side varies depending to the shape of the distribution of the data represented by the box-and-whisker plot. Only when the distribution is symmetric that the length of each side is equal. Otherwise, the distribution is skewed.
c. Sometimes true
There are qualitative data that appears to be numerical. An example is grade levels. Grade levels appear to be numerical but applying mathematical operations with it do not make sense.

Question 5.
Find the mean, median, mode, range, and standard deviation of the prices.

Answer:
Mean = 15, median = 11.425, no mode
Range = 15.7
Standard deviation = 5.46

Explanation:
Mean = \(\frac { 15.5+7.8+18.9+23.5+10.6+9.75+12.25+21.7 }{ 8 } \) = 15
Arrange the data
7.80, 9.75, 10.6, 12.25, 15.5, 18.9, 21.70, 23.50
Median = 10.6 + 12.25/2 = 11.425
No mode
Range = 23.50 – 7.8 = 15.7

x	μ	x – μ	(x-μ)²
15.5	15	0.5	0.25
7.8	15	-7.2	51.84
18.9	15	3.9	15.21
23.5	15	8.5	72.25
10.6	15	-4.4	19.36
9.75	15	-5.25	27.56
12.25	15	-2.75	7.56
21.7	15	6.7	44.89

Variance = 29.86
Standard deviation = 5.46

Question 6.
Repeat Exercise 5 when all the shirts in the clothing store are 20% off.

Answer:
Mean = 12
Median = 9.14
no mode
range = 15.7
Standard deviation = 5.46

Explanation:
Mean = 15 – 3 = 12
Median = 11.425 – 2.285 = 9.14
no mode
range = 15.7
Standard deviation = 5.46

Question 7.
Which data display best represents the data, a histogram or a stem-and-leaf plot? Explain.
15, 21, 18, 10, 12, 11, 17, 18, 16, 12, 20, 12, 17, 16

Answer:

Explanation:
The stem-and-leaf plot would be better than the histogram to represent the given data
This is because of the following:
There are few values in the data
The stem-and-leaf organizes and orders the data
When views sideways, it will resemble the corresponding histogram

Question 8.
The tables show the battery lives (in hours) of two brands of laptops.

a. Make a double box-and-whisker plot that represents the data.
b. Identify the shape of each distribution.
c. Which brand’s battery lives are more spread out? Explain.
d. Compare the distributions using their shapes and appropriate measures of center and variation.

Answer:
a. Brand A

Lowest value = 8.5
highest value = 20.75
lowest quartile = 13.5
upper quartile = 16.75
median = 15
Brand B

Lowest value = 7.8
highest value = 12.5
lowest quartile = 8.5
upper quartile = 10.25
median = 9.25
b. For both brands distribution is skewed right.
c. Brand A having lives more spread out.
d. As distributions are skewed, median should be used as a measure of center and five number summery for measuring variation. For brand A median is 15 while for brand B median is 9.25.

Question 9.
The table shows the results of a survey that asked students their preferred method of exercise. Analyze the data and then create a display that best represents the data. Explain your reasoning.

Answer:

Question 10.
You conduct a survey that asks 271 students in your class whether they are attending the class field trip. One hundred twenty-one males respond 92 of which are attending the field trip. Thirty-one females are not attending the field trip.
a. Organize the results in a two-way table. Find and interpret the marginal frequencies.
b. What percent of females are attending the class field trip?

Answer:
a.
From the table,
121 male students were surveyed
60 female students were surveyed
271 students were surveyed
211 students will attend the field trip
60 students will not attend the field trip
(b) 119/150 = 0.79 or 79% of the females are attending the class field trip.

Data Analysis and Displays Cumulative Assessment

Question 1.
You ask all the students in your grade whether they have a cell phone. The results are shown in the two-way table. Your friend claims that a greater percent of males in your grade have cell phones than females. Do you support your friend’s claim? Justify your answer.

Answer:

Question 2.
Use the graphs of the functions to answer each question.
a. Are there any values of x greater than 0 where f (x) > h(x)? Explain.
b. Are there any values of x greater than 1 where g(x) > f(x)? Explain.
c. Are there any values of x greater than 0 where g(x) > h(x)? Explain.

Answer:

Question 3.
Classify the shape of each distribution as symmetric, skewed left, or skewed right.

Answer:
a. The distribution is skewed left
b. The distribution is symmetric
c. The distribution is skewed left
d. the distribution is skewed right.

Question 4.
Complete the equation so that the solutions of the system of equations are (-2, 4) and (1, -5).

Answer:
Slope = \(\frac { 4 + 5 }{ 1 + 2 } \) = 3
y – 4 = 3(x + 2)
y – 4 = 3x + 6
y = 3x + 6 + 4
y = 3x + 10

Question 5.
Pair each function with its inverse.

Answer:
y = -3x² and y = √\(\frac { -x }{ 3 } \)
y = -x + 7 and y = -x + 7
y = 2x – 4 and y = 0.5x + 2
y = x² – 5 and y = √x + 5

Explanation:
y = -3x², x² = -y/3
x = √\(\frac { -y }{ 3 } \)
y = -x + 7
-x = y – 7
x = 7 – y
y = 2x – 4
y + 4 = 2x
x = y + 4/2

Question 6.
The box-and-whisker plot represents the lengths (in minutes) of project presentations at a science fair. Find the interquartile range of the data. What does this represent in the context of the situation?

A. 7; The middle half of the presentation lengths vary by no more than 7 minutes.
B. 3; The presentation lengths vary by no more than 3 minutes.
C. 3; The middle half of the presentation lengths vary by no more than 3 minutes.
D. 7; The presentation lengths vary by no more than 7 minutes.

Answer:
C. 3; The middle half of the presentation lengths vary by no more than 3 minutes.

Explanation:
Interquartile range = 7 – 4 = 3

Question 7.
Scores in a video game can be between 0 and 100. Use the data set shown to fill in a value for x so that each statement is true.

a. When x = ____, the mean of the scores is 45.5.
b. When x = ____, the median of the scores is 47.
c. When x = ____, the mode of the scores is 63.
d. When x = ____, the range of the scores is 71.

Answer:
a. 38
b. 47
c. 63
d. 99

Explanation:
a. Mean = 45.5
Sum = 326 + x
45.5 x 8 = 326 + x
x = 38
b. Arrange the data
28, 36, 42, 48, 52, 57, 63, x
x = 47

Question 8.
Select all the numbers that are in the range of the function shown.

Answer:

Question 9.
A traveler walks and takes a shuttle bus to get to a terminal of an airport. The function y = D(x) represents the traveler’s distance (in feet) after x minutes.

a. Estimate and interpret D(2).
b. Use the graph to find the solution of the equation D(x) = 3500. Explain the meaning of the solution.
c. How long does the traveler wait for the shuttle bus?
d. How far does the traveler ride on the shuttle bus?
e. What is the total distance that the traveler walks before and after riding the shuttle bus?

Answer:
a. D(2) = 500
b. D(15) = 3500
c. waiting time is 8 minutes
d. The distance covered by the traveler on shuttle bus is 2000 ft
e. The total distance that the traveler walks before and after riding the shuttle bus is 1500 ft

Explanation:
b. D(x) = 3500
D(15) = 3500
c. Waiting time = 12 – 4 = 8
d. Distance covered on shuttle bus = 3000 – 1000 = 2000
e. distance covered by foot = (1000 – 0) + (3500 – 3000) = 1500 ft

Exponential Functions and Sequences Chapter of Big Ideas Math Algebra 1 Answers provided helps students to learn the fundamentals associated with Exponential Functions and Sequences. Each of them is quite simple and is sequenced as per the BIM Textbook Ch 6 Exponential Functions and Sequences. In fact, our experts have covered questions belonging to Exercises, Cumulative Assessments, Chapter Tests, Review Test, Quiz, Practice Tests, etc.

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Big Ideas Math Book Algebra 1 Answer Key Chapter 6 Exponential Functions and Sequences

You can improve your subject knowledge by taking the help of the BIM Algebra 1 Solution Key and stand out from the crowd. All you have to do is practice the respective concept by availing the quick links present below. Allot time to the areas you feel difficulty and solve accordingly. Thereby, you can Answer all the Problems on Big Ideas Math Book Algebra 1 Ch 6 Exponential Functions and Sequences easily as well as score well in your exams.

• Exponential Functions and Sequences Maintaining Mathematical Proficiency – Page 289 
• Exponential Functions and Sequences Mathematical Practices – Page 290
• Lesson 6.1 Properties of Exponents – Page(291-298) 
• Properties of Exponents 6.1 Exercises – Page(296-298)
• Lesson 6.2 Radicals and Rational Exponents – Page(299-304)
• Radicals and Rational Exponents 6.2 Exercises – Page(303-304)
• Lesson 6.3 Exponential Functions – Page(305-312)
• Exponential Functions 6.3 Exercises – Page(310-312)
• Lesson 6.4 Exponential Growth and Decay – Page(313-322)
• Exponential Growth and Decay 6.4 Exercises – Page(319-322)
• Exponential Functions and Sequences Study Skills: Analyzing Your Errors – Page 323
• Exponential Functions 6.1 – 6.4 Quiz – Page 324
• Lesson 6.5 Solving Exponential Functions – Page(325-330)
• Solving Exponential Functions 6.5 Exercises – Page(329-330)
• Lesson 6.6 Geometric Sequences – Page(331-338)
• Geometric Sequences 6.6 Exercises – Page(336-338)
• Lesson 6.7 Recursively Defined Sequences – Page(339-346)
• Recursively Defined Sequences 6.7 Exercises – Page(344-346)
• Exponential Functions and Sequences Performance Task: The New Car – Page 347
• Exponential Functions and Sequences Chapter Review – Page(348-350)
• Exponential Functions and Sequences Chapter Test – Page 351
• Exponential Functions and Sequences Cumulative Assessment – Page(352-353)

Exponential Functions and Sequences Maintaining Mathematical Proficiency

Evaluate the expression.

Question 1.
12(\(\frac{14}{2}\)) – 3³ + 15 – 9²

Question 2.
5³ • 8 ÷ 2² + 20 • 3 – 4

Question 3.
-7 + 16 ÷ 2⁴ + (10 – 4²)

Find the square root(s).

Question 4.
\(\sqrt{64}\)

Question 5.
–\(\sqrt{4}\)

Question 6.
–\(\sqrt{25}\)

Question 7.
±\(\sqrt{21}\)

Question 8.
12, 14, 16, 18, . . .

Question 9.
6, 3, 0, -3, . . .

Question 10.
22, 15, 8, 1, . . .

Question 11.
ABSTRACT REASONING
Recall that a perfect square is a number with integers as its square roots. Is the product of two perfect squares always a perfect square? Is the quotient of two perfect squares always a perfect square? Explain your reasoning.

Exponential Functions and Sequences Mathematical Practices

Monitoring Progress

Question 1.
A rabbit population over 8 consecutive years is given by 50, 80, 128, 205, 328, 524, 839, 1342. Find the population in the tenth year.

Question 2.
The sums of the numbers in the first eight rows of Pascal’s Triangle are 1, 2, 4, 8, 16, 32, 64, 128. Find the sum of the numbers in the tenth row.

Lesson 6.1 Properties of Exponents

Essential Question
How can you write general rules involving properties of exponents?

EXPLORATION 1
Writing Rules for Properties of Exponents
Work with a partner.
a. What happens when you multiply two powers with the same base? Write the product of the two powers as a single power. Then write a general rule for finding the product of two powers with the same base.


b. What happens when you divide two powers with the same base? Write the quotient of the two powers as a single power. Then write a general rule for finding the quotient of two powers with the same base.

c. What happens when you find a power of a power? Write the expression as a single power. Then write a general rule for finding a power of a power.

d. What happens when you find a power of a product? Write the expression as the product of two powers. Then write a general rule for finding a power of a product.

e. What happens when you find a power of a quotient? Write the expression as the quotient of two powers. Then write a general rule for finding a power of a quotient.

Communicate Your Answer

Question 2.
How can you write general rules involving properties of exponents?

Question 3.
There are 3³ small cubes in the cube below. Write an expression for the number of small cubes in the large cube at the right.

6.1 Lesson

Monitoring Progress

Evaluate the expression.

Question 1.
(-9)°

Question 2.
3^-3

Question 3.

Question 4.
Simplify the expression . Write your answer using only positive exponents.

Monitoring Progress

Simplify the expression. Write your answer using only positive exponents.

Question 5.
10⁴ • 10^-6

Question 6.
x⁹ • x^-9

Question 7.
\(\frac{-5^{8}}{-5^{4}}\)

Question 8.
\(\frac{y^{6}}{y^{7}}\)

Question 9.
(6^-2)^-1

Question 10.
(w¹²)⁵

Monitoring Progress

Simplify the expression. Write your answer using only positive exponents.

Question 11.
(10y)^-3

Question 12.
(\(-\frac{4}{n}\))⁵

Question 13.
(\(\frac{1}{2 k^{2}}\))⁵

Question 14.
(\(\frac{6 c}{7}\))^-2

Monitoring Progress

Question 15.
Write two expressions that represent the area of a base of the cylinder in Example 5.

Question 16.
It takes the Sun about 2.3 × 10⁸ years to orbit the center of the Milky Way. It takes Pluto about 2.5 × 10² years to orbit the Sun. How many times does Pluto orbit the Sun while the Sun completes one orbit around the center of the Milky Way? Write your answer in scientific notation.

Properties of Exponents 6.1 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Which definitions or properties would you use to simplify the expression (4⁸ • 4^-4)^-2? Explain.
Answer:

Question 2.
WRITING
Explain when and how to use the Power of a Product Property.
Answer:

Question 3.
WRITING
Explain when and how to use the Quotient of Powers Property.
Answer:

Question 4.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer:

In Exercises 5–12, evaluate the expression. (See Example 1.)

Question 5.
(-7)°
Answer:

Question 6.
4°
Answer:

Question 7.
5^-4
Answer:

Question 8.
(-2)^-5

Question 9.
\(\frac{2^{-4}}{4^{0}}\)
Answer:

Question 10.
\(\frac{5^{-1}}{-9^{0}}\)
Answer:

Question 11.
\(\frac{-3^{-3}}{6^{-2}}\)
Answer:

Question 12.
\(\frac{(-8)^{-2}}{3^{-4}}\)
Answer:

In Exercises 13–22, simplify the expression. Write your answer using only positive exponents.

Question 13.
x^-7
Answer:

Question 14.
y°
Answer:

Question 15.
9x°y^-3
Answer:

Question 16.
15c^-8d°
Answer:

Question 17.
\(\frac{2^{-2} m^{-3}}{n^{0}}\)
Answer:

Question 18.
\(\frac{10^{0} r^{-11} s}{3^{2}}\)
Answer:

Question 19.
\(\frac{4^{-3} a^{0}}{b^{-7}}\)
Answer:

Question 20.
\(\frac{p^{-8}}{7^{-2} q^{-9}}\)
Answer:

Question 21.
\(\frac{2^{2} y^{-6}}{8^{-1} z^{0} x^{-7}}\)
Answer:

Question 22.
\(\frac{13 x^{-5} y^{0}}{5^{-3} z^{-10}}\)
Answer:

In Exercises 23–32, simplify the expression. Write your answer using only positive exponents.

Question 23.
\(\frac{5^{6}}{5^{2}}\)
Answer:

Question 24.
\(\frac{(-6)^{8}}{(-6)^{5}}\)
Answer:

Question 25.
(-9)² • (-9)²
Answer:

Question 26.
4^-5 • 4⁵
Answer:

Question 27.
(p⁶)⁴
Answer:

Question 28.
(s^-5)³
Answer:

Question 29.
6^-8 • 6⁵
Answer:

Question 30.
-7 • (-7)^-4

Question 31.
\(\frac{x^{5}}{x^{4}}\) • x
Answer:

Question 32.
\(\frac{z^{8} \cdot z^{2}}{z^{5}}\)
Answer:

Question 33.
USING PROPERTIES
A microscope magnifies an object 10⁵ times. The length of an object is 10^-7 meter. What is its magnified length?

Answer:

Question 34.
USING PROPERTIES
The area of the rectangular computer chip is 112³b² square microns. What is the length?

Answer:

ERROR ANALYSIS
In Exercises 35 and 36, describe and correct the error in simplifying the expression.

Question 35.

Answer:

Question 36.

Answer:

In Exercises 37–44, simplify the expression. Write your answer using only positive exponents.

Question 37.
(-5z)³
Answer:

Question 38.
(4x)^-4
Answer:

Question 39.
(\(\frac{6}{n}\))^-2

Answer:

Question 40.
(\(\frac{-t}{3}\))²
Answer:

Question 41.
(3s⁸)^-5

Answer:

Question 42.
(-5p³)³
Answer:

Question 43.
(\(-\frac{w^{3}}{6}\))^-2

Answer:

Question 44.
(\(\frac{1}{2 r^{6}}\))^-6

Question 45.
USING PROPERTIES
Which of the expressions represent the volume of the sphere? Explain.


Answer:

Question 46.
MODELING WITH MATHEMATICS
Diffusion is the movement of molecules from one location to another. The time t (in seconds) it takes molecules to diffuse a distance of x centimeters is given by t = \(\frac{x^{2}}{2 D}\), where D is the diffusion coefficient. The diffusion coefficient for a drop of ink in water is about 10^-5 square centimeters per second. How long will it take the ink to diffuse 1 micrometer (10^-4 centimeter)?
Answer:

In Exercises 47–50, simplify the expression. Write your answer using only positive exponents.

Question 47.

Answer:

Question 48.

Answer:

Question 49.

Answer:

Question 50.

Answer:

In Exercises 51–54, evaluate the expression. Write your answer in scientific notation and standard form.

Question 51.
(3 × 10²)(1.5 × 10^-5)
Answer:

Question 52.
(6.1 × 10^-3)(8 × 10⁹)
Answer:

Question 53.

Answer:

Question 54.

Answer:

Question 55.
PROBLEM SOLVING
In 2012, on average, about 9.46 × 10^-1 pound of potatoes was produced for every 2.3 × 10^-5 acre harvested. How many pounds of potatoes on average were produced for each acre harvested? Write your answer in scientific notation and in standard form.
Answer:

Question 56.
PROBLEM SOLVING
The speed of light is approximately 3 × 10⁵ kilometers per second. How long does it take sunlight to reach Jupiter? Write your answer in scientific notation and in standard form.

Answer:

Question 57.
MATHEMATICAL CONNECTIONS
Consider Cube A and Cube B.

a. Which property of exponents should you use to simplify an expression for the volume of each cube?
b. How can you use the Power of a Quotient Property to find how many times greater the volume of Cube B is than the volume of Cube A?
Answer:

Question 58.
PROBLEM SOLVING
A byte is a unit used to measure a computer’s memory. The table shows the numbers of bytes in several units of measure.

a. How many kilobytes are in 1 terabyte?
b. How many megabytes are in 16 gigabytes?
c. Another unit used to measure a computer’s memory is a bit. There are 8 bits in a byte. How can you convert the number of bytes in each unit of measure given in the table to bits? Can you still use a base of 2? Explain.
Answer:

REWRITING EXPRESSIONS
In Exercises 59–62, rewrite the expression as a power of a product.

questions 59.
8a³b³
Answer:

Question 60.
16r²s²
Answer:

Question 61.
64w¹⁸z¹²
Answer:

Question 62.
81x⁴y⁸
Answer:

Question 63.
USING STRUCTURE
The probability of rolling a 6 on a number cube is \(\frac{1}{6}\). The probability of rolling a 6 twice in a row is (\(\frac{1}{6}\))² = \(\frac{1}{36}\).
a. Write an expression that represents the probability of rolling a 6 n times in a row.

b. What is the probability of rolling a 6 four times in a row?
c. What is the probability of flipping heads on a coin five times in a row? Explain.
Answer:

Question 64.
HOW DO YOU SEE IT?
The shaded part of Figure n represents the portion of a piece of paper visible after folding the paper in half n times.

a. What fraction of the original piece of paper is each shaded part?
b. Rewrite each fraction from part (a) in the form 2x.
Answer:

Question 65.
REASONING
Find x and y when \(\frac{b^{x}}{b^{y}}\) = b⁹ and \(\frac{b^{x} \cdot b^{2}}{b^{3 y}}\) = b¹³. Explain how you found your answer.
Answer:

Question 66.
THOUGHT PROVOKING
Write expressions for r and h so that the volume of the cone can be represented by the expression 27πx⁸. Find r and h.

Answer:

Question 67.
MAKING AN ARGUMENT
One of the smallest plant seeds comes from an orchid, and one of the largest plant seeds comes from a double coconut palm. A seed from an orchid has a mass of 10^-6 gram. The mass of a seed from a double coconut palm is 1010 times the mass of the seed from the orchid. Your friend says that the seed from the double coconut palm has a mass of about 1 kilogram. Is your friend correct? Explain.
Answer:

Question 68.
CRITICAL THINKING
Your school is conducting a survey. Students can answer the questions in either part with “agree” or “disagree.”

a. What power of 2 represents the number of different ways that a student can answer all the questions in Part 1?
b. What power of 2 represents the number of different ways that a student can answer all the questions on the entire survey?
c. The survey changes, and students can now answer “agree,” “disagree,” or “no opinion.” How does this affect your answers in parts (a) and (b)?
Answer:

Question 69.
ABSTRACT REASONING
Compare the values of aⁿ and a^-n when n < 0, when n = 0, and when n > 0 for
(a) a > 1 and
(b) 0 < a < 1. Explain your reasoning.
Answer:

Maintaining Mathematical Proficiency

Find the square root(s).

Question 70.
\(\sqrt{25}\)
Answer:

Question 71.
–\(\sqrt{100}\)
Answer:

Question 72.
± \(\sqrt{\frac{1}{64}}\)
Answer:

Classify the real number in as many ways as possible.

Question 73.
12
Answer:

Question 74.
\(\frac{65}{9}\)
Answer:

Question 75.
\(\frac{\pi}{4}\)
Answer:

Lesson 6.2 Radicals and Rational Exponents

Essential Question
How can you write and evaluate an nth root of a number?
Recall that you cube a number as follows.

To “undo” cubing a number, take the cube root of the number.

EXPLORATION 1
Finding Cube Roots
Work with a partner. Use a cube root symbol to write the side length of each cube. Then find the cube root. Check your answers by multiplying. Which cube is the largest? Which two cubes are the same size? Explain your reasoning.

a. Volume = 27 ft³
b. Volume = 125 cm³
c. Volume = 3375 in.³
d. Volume = 3.375 m³
e. Volume = 1 yd³
f. Volume = \(\frac{125}{8}\) mm³

EXPLORATION 2
Estimating nth Roots
Work with a partner. Estimate each positive nth root. Then match each nth root with the point on the number line. Justify your answers.

Communicate Your Answer

Question 3.
How can you write and evaluate an nth root of a number?

Question 4.
The body mass m (in kilograms) of a dinosaur that walked on two feet can be modeled by
m = (0.00016)C^2.73
where C is the circumference (in millimeters) of the dinosaur’s femur. The mass of a Tyrannosaurus rex was 4000 kilograms. Use a calculator to approximate the circumference of its femur.

6.2 Lesson

Monitoring Progress

Find the indicated real nth root(s) of a.

Question 1.
n = 3, a = -125

Question 2.
n = 6, a = 64

Evaluate the expression.

Question 3.
\(\sqrt[3]{-125}\)

Question 4.
(-64)^{2 / 3}

Question 5.
9^{5 / 2}

Question 6.
256^{3 / 4}

Question 7.
WHAT IF? In Example 4, the volume of the beach ball is 17,000 cubic inches. Find the radius to the nearest inch. Use 3.14 for π.

Question 8.
The average cost of college tuition increases from $8500 to $13,500 over a period of 8 years. Find the annual inflation rate to the nearest tenth of a percent.

Radicals and Rational Exponents 6.2 Exercises

Monitoring Progress and Modeling with Mathematics

Question 1.
WRITING
Explain how to evaluate 81^{1 / 4}.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG?
Which expression does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, rewrite the expression in rational exponent form.

Question 3.
\(\sqrt{10}\)
Answer:

Question 4.
\(\sqrt[5]{34}\)
Answer:

In Exercises 5 and 6, rewrite the expression in radical form.

Question 5.
15^{1 / 3}
Answer:

Question 6.
140^{1 / 8}
Answer:

In Exercises 7–10, find the indicated real nth root(s) of a.

Question 7.
n = 2, a = 36
Answer:

Question 8.
n = 4, a = 81
Answer:

Question 9.
n = 3, a = 1000
Answer:

Question 10.
n = 9, a = -512
Answer:

MATHEMATICAL CONNECTIONS
In Exercises 11 and 12, find the dimensions of the cube. Check your answer.

Question 11.

Answer:

Question 12.

Answer:

In Exercises 13–18, evaluate the expression.

Question 13.
\(\sqrt[4]{256}\)
Answer:

Question 14.
\(\sqrt[3]{-216}\)
Answer:

Question 15.
\(\sqrt[3]{-343}\)
Answer:

Question 16.
–\(\sqrt[5]{1024}\)
Answer:

Question 17.
128^{1 / 7}
Answer:

Question 18.
(-64)^{1 / 2}
Answer:

In Exercises 19 and 20, rewrite the expression in rational exponent form.

Question 19.
(\(\sqrt[5]{8}\))⁴
Answer:

Question 20.
(\(\sqrt[5]{-21}\))⁶
Answer:

In Exercises 21 and 22, rewrite the expression in radical form.

Question 21.
(-4)^{2 / 7}
Answer:

Question 22.
9^{5 / 2}
Answer:

In Exercises 23–28, evaluate the expression.

Question 23.
32^{3 / 5}
Answer:

Question 24.
125^{2 / 3}
Answer:

Question 25.
(-36)^{3 / 2}
Answer:

Question 26.
(-243)^{2 / 5}
Answer:

Question 27.
(-128)^{5 / 7}
Answer:

Question 28.
343^{4 / 3}
Answer:

Question 29.
ERROR ANALYSIS
Describe and correct the error in rewriting the expression in rational exponent form.

Answer:

Question 30.
ERROR ANALYSIS Describe and correct the error in evaluating the expression.

Answer:

In Exercises 31–34, evaluate the expression.

Question 31.
(\(\frac{1}{1000}\))^{1 / 3}
Answer:

Question 32.
(\(\frac{1}{64}\))^{1 / 6}
Answer:

Question 33.
(27)-^{2 / 3}
Answer:

Question 34.
(9)-^{5 / 2}
Answer:

Question 35.
PROBLEM SOLVING
A math club is having a bake sale. Find the area of the bake sale sign.

Answer:

Question 36.
PROBLEM SOLVING
The volume of a cube-shaped box is 27⁵ cubic millimeters. Find the length of one side of the box.
Answer:

Question 37.
MODELING WITH MATHEMATICS The radius r of the base of a cone is given by the equation r = (\(\frac{3 V}{\pi h}\))^{1 / 2}

where V is the volume of the cone and h is the height of the cone. Find the radius of the paper cup to the nearest inch. Use 3.14 for π.
Answer:

Question 38.
MODELING WITH MATHEMATICS
The volume of a sphere is given by the equation V = \(\frac{1}{6 \sqrt{\pi}}\)S^{3 / 2}, where S is the surface area of the sphere. Find the volume of a sphere, to the nearest cubic meter, that has a surface area of 60 square meters. Use 3.14 for π.
Answer:

Question 39.
WRITING
Explain how to write (\(\sqrt[n]{a}\))^m in rational exponent form.
Answer:

Question 40.
HOW DO YOU SEE IT?
Write an expression in rational exponent form that represents the side length of the square.

Answer:

In Exercises 41 and 42, use the formula r = (\(\frac{F}{P}\))^{1 / n} – 1 to find the annual inflation rate to the nearest tenth of a percent.

Question 41.
A farm increases in value from $800,000 to $1,100,000 over a period of 6 years.
Answer:

Question 42.
The cost of a gallon of gas increases from $1.46 to $3.53 over a period of 10 years.
Answer:

Question 43.
REASONING
For what values of x is x = x^{1 / 5}?
Answer:

Question 44.
MAKING AN ARGUMENT
Your friend says that for a real number a and a positive integer n, the value of \(\sqrt[n]{a}\) is always positive and the value of –\(\sqrt[n]{a}\) is always negative. Is your friend correct? Explain.
Answer:

In Exercises 45–48, simplify the expression.

Question 45.
(y^{1 / 6})³ • x
Answer:

Question 46.
(y • y^{1 / 3})^{3 / 2}
Answer:

Question 47.
x • \(\sqrt[3]{y^{6}}\) + y² • \(\sqrt[3]{x^{3}}\)
Answer:

Question 48.
(x^{1 / 3} • y^{1 / 2})⁹ • \(\sqrt{y}\)
Answer:

Question 49.
PROBLEM SOLVING
The formula for the volume of a regular dodecahedron is V ≈ 7.66ℓ³, where ℓ is the length of an edge. The volume of the dodecahedron is 20 cubic feet. Estimate the edge length.

Answer:

Question 50.
THOUGHT PROVOKING
Find a formula (for instance, from geometry or physics) that contains a radical. Rewrite the formula using rational exponents.
Answer:

ABSTRACT REASONING
In Exercises 51–56, let x be a non negative real number. Determine whether the statement is always, sometimes, or never true. Justify your answer.
Answer:

Question 51.
(x^{1 / 3})³ = x
Answer:

Question 52.
x^{1 / 3} = x^-3
Answer:

Question 53.
x^{1 / 3} = \(\sqrt[3]{x}\)
Answer:

Question 54.
x = x^{1 / 3} • x³
Answer:

Question 55.

Answer:

Maintaining Mathematical Proficiency

Evaluate the function when x = −3, 0, and 8.(Section 3.3)

Question 57.
f(x) = 2x – 10
Answer:

Question 58.
w(x) = -5x – 1
Answer:

Question 59.
h(x) = 13 – x
Answer:

Question 60.
g(x) = 8x + 16
Answer:

Lesson 6.3 Exponential Functions

Essential Question
What are some of the characteristics of the graph of an exponential function?
EXPLORATION 1
Exploring an Exponential Function
Work with a partner. Copy and complete each table for the exponential function y = 16(2)^x. In each table, what do you notice about the values of x? What do you notice about the values of y?

EXPLORATION 2
Exploring an Exponential Function
Work with a partner. Repeat Exploration 1 for the exponential function y = 16(\(\frac{1}{2}\))^x. Do you think the statement below is true for any exponential function? Justify your answer.

“As the independent variable x changes by a constant amount, the dependent variable y is multiplied by a constant factor.”

EXPLORATION 3
Graphing Exponential Functions
Work with a partner. Sketch the graphs of the functions given in Explorations 1 and 2. How are the graphs similar? How are they different?

Communicate Your Answer

Question 4.
What are some of the characteristics of the graph of an exponential function?

Question 5.
Sketch the graph of each exponential function. Does each graph have the characteristics you described in Question 4? Explain your reasoning.
a. y = 2^x
b. y = 2(3)^x
c. y = 3(1.5)^x
d. y = (\(\frac{1}{2}\))^x
e. y = 3(\(\frac{1}{2}\))^x
f. y = 2(\(\frac{3}{4}\))^x

6.3 Lesson

Monitoring Progress

Does the table represent a linear or an exponential function? Explain.

Question 1.

Question 2.

Evaluate the function when x = −2, 0, and \(\frac{1}{2}\).

Question 3.
y = 2(9)^x

Question 4.
y = 1.5(2)^x

Monitoring Progress

Graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f.

Question 5.
f(x) = -2(4)^x

Question 6.
f(x) = 2(\(\frac{1}{4}\))^x

Graph the function. Describe the domain and range.

Question 7.
y = -2(3)^{x + 2} – 1

Question 8.
f(x) = (0.25)^x + 3

Question 9.
WHAT IF? In Example 6, the dependent variable of g is multiplied by 3 for every 1 unit the independent variable x increases. Graph g when g(0) = 4. Compare g and the function f from Example 3 over the interval x = 0 to x = 2.

Question 10.
A bacterial population y after x days can be represented by an exponential function whose graph passes through (0, 100) and (1, 200).
(a) Write a function that represents the population.
(b) Find the population after 6 days.
(c) Does this bacterial population grow faster than the bacterial population in Example 7? Explain.

Exponential Functions 6.3 Exercises

Vocabulary and Core Concept Check

Question 1.
OPEN-ENDED
Sketch an increasing exponential function whose graph has a y-intercept of 2.
Answer:

Question 2.
REASONING
Why is a the y-intercept of the graph of the function y = ab^x?
Answer:

Question 3.
WRITING
Compare the graph of y = 2(5)^x with the graph of y = 5^x.
Answer:

Question 4.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–10, determine whether the equation represents an exponential function. Explain.

Question 5.
y = 4(7)^x
Answer:

Question 6.
y = -6x
Answer:

Question 7.
y = 2x³
Answer:

Question 8.
y = -3x
Answer:

Question 9.
y = 9(-5)^x
Answer:

Question 10.
y = \(\frac{1}{2}\)(1)^x
Answer:

In Exercises 11–14, determine whether the table represents a linear or an exponential function. Explain. 

Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Answer:

In Exercises 15–20, evaluate the function for the given value of x.

Question 15.
y = 3^x; x = 2
Answer:

Question 16.
f(x) = 3(2)^x; x = -1
Answer:

Question 17.
y = -4(5)^x; x = 2
Answer:

Question 18.
f(x) = 0.5^x; x = -3
Answer:

Question 19.
f(x) = \(\frac{1}{3}\)(6)^x; x = 3
Answer:

Question 20.
y = \(\frac{1}{4}\)(4)^x; x = \(\frac{3}{2}\)
Answer:

USING STRUCTURE
In Exercises 21–24, match the function with its graph.

Question 21.
f(x) = 2(0.5)^x
Answer:

Question 22.
y = -2(0.5)^x
Answer:

Question 23.
y = 2(2)^x
Answer:

Question 24.
f(x) = -2(2)^x

Answer:

In Exercises 25–30, graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f.

Question 25.
f (x) = 3(0.5)^x
Answer:

Question 26.
f(x) = -4x
Answer:

Question 27.
f(x) = -2(7)^x
Answer:

Question 28.
f(x) = 6 (\(\frac{1}{3}\))^x
Answer:

Question 29.
f(x) = \(\frac{1}{2}\)(8)^x
Answer:

Question 30.
f (x) = \(\frac{3}{2}\)(0.25)^x
Answer:

In Exercises 31–36, graph the function. Describe the domain and range.

Question 31.
f(x) = 3^x – 1
Answer:

Question 32.
f(x) = 4^{x + 3}
Answer:

Question 33.
y = 5^{x – 2} + 7
Answer:

Question 34.
y = – (\(\frac{1}{2}\))^{x + 1} – 3
Answer:

Question 35.
y = -8(0.75)^{x + 2} – 2
Answer:

Question 36.
f(x) = 3(6)^{x – 1}
Answer:

In Exercises 37–40, compare the graphs. Find the value of h, k, or a.

Question 37.

Answer:

Question 38.

Answer:

Question 39.

Answer:

Question 40.

Answer:

Question 41.
ERROR ANALYSIS
Describe and correct the error in evaluating the function.

Answer:

Question 42.
ERROR ANALYSIS
Describe and correct the error in finding the domain and range of the function.

Answer:

In Exercises 43 and 44, graph the function with the given description. Compare the function to f (x) = 0.5(4)^x over the interval x = 0 to x = 2.

Question 43.
An exponential function g models a relationship in which the dependent variable is multiplied by 2.5 for every 1 unit the independent variable x increases. The value of the function at 0 is 8.
Answer:

Question 44.
An exponential function h models a relationship in which the dependent variable is multiplied by \(\frac{1}{2}\) for every 1 unit the independent variable x increases. The value of the function at 0 is 32.
Answer:

Question 45.
MODELING WITH MATHEMATICS
You graph an exponential function on a calculator. You zoom in repeatedly to 25% of the screen size. The function y = 0.25^x represents the percent (in decimal form) of the original screen display that you see, where x is the number of times you zoom in.
a. Graph the function. Describe the domain and range.
b. Find and interpret the y-intercept.
c. You zoom in twice. What percent of the original screen do you see?
Answer:

Question 46.
MODELING WITH MATHEMATICS
A population y of coyotes in a national park triples every 20 years. The function y = 15(3)^x represents the population, where x is the number of 20-year periods.

a. Graph the function. Describe the domain and range.
b. Find and interpret the y-intercept.
c. How many coyotes are in the national park in 40 years?
Answer:

In Exercises 47–50, write an exponential function represented by the table or graph.

Question 47.

Answer:

Question 48.

Answer:

Question 49.

Answer:

Question 50.

Answer:

Question 51.
MODELING WITH MATHEMATICS
The graph represents the number y of visitors to a new art gallery after x months.

a. Write an exponential function that represents this situation.
b. Approximate the number of visitors after 5 months.
Answer:

Question 52.
PROBLEM SOLVING
A sales report shows that 3300 gas grills were purchased from a chain of hardware stores last year. The store expects grill sales to increase 6% each year. About how many grills does the store expect to sell in Year 6? Use an equation to justify your answer.
Answer:

Question 53.
WRITING
Graph the function f(x) = -2^x. Then graph g(x) = -2^x – 3. How are the y-intercept, domain, and range affected by the translation?
Answer:

Question 54.
MAKING AN ARGUMENT
Your friend says that the table represents an exponential function because y is multiplied by a constant factor. Is your friend correct? Explain.

Answer:

Question 55.
WRITING
Describe the effect of a on the graph of y = a • 2^x when a is positive and when a is negative.
Answer:

Question 56.
OPEN-ENDED
Write a function whose graph is a horizontal translation of the graph of h(x) = 4^x.
Answer:

Question 57.
USING STRUCTURE
The graph of g is a translation 4 units up and 3 units right of the graph of f(x) = 5^x. Write an equation for g.
Answer:

Question 58.
HOW DO YOU SEE IT? The exponential function y = V(x) represents the projected value of a stock x weeks after a corporation loses an important legal battle. The graph of the function is shown.

a. After how many weeks will the stock be worth $20?
b. Describe the change in the stock price from Week 1 to Week 3.
Answer:

Question 59.
USING GRAPHS
The graph represents the exponential function f. Find f(7).

Answer:

Question 60.
THOUGHT PROVOKING
Write a function of the form y = ab^x that represents a real-life population. Explain the meaning of each of the constants a and b in the real-life context.
Answer:

Question 61.
REASONING
Let f(x) = ab^x. Show that when x is increased by a constant k, the quotient \(\frac{f(x+k)}{f(x)}\) is always the same regardless of the value of x.
Answer:

Question 62.
PROBLEM SOLVING
A function g models a relationship in which the dependent variable is multiplied by 4 for every 2 units the independent variable increases. The value of the function at 0 is 5. Write an equation that represents the function.
Answer:

Question 63.
PROBLEM SOLVING
Write an exponential function f so that the slope from the point (0, f(0)) to the point (2, f(2)) is equal to 12.
Answer:

Maintaining Mathematical Proficiency

Write the percent as a decimal.

Question 64.
4%
Answer:

Question 65.
35%
Answer:

Question 66.
128%
Answer:

Question 67.
250%
Answer:

Lesson 6.4 Exponential Growth and Decay

Essential Question
What are some of the characteristics of exponential growth and exponential decay functions?

EXPLORATION 1
Predicting a Future Event
Work with a partner. It is estimated, that in 1782, there were about 100,000 nesting pairs of bald eagles in the United States. By the 1960s, this number had dropped to about 500 nesting pairs. In 1967, the bald eagle was declared an endangered species in the United States. With protection, the nesting pair population began to increase. Finally, in 2007, the bald eagle was removed from the list of endangered and threatened species.
Describe the pattern shown in the graph. Is it exponential growth? Assume the pattern continues. When will the population return to that of the late 1700s? Explain your reasoning.

EXPLORATION 2
Describing a Decay Pattern
Work with a partner. A forensic pathologist was called to estimate the time of death of a person. At midnight, the body temperature was 80.5°F and the room temperature was a constant 60°F. One hour later, the body temperature was 78.5°F.
a. By what percent did the difference between the body temperature and the room temperature drop during the hour?
b. Assume that the original body temperature was 98.6°F. Use the percent decrease found in part (a) to make a table showing the decreases in body temperature. Use the table to estimate the time of death.

Communicate Your Answer

Question 3.
What are some of the characteristics of exponential growth and exponential decay functions?

Question 4.
Use the Internet or some other reference to find an example of each type of function. Your examples should be different than those given in Explorations 1 and 2.
a. exponential growth
b. exponential decay

6.4 Lesson

Monitoring Progress

Question 1.
A website has 500,000 members in 2010. The number y of members increases by 15% each year.
(a) Write an exponential growth function that represents the website membership t years after 2010.
(b) How many members will there be in 2016? Round your answer to the nearest ten thousand.

Monitoring Progress

Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain.

Question 2.

Question 3.

Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change.

Question 4.
y = 2(0.92)^t

Question 5.
f(t) = (1.2)^t

Rewrite the function to determine whether it represents exponential growth or exponential decay.

Question 6.
f(t) = 3(1.02)^10t

Question 7.
y = (0.95)^{t + 2}

Question 8.
You deposit $500 in a savings account that earns 9% annual interest compounded monthly. Write and graph a function that represents the balance y (in dollars) after t years.

Question 9.
WHAT IF? The car loses 9% of its value every year.
(a) Write a function that represents the value y (in dollars) of the car after t years.
(b) Find the approximate monthly percent decrease in value.
(c) Graph the function from part (a). Use the graph to estimate the value of the car after 12 years. Round your answer to the nearest thousand.

Exponential Growth and Decay 6.4 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
In the exponential growth function y = a(1 + r)^t, the quantity r is called the ________.
Answer:

Question 2.
VOCABULARY
What is the decay factor in the exponential decay function y = a(1 – r)^t?
Answer:

Question 3.
VOCABULARY
Compare exponential growth and exponential decay.
Answer:

Question 4.
WRITING
When does the function y = ab^x represent exponential growth? exponential decay?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, identify the initial amount a and the rate of growth r (as a percent) of the exponential function. Evaluate the function when t = 5. Round your answer to the nearest tenth.

Question 5.
y = 350(1 + 0.75)^t
Answer:

Question 6.
y = 10(1 + 0.4)^t
Answer:

Question 7.
y = 25(1.2)^t
Answer:

Question 8.
y = 12(1.05)^t
Answer:

Question 9.
f(t) = 1500(1.074)^t
Answer:

Question 10.
h(t) = 175(1.028)^t
Answer:

Question 11.
g(t) = 6.72(2)^t
Answer:

Question 12.
p(t) = 1.8^t
Answer:

In Exercises 13–16, write a function that represents the situation.

Question 13.
Sales of $10,000 increase by 65% each year.
Answer:

Question 14.
Your starting annual salary of $35,000 increases by 4% each year.
Answer:

Question 15.
A population of 210,000 increases by 12.5% each year.
Answer:

Question 16.
An item costs $4.50, and its price increases by 3.5% each year.
Answer:

Question 17.
MODELING WITH MATHEMATICS
The population of a city has been increasing by 2% annually. The sign shown is from the year 2000.
a. Write an exponential growth function that represents the population t years after 2000.

b. What will the population be in 2020? Round your answer to the nearest thousand.
Answer:

Question 18.
MODELING WITH MATHEMATICS
A young channel catfish weighs about 0.1 pound. During the next 8 weeks, its weight increases by about 23% each week.
a. Write an exponential growth function that represents the weight of the catfish after t weeks during the 8-week period.
b. About how much will the catfish weigh after 4 weeks? Round your answer to the nearest thousandth.

Answer:

In Exercises 19–26, identify the initial amount a and the rate of decay r (as a percent) of the exponential function. Evaluate the function when t = 3. Round your answer to the nearest tenth.

Question 19.
y = 575(1 – 0.6)^t
Answer:

Question 20.
y = 8(1 – 0.15)^t
Answer:

Question 21.
g(t) = 240(0.75)^t
Answer:

Question 22.
f(t) = 475(0.5)^t
Answer:

Question 23.
w(t) = 700(0.995)^t
Answer:

Question 24.
h(t) = 1250(0.865)^t
Answer:

Question 25.
y = (\(\frac{7}{8}\))^t
Answer:

Question 26.
y = 0.5 (\(\frac{3}{4}\))^t
Answer:

In Exercises 27–30, write a function that represents the situation.

Question 27.
A population of 100,000 decreases by 2% each year.
Answer:

Question 28.
A $900 sound system decreases in value by 9% each year.
Answer:

Question 29.
A stock valued at $100 decreases in value by 9.5% each year.
Answer:

Question 30.
A company profit of $20,000 decreases by 13.4% each year.
Answer:

Question 31.
ERROR ANALYSIS The growth rate of a bacterial culture is 150% each hour. Initially, there are 10 bacteria. Describe and correct the error in finding the number of bacteria in the culture after 8 hours.

Answer:

Question 32.
ERROR ANALYSIS You purchase a car in 2010 for $25,000. The value of the car decreases by 14% annually. Describe and correct the error in finding the value of the car in 2015.

Answer:

In Exercises 33–38, determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain.

Question 33.

Answer:

Question 34.

Answer:

Question 35.

Answer:

Question 36.

Answer:

Question 37.

Answer:

Question 38.

Answer:

Question 39.
ANALYZING RELATIONSHIPS
The table shows the value of a camper t years after it is purchased.

a. Determine whether the table represents an exponential growth function, an exponential decay function, or neither.
b. What is the value of the camper after 5 years?
Answer:

Question 40.
ANALYZING RELATIONSHIPS
The table shows the total numbers of visitors to a website t days after it is online.

a. Determine whether the table represents an exponential growth function, an exponential decay function, or neither.

b. How many people will have visited the website after it is online 47 days?
Answer:

In Exercises 41–48, determine whether each function represents exponential growth or exponential decay. Identify the percent rate of change.

Question 41.
y = 4(0.8)^t
Answer:

Question 42.
y = 15(1.1)^t
Answer:

Question 43.
y = 30(0.95)^t
Answer:

Question 44.
y = 5(1.08)^t
Answer:

Question 45.
r(t) = 0.4(1.06)^t
Answer:

Question 46.
s(t) = 0.65(0.48)^t
Answer:

Question 47.
g(t) = 2 (\(\frac{5}{4}\))^t
Answer:

Question 48.
m(t) = (\(\frac{4}{5}\))^t
Answer:

In Exercises 49–56, rewrite the function to determine whether it represents exponential growth or exponential decay.

Question 49.
y = (0.9)^{t – 4}
Answer:

Question 50.
y = (1.4)^{t + 8}
Answer:

Question 51.
y = 2(1.06)^9t
Answer:

Question 52.
y = 5(0.82)^t/5
Answer:

Question 53.
x(t) = (1.45)^t/2
Answer:

Question 54.
f(t) = 0.4(1.16)^{t – 1}
Answer:

Question 55.
b(t) = 4(0.55)^{t + 3}
Answer:

Question 56.
r(t) = (0.88)^4r
Answer:

In Exercises 57–60, write a function that represents the balance after t years. 

Question 57.
$2000 deposit that earns 5% annual interest compounded quarterly
Answer:

Question 58.
$1400 deposit that earns 10% annual interest compounded semiannually
Answer:

Question 59.
$6200 deposit that earns 8.4% annual interest compounded monthly
Answer:

Question 60.
$3500 deposit that earns 9.2% annual interest compounded quarterly
Answer:

Question 61.
PROBLEM SOLVING
The cross-sectional area of a tree 4.5 feet from the ground is called its basal area. The table shows the basal areas (in square inches) of Tree A over time.

a. Write functions that represent the basal areas of the trees after t years.
b. Graph the functions from part (a) in the same coordinate plane. Compare the basal areas.
Answer:

Question 62.
PROBLEM SOLVING
You deposit $300 into an investment account that earns 12% annual interest compounded quarterly. The graph shows the balance of a savings account over time.

a. Write functions that represent the balances of the accounts after t years.
b. Graph the functions from part (a) in the same coordinate plane. Compare the account balances.
Answer:

Question 63.
PROBLEM SOLVING A city has a population of 25,000. The population is expected to increase by 5.5% annually for the next decade.

a. Write a function that represents the population y after t years.
b. Find the approximate monthly percent increase in population.
c. Graph the function from part (a). Use the graph to estimate the population after 4 years.
Answer:

Question 64.
PROBLEM SOLVING
Plutonium-238 is a material that generates steady heat due to decay and is used in power systems for some spacecraft. The function y = a(0.5)^t/x represents the amount y of a substance remaining after t years, where a is the initial amount and x is the length of the half-life (in years).

a. A scientist is studying a 3-gram sample. Write a function that represents the amount y of plutonium-238 after t years.
b. What is the yearly percent decrease of plutonium-238?
c. Graph the function from part (a). Use the graph to estimate the amount remaining after 12 years.
Answer:

Question 65.
COMPARING FUNCTIONS
The three given functions describe the amount y of ibuprofen (in milligrams) in a person’s bloodstream t hours after taking the dosage.
y ≈ 800(0.71)t
y ≈ 800(0.9943)^60t
y ≈ 800(0.843)^2t
a. Show that these expressions are approximately equivalent.
b. Describe the information given by each of the functions.
Answer:

Question 66.
COMBINING FUNCTIONS You deposit $9000 in a savings account that earns 3.6% annual interest compounded monthly. You also save $40 per month in a safe at home. Write a function C(t) = b(t) + h(t), where b(t) represents the balance of your savings account and h(t) represents the amount in your safe after t years. What does C(t) represent?
Answer:

Question 67.
NUMBER SENSE During a flu epidemic, the number of sick people triples every week. What is the growth rate as a percent? Explain your reasoning.
Answer:

Question 68.
HOW DO YOU SEE IT? Match each situation with its graph. Explain your reasoning.
a. A bacterial population doubles each hour.
b. The value of a computer decreases by 18% each year.
c. A deposit earns 11% annual interest compounded yearly.
d. A radioactive element decays 5.5% each year.

Answer:

Question 69.
WRITING Give an example of an equation in the form y = ab^x that does not represent an exponential growth function or an exponential decay function. Explain your reasoning.
Answer:

Question 70.
THOUGHT PROVOKING Describe two account options into which you can deposit $1000 and earn compound interest. Write a function that represents the balance of each account after t years. Which account would you rather use? Explain your reasoning.
Answer:

Question 71.
MAKING AN ARGUMENT A store is having a sale on sweaters. On the first day, the prices of the sweaters are reduced by 20%. The prices will be reduced another 20% each day until the sweaters are sold. Your friend says the sweaters will be free on the fifth day. Is your friend correct? Explain.

Answer:

Question 72.
COMPARING FUNCTIONS The graphs of f and g are shown.

a. Explain why f is an exponential growth function. Identify the rate of growth.
b. Describe the transformation from the graph of f to the graph of g. Determine the value of k.
c. The graph of g is the same as the graph of h(t) = f (t + r). Use properties of exponents to find the value of r.
Answer:

Maintaining Mathematical Proficiency

Solve the equation. Check your solution.(Section 1.3)

Question 73.
8x + 12 = 4x
Answer:

Question 74.
5 – t = 7t + 21
Answer:

Question 75.
6(r – 2) = 2r + 8
Answer:

Find the slope and the y-intercept of the graph of the linear equation.(Section 3.5)

Question 76.
y = -6x + 7
Answer:

Question 77.
y = \(\frac{1}{4}\)x + 7
Answer:

Question 78.
3y = 6x – 12
Answer:

Question 79.
2y + x = 8
Answer:

Exponential Functions and Sequences Study Skills: Analyzing Your Errors

6.1–6.4 What Did You Learn?

Core Vocabulary

Core Concepts
Section 6.1

Section 6.2

Section 6.3

Section 6.4

Mathematical Practices

Question 1.
How did you apply what you know to simplify the complicated situation in Exercise 56 on page 297?

Question 2.
How can you use previously established results to construct an argument in Exercise 44 on page 304?

Question 3.
How is the form of the function you wrote in Exercise 66 on page 322 related to the forms of other types of functions you have learned about in this course?

Study Skills

Analyzing Your Errors

Misreading Directions

• What Happens: You incorrectly read or do not understand directions.
• How to Avoid This Error: Read the instructions for exercises at least twice and make sure you understand what they mean. Make this a habit and use it when taking tests.

Exponential Functions 6.1 – 6.4 Quiz

Simplify the expression. Write your answer using only positive exponents.(Section 6.1)

Question 1.
3² • 3⁴

Question 2.
(k⁴)^-3

Question 3.

Question 4.

Evaluate the expression.(Section 6.2)

Question 5.
\(\sqrt[3]{27}\)

Question 6.
\(\frac{1}{16}\)^1/4

Question 7.
512^2/3

Question 8.
\(\sqrt{4}\)⁵

Graph the function. Describe the domain and range.(Section 6.3)

Question 9.
y = 5^x

Question 10.
y = -2(\(\frac{1}{6}\))^x

Question 11.
y = 6(2)^{x – 4} – 1

Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain.(Section 6.4)

Question 12.

Question 13.

Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change.(Section 6.4)

Question 14.
y = 3(1.88)^t

Question 15.
f(t) = \(\frac{1}{3}\)(1.26)^t

Question 16.
f(t) = 80(\(\frac{3}{5}\))^t

Question 17.
The table shows several units of mass. (Section 6.1)

a. How many times larger is a kilogram than a nanogram? Write your answer using only positive exponents.
b. How many times smaller is a milligram than a hectogram? Write your answer using only positive exponents.
c. Which is greater, 10,000 milligrams or 1000 decigrams? Explain your reasoning.

Question 18.
You store blankets in a cedar chest. What is the volume of the cedar chest? (Section 6.2)

Question 19.
The function f(t) = 5(4)^t represents the number of frogs in a pond after t years. (Section 6.3 and Section 6.4)
a. Does the function represent exponential growth or exponential decay? Explain.
b. Graph the function. Describe the domain and range.
c. What is the yearly percent change? the approximate monthly percent change?
d. How many frogs are in the pond after 4 years?

Lesson 6.5 Solving Exponential Functions

Essential Question
How can you solve an exponential equation graphically?
EXPLORATION 1
Solving an Exponential Equation Graphically
Work with a partner. Use a graphing calculator to solve the exponential equation 2.5^{x – 3} = 6.25 graphically. Describe your process and explain how you determined the solution.

EXPLORATION 2
The Number of Solutions of an Exponential Equation
Work with a partner.
a. Use a graphing calculator to graph the equation y = 2^x.

b. In the same viewing window, graph a linear equation (if possible) that does not intersect the graph of y = 2^x.
c. In the same viewing window, graph a linear equation (if possible) that intersects the graph of y = 2^x in more than one point.
d. Is it possible for an exponential equation to have no solution? more than one solution? Explain your reasoning.

EXPLORATION 3
Solving Exponential Equations Graphically
Work with a partner. Use a graphing calculator to solve each equation.

a. 2^x = \(\frac{1}{2}\)
b. 2^{x + 1} = 0
c. 2^x = \(\sqrt{2}\)
d. 3^x = 9
e. 3^{x – 1} = 0
f. 4^2x = 2
g. 2^x/2 = \(\frac{1}{4}\)
h. 3^{x + 2} = \(\frac{1}{9}\)
i. 2^{x – 2} = \(\frac{3}{2}\)x – 2

Communicate Your Answer

Question 4.
How can you solve an exponential equation graphically?

Question 5.
A population of 30 mice is expected to double each year. The number p of mice in the population each year is given by p = 30(2ⁿ). In how many years will there be 960 mice in the population?

Monitoring Progress

Solve the equation. Check your solution.

Question 1.
2^2x = 2⁶

Question 2.
5^2x = 5^{x + 1}

Question 3.
7^{3x + 5} = 7^{x + 1}

Solve the equation. Check your solution.

Question 4.
4^x = 256

Question 5.
9^2x = 3^{x – 6}

Question 6.
4^3x = 8^{x + 1}

Question 7.
(\(\frac{1}{3}\))^{x – 1} = 27

Use a graphing calculator to solve the equation.

Question 8.
2^x = 1.8

Question 9.
4^{x – 3} = x + 2

Question 10.
(\(\frac{1}{4}\))^x = -2x – 3

Solving Exponential Functions 6.5 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
Describe how to solve an exponential equation with unlike bases.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, solve the equation. Check your solution.

Question 3.
4^5x = 4¹⁰
Answer:

Question 4.
7^{x – 4} = 7⁸
Answer:

Question 5.
3^9x = 3^{7x + 8}
Answer:

Question 6.
2^4x = 2^{x + 9}
Answer:

Question 7.
2^x = 64
Answer:

Question 8.
3x = 243
Answer:

Question 9.
7^{x – 5} = 49^x
Answer:

Question 10.
216^x = 6^{x + 10}
Answer:

Question 11.
64^{2x + 4} = 16^5x
Answer:

Question 12.
27^x = 9^{x – 2}
Answer:

In Exercises 13–18, solve the equation. Check your solution.

Question 13.
(\(\frac{1}{5}\))^x = 125
Answer:

Question 14.
(\(\frac{1}{4}\))^x = 256
Answer:

Question 15.
\(\frac{1}{128}\) = 2^{5x + 3}
Answer:

Question 16.
3^{4x – 9} = \(\frac{1}{243}\)
Answer:

Question 17.
36^{-3x + 3} = (\(\frac{1}{216}\))^{x + 1}
Answer:

Question 18.
(\(\frac{1}{27}\))^4-x = 9^2x-1
Answer:

ERROR ANALYSIS In Exercises 19 and 20, describe and correct the error in solving the exponential equation.

Question 19.

Answer:

Question 20.

Answer:

Question 21.
2^x = 6
Answer:

Question 22.
4^{2x – 5} = 6
Answer:

Question 23.
5^{x + 2} = 6
Answer:

Question 24.
3^{-x – 1} = 6

Answer:

In Exercises 25–36, use a graphing calculator to solve the equation.

Question 25.
6^{x + 2} = 12
Answer:

Question 26.
5^{x – 4} = 8
Answer:

Question 27.
(\(\frac{1}{2}\))^{7x + 1} = -9
Answer:

Question 28.
(\(\frac{1}{3}\))^{x + 3} = 10
Answer:

Question 29.
2^{x + 6} = 2x + 15
Answer:

Question 30.
3x – 2 = 5^{x – 1}
Answer:

Question 31.
\(\frac{1}{2}\)x – 1 = (\(\frac{1}{3}\))^{2x – 1}
Answer:

Question 32.
2^{-x + 1} = –\(\frac{3}{4}\)x + 3
Answer:

Question 33.
5^x = -4^{-x + 4}
Answer:

Question 34.
7^{x – 2} = 2^-x
Answer:

Question 35.
2^{-x – 3} = 3^{x + 1}
Answer:

Question 36.
5^{-2x + 3} = -6^{x + 5}
Answer:

In Exercises 37–40, solve the equation by using the Property of Equality for Exponential Equations.

Question 37.
30 • 5^{x + 3} = 150
Answer:

Question 38.
12 • 2^{x – 7} = 24
Answer:

Question 39.
4(3^{-2x – 4}) = 36
Answer:

Question 40.
2(4^{2x + 1}) = 128
Answer:

Question 41.
MODELING WITH MATHEMATICS
You scan a photo into a computer at four times its original size. You continue to increase its size repeatedly by 100% using the computer. The new size of the photo y in comparison to its original size after x enlargements on the computer is represented by y = 2^{x + 2}. How many times must the photo be enlarged on the computer so the new photo is 32 times the original size?
Answer:

Question 42.
MODELING WITH MATHEMATICS A bacterial culture quadruples in size every hour. You begin observing the number of bacteria 3 hours after the culture is prepared. The amount y of bacteria x hours after the culture is prepared is represented by y = 192(4^{x – 3}). When will there be 200,000 bacteria?

Answer:

In Exercises 43–46, solve the equation.

Question 43.
3^{3x + 6} = 27^{x + 2}
Answer:

Question 44.
3^{4x + 3} = 81^x
Answer:

Question 45.
4^{x + 3} = 2^{2(x + 1)}
Answer:

Question 46.
5^{8(x – 1)} = 625^{2x – 2}
Answer:

Question 47.
NUMBER SENSE
Explain how you can use mental math to solve the equation 8^{x – 4} = 1.
Answer:

Question 48.
PROBLEM SOLVING
There are a total of 128 teams at the start of a citywide 3-on-3 basketball tournament. Half the teams are eliminated after each round. Write and solve an exponential equation to determine after which round there are 16 teams left.
Answer:

Question 49.
PROBLEM SOLVING
You deposit $500 in a savings account that earns 6% annual interest compounded yearly. Write and solve an exponential equation to determine when the balance of the account will be $800.
Answer:

Question 50.
HOW DO YOU SEE IT? The graph shows the annual attendance at two different events. Each event began in 2004.

a. Estimate when the events will have about the same attendance.
b. Explain how you can verify your answer in part(a).
Answer:

Question 51.
REASONING
Explain why the Property of Equality for Exponential Equations does not work when b = 1. Give an example to justify your answer.
Answer:

Question 52.
THOUGHT PROVOKING
Is it possible for an exponential equation to have two different solutions? If not, explain your reasoning. If so, give an example.
Answer:

USING STRUCTURE
In Exercises 53–58, solve the equation.

Question 53.
8^{x – 2} = \(\sqrt{8}\)
Answer:

Question 54.
\(\sqrt{5}\) = 5^{x + 4}
Answer:

Question 55.
(\(\sqrt[5]{7}\))^x = 5^{2x + 3}
Answer:

Question 56.

Answer:

Question 57.

Answer:

Question 58.

Answer:

Question 59.
MAKING AN ARGUMENT
Consider the equation (\(\frac{1}{a}\))^x = b, where a > 1 and b > 1. Your friend says the value of x will always be negative. Is your friend correct? Explain.
Answer:

Maintaining Mathematical Proficiency

Determine whether the sequence is arithmetic. If so, find the common difference.(Section 4.6)

Question 60.
-20, -26, -32, -38, . . .
Answer:

Question 61.
9, 18, 36, 72, . . .
Answer:

Question 62.
-5, -8, -12, -17, . . .
Answer:

Question 63.
10, 20, 30, 40, . . .
Answer:

Lesson 6.6 Geometric Sequences

Essential Question

How can you use a geometric sequence to describe a pattern?

In a geometric sequence, the ratio between each pair of consecutive terms is the same. This ratio is called the common ratio.

EXPLORATION 1
Describing Calculator Patterns
Work with a partner. Enter the keystrokes on a calculator and record the results in the table. Describe the pattern.

c. Use a calculator to make your own sequence. Start with any number and multiply by 3 each time. Record your results in the table.

d. Part (a) involves a geometric sequence with a common ratio of 2. What is the common ratio in part (b)? part (c)?

EXPLORATION 2
Folding a Sheet of Paper
Work with a partner. A sheet of paper is about 0.1 millimeter thick.
a. How thick will it be when you fold it in half once? twice? three times?
b. What is the greatest number of times you can fold a piece of paper in half? How thick is the result?

c. Do you agree with the statement below? Explain your reasoning.“If it were possible to fold the paper in half 15 times, it would be taller than you.”

Communicate Your Answer

Question 3.
How can you use a geometric sequence to describe a pattern?

Question 4.
Give an example of a geometric sequence from real life other than paper folding.

6.6 Lesson

Monitoring Progress

Decide whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.

Question 1.
5, 1, -3, -7, . . .

Question 2.
1024, 128, 16, 2, . . .

Question 3.
2, 6, 10, 16, . .

Write the next three terms of the geometric sequence. Then graph the sequence.

Question 4.
1, 3, 9, 27, . . .

Question 5.
2500, 500, 100, 20, . . .

Question 6.
80, -40, 20, -10, . . .

Question 7.
-2, 4, -8, 16, . . .

Write an equation for the nth term of the geometric sequence. Then find a₇.

Question 8.
1, -5, 25, -125, . . .

Question 9.
13, 26, 52, 104, . . .

Question 10.
432, 72, 12, 2, . . .

Question 11.
4, 10, 25, 62.5, . . .

Question 12.
WHAT IF? After how many clicks on the zoom-out button is the side length of the map 2560 miles?

Geometric Sequences 6.6 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING Compare the two sequences.

Answer:

Question 2.
CRITICAL THINKING Why do the points of a geometric sequence lie on an exponential curve only when the common ratio is positive?
Answer:

In Exercises 3–8, find the common ratio of the geometric sequence.

Question 3.
4, 12, 36, 108, . . .
Answer:

Question 4.
36, 6, 1, \(\frac{1}{6}\), . . .
Answer:

Question 5.
\(\frac{3}{8}\), -3, 24, -192, . . .
Answer:

Question 6.
0.1, 1, 10, 100, . . .
Answer:

Question 7.
128, 96, 72, 54, . . .
Answer:

Question 8.
-162, 54, -18, 6, . . .
Answer:

In Exercises 9–14, determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.

Question 9.
-8, 0, 8, 16, . . .
Answer:

Question 10.
-1, 4, -7, 10, . . .
Answer:

Question 11.
9, 14, 20, 27, . . .
Answer:

Question 12.
\(\frac{3}{49}\), \(\frac{3}{7}\), 3, 21, . . .
Answer:

Question 13.
192, 24, 3, \(\frac{3}{8}\), , . . .
Answer:

Question 14.
-25, -18, -11, -4, . . .
Answer:

In Exercises 15–18, determine whether the graph represents an arithmetic sequence, a geometric sequence, or neither. Explain your reasoning.

Question 15.

Answer:

Question 16.

Answer:

Question 17.

Answer:

Question 18.

Answer:

In Exercises 19–24, write the next three terms of the geometric sequence. Then graph the sequence.

Question 19.
5, 20, 80, 320, . . .
Answer:

Question 20.
-3, 12, -48, 192, . . .
Answer:

Question 21.
81, -27, 9, -3, . . .
Answer:

Question 22.
-375, -75, -15, -3, . . .
Answer:

Question 23.
32, 8, 2, \(\frac{1}{2}\), . . .
Answer:

Question 24.
\(\frac{16}{9}\), \(\frac{8}{3}\), 4, 6, . . .
Answer:

In Exercises 25–32, write an equation for the nth termof the geometric sequence. Then find a₆.

Question 25.
2, 8, 32, 128, . . .
Answer:

Question 26.
0.6, -3, 15, -75, . . .
Answer:

Question 27.
–\(\frac{1}{8}\), –\(\frac{1}{4}\), –\(\frac{1}{2}\), -1, . . .
Answer:

Question 28.
0.1, 0.9, 8.1, 72.9, . . .
Answer:

Question 29.

Answer:

Question 30.

Answer:

Question 31.

Answer:

Question 32.

Answer:

Question 33.
PROBLEM SOLVING
A badminton tournament begins with 128 teams. After the first round, 64 teams remain. After the second round, 32 teams remain. How many teams remain after the third, fourth, and fifth rounds?
Answer:

Question 34.
PROBLEM SOLVING
The graphing calculator screen displays an area of 96 square units. After you zoom out once, the area is 384 square units. After you zoom out a second time, the area is 1536 square units. What is the screen area after you zoom out four times?

Answer:

Question 35.
ERROR ANALYSIS
Describe and correct the error in writing the next three terms of the geometric sequence.

Answer:

Question 36.
ERROR ANALYSIS Describe and correct the error in writing an equation for the nth term of the geometric sequence.

Answer:

Question 37.
MODELING WITH MATHEMATICS The distance (in millimeters) traveled by a swinging pendulum decreases after each swing, as shown in the table.

a. Write a function that represents the distance the pendulum swings on its nth swing.
b. After how many swings is the distance 256 millimeters?
Answer:

Question 38.
MODELING WITH MATHEMATICS You start a chain email and send it to six friends. The next day, each of your friends forwards the email to six people. The process continues for a few days.
a. Write a function that represents the number of people who have received the email after n days.
b. After how many days will 1296 people have received the email?

Answer:

MATHEMATICAL CONNECTIONS In Exercises 39 and 40, (a) write a function that represents the sequence of figures and (b) describe the 10th figure in the sequence.

Question 39.

Answer:

Question 40.

Answer:

Question 41.
REASONING Write a sequence that represents the number of teams that have been eliminated in round n of the badminton tournament in Exercise 33. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.
Answer:

Question 42.
REASONING Write a sequence that represents the perimeter of the graphing calculator screen in Exercise 34 after you zoom out n times. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.
Answer:

Question 43.
WRITING Compare the graphs of arithmetic sequences to the graphs of geometric sequences.
Answer:

Question 44.
MAKING AN ARGUMENT You are given two consecutive terms of a sequence.

Your friend says that the sequence is not geometric. A classmate says that is impossible to know given only two terms. Who is correct? Explain.
Answer:

Question 45.
CRITICAL THINKING Is the sequence shown an arithmetic sequence? a geometric sequence? Explain your reasoning.

Answer:

Question 46.
HOW DO YOU SEE IT? Without performing any calculations, match each equation with its graph. Explain your reasoning.

Answer:

Question 47.
REASONING What is the 9th term of the geometric sequence where a₃ = 81 and r = 3?
Answer:

Question 48.
OPEN-ENDED Write a sequence that has a pattern but is not arithmetic or geometric. Describe the pattern.

Question 49.
ATTENDING TO PRECISION Are the terms of a geometric sequence independent or dependent? Explain your reasoning.
Answer:

Question 50.
DRAWING CONCLUSIONS A college student makes a deal with her parents to live at home instead of living on campus. She will pay her parents $0.01 for the first day of the month, $0.02 for the second day, $0.04 for the third day, and so on.
a. Write an equation that represents the nth term of the geometric sequence.
b. What will she pay on the 25th day?
c. Did the student make a good choice or should she have chosen to live on campus? Explain.
Answer:

Question 51.
REPEATED REASONING A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved for the next day.

a. Write the first five terms of the sequence of the number of fluid ounces of soup left each day.
b. Write an equation that represents the nth term of the sequence.
c. When is all the soup gone? Explain.
Answer:

Question 52.
THOUGHT PROVOKING Find the sum of the terms of the geometric sequence.

Explain your reasoning. Write a different infinite geometric sequence that has the same sum.
Answer:

Question 53.
OPEN-ENDED Write a geometric sequence in which a₂ < a₁ < a₃.
Answer:

Question 54.
NUMBER SENSE Write an equation that represents the nth term of each geometric sequence shown.

a. Do the terms a₁ – b₁, a₂ – b₂, a₃ – b₃, . . . form a geometric sequence? If so, how does the common ratio relate to the common ratios of the sequences above?
b. Do the terms form a geometric sequence? If so, how does the common ratio relate to the common ratios of the sequences above?
Answer:

Maintaining Mathematical Proficiency

Use residuals to determine whether the model is a good fit for the data in the table. Explain.

Question 55.
y = 3x – 8

Answer:

Question 56.
y = -5x + 1

Answer:

Lesson 6.7 Recursively Defined Sequences

Essential Question

How can you define a sequence recursively?
A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how a_n is related to one or more preceding terms.

EXPLORATION 1
Describing a Pattern
Work with a partner. Consider a hypothetical population of rabbits. Start with one breeding pair. After each month, each breeding pair produces another breeding pair. The total number of rabbits each month follows the exponential pattern 2, 4, 8, 16, 32,. . .. Now suppose that in the first month after each pair is born, the pair is too young to reproduce. Each pair produces another pair after it is 2 months old. Find the total number of pairs in months 6, 7, and 8.

EXPLORATION 2
Using a Recursive Equation
Work with a partner. Consider the following recursive equation.

Each term in the sequence is the sum of the two preceding terms.
Copy and complete the table. Compare the results with the sequence of the number of pairs in Exploration 1.

Communicate Your Answer

Question 3.
How can you define a sequence recursively?

Question 4.
Use the Internet or some other reference to determine the mathematician who first described the sequences in Explorations 1 and 2.

6.7 Lesson

Monitoring Progress

Write a recursive rule for the sequence.

Question 5.
8, 3, -2, -7, -12, . . .

Question 6.
1.3, 2.6, 3.9, 5.2, 6.5, . . .

Question 7.
4, 20, 100, 500, 2500, . . .

Question 8.
128, -32, 8, -2, 0.5, . . .

Question 9.
Write a recursive rule for the height of the sunflower over time.

Monitoring Progress

Write an explicit rule for the recursive rule.

Question 10.
a₁ = -45, a_n = a_{n – 1} + 20

Question 11.
a₁ = 13, a_n = -3a_{n – 1}

Write a recursive rule for the explicit rule.

Question 12.
a_n = -n + 1

Question 13.
a_n = -2.5(4)_{n – 1}

Monitoring Progress

Write a recursive rule for the sequence. Then write the next three terms of the sequence.

Question 14.
5, 6, 11, 17, 28, . . .

Question 15.
-3, -4, -7, -11, -18, . . .

Question 16.
1, 1, 0, -1, -1, 0, 1, 1, . . .

Question 17.
4, 3, 1, 2, -1, 3, -4, . . .

Recursively Defined Sequences 6.7 Exercises

Question 1.
COMPLETE THE SENTENCE A recursive rule gives the beginning term(s) of a sequence and a(n)_____________ that tells how a_n is related to one or more preceding terms.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG? Which rule does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

Vocabulary and Core Concept Check

In Exercises 3–6, determine whether the recursive rule represents an arithmetic sequence or a geometric sequence.

Question 3.
a₁ = 2, a_n = 7a_{n – 1}
Answer:

Question 4.
a₁ = 18, a_n = a_{n – 1} + 1
Answer:

Question 5.
a₁ = 5, a_n = a_{n – 1} – 4
Answer:

Question 6.
a₁ = 3, a_n = -6a_{n – 1}
Answer:

In Exercises 7–12, write the first six terms of the sequence. Then graph the sequence.

Question 7.
a₁ = 0, a_n = a_{n – 1} + 2
Answer:

Question 8.
a₁ = 10, a_n = a_{n – 1} – 5
Answer:

Question 9.
a₁ = 2, a_n = 3a_{n – 1}
Answer:

Question 10.
a₁ = 8, a_n = 1.5a_{n – 1}
Answer:

Question 11.
a₁ = 80, a_n = –\(\frac{1}{2}\)a_{n – 1}
Answer:

Question 12.
a₁ = -7, a_n = -4a_{n – 1}
Answer:

In Exercises 13–20, write a recursive rule for the sequence.

Question 13.

Answer:

Question 14.

Answer:

Question 15.
243, 81, 27, 9, 3, . . .
Answer:

Question 16.
3, 11, 19, 27, 35, . . .
Answer:

Question 17.
0, -3, -6, -9, -12, . . .
Answer:

Question 18.
5, -20, 80, -320, 1280, . . .
Answer:

Question 19.

Answer:

Question 20.

Answer:

Question 21.
MODELING WITH MATHEMATICS Write a recursive rule for the number of bacterial cells over time.

Answer:

Question 22.
MODELING WITH MATHEMATICS Write a recursive rule for the length of the deer antler over time.

Answer:

In Exercises 23–28, write an explicit rule for the recursive rule.

Question 23.
a₁ = -3, a_n = a_{n – 1} + 3
Answer:

Question 24.
a₁ = 8, a_n = a_{n – 1} – 12
Answer:

Question 25.
a₁ = 16, a_n = 0.5a_{n – 1}
Answer:

Question 26.
a₁ = -2, a_n = 9a_{n – 1}
Answer:

Question 27.
a₁ = 4, a_n = a_{n – 1} + 17
Answer:

Question 28.
a₁ = 5, a_n = -5a_{n – 1}
Answer:

In Exercises 29–34, write a recursive rule for the explicit rule.

Question 29.
a_n = 7(3)^{n – 1}
Answer:

Question 30.
a_n = -4n + 2
Answer:

Question 31.
a_n = 1.5n + 3
Answer:

Question 32.
a_n = 6n – 20
Answer:

Question 33.
a_n = (-5)^{n – 1}
Answer:

Question 34.
a_n = -81(\(\frac{2}{3}\))^{n – 1}
Answer:

In Exercises 35–38, graph the first four terms of the sequence with the given description. Write a recursive rule and an explicit rule for the sequence.

Question 35.
The first term of a sequence is 5. Each term of the sequence is 15 more than the preceding term.
Answer:

Question 36.
The first term of a sequence is 16. Each term of the sequence is half the preceding term.
Answer:

Question 37.
The first term of a sequence is -1. Each term of the sequence is -3 times the preceding term.
Answer:

Question 38.
The first term of a sequence is 19. Each term of the sequence is 13 less than the preceding term.
Answer:

In Exercises 39–44, write a recursive rule for the sequence. Then write the next two terms of the sequence.

Question 39.
1, 3, 4, 7, 11, . . .
Answer:

Question 40.
10, 9, 1, 8, -7, 15, . . .
Answer:

Question 41.
2, 4, 2, -2, -4, -2, . . .
Answer:

Question 42.
6, 1, 7, 8, 15, 23, . . .
Answer:

Question 43.

Answer:

Question 44.

Answer:

Question 45.
ERROR ANALYSIS Describe and correct the error in writing an explicit rule for the recursive rule a₁ = 6, a_n = a_{n – 1} – 12.

Answer:

Question 46.
ERROR ANALYSIS Describe and correct the error in writing a recursive rule for the sequence 2, 4, 6, 10, 16, . . ..

Answer:

In Exercises 47–51, the function f represents a sequence. Find the 2nd, 5th, and 10th terms of the sequence.

Question 47.
f(1) = 3, f (n) = f(n – 1) + 7
Answer:

Question 48.
f(1) = -1, f(n) = 6f(n – 1)
Answer:

Question 49.
f(1) = 8, f(n) = -f(n – 1)
Answer:

Question 50.
f(1) = 4, f(2) = 5, f(n) = f(n – 2) + f(n – 1)
Answer:

Question 51.
f(1) = 10, f(2) = 15, f(n) = f(n – 1) – f(n – 2)
Answer:

Question 52.
MODELING WITH MATHEMATICS The X-ray shows the lengths (in centimeters) of bones in a human hand.

a. Write a recursive rule for the lengths of the bones.
b. Measure the lengths of different sections of your hand. Can the lengths be represented by a recursively defined sequence? Explain.
Answer:

Question 53.
USING TOOLS You can use a spreadsheet to generate the terms of a sequence.

a. To generate the terms of the sequence a₁ = 3, a_n = a_{n – 1} + 2, enter the value of a₁, 3, into cell A1. Then enter “=A1+2” into cell A2, as shown. Use the fill down feature to generate the first 10 terms of the sequence.
b. Use a spreadsheet to generate the first 10 terms of the sequence a₁ = 3, a_n = 4a_{n – 1}. (Hint: Enter “=4*A1” into cell A2.)
c. Use a spreadsheet to generate the first 10 terms of the sequence a₁ = 4, a₂ = 7, a_n = a_{n – 1} – a_{n – 2}. (Hint: Enter “=A2-A1” into cell A3.)
Answer:

Question 54.
HOW DO YOU SEE IT? Consider Squares 1–6 in the diagram.

a. Write a sequence in which each term an is the side length of square n.
b. What is the name of this sequence? What is the next term of this sequence?
c. Use the term in part (b) to add another square to the diagram and extend the spiral.
Answer:

Question 55.
REASONING Write the first 5 terms of the sequence a₁ = 5, a_n = 3a_{n – 1} + 4. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.
Answer:

Question 56.
THOUGHT PROVOKING Describe the pattern for the numbers in Pascal’s Triangle, shown below. Write a recursive rule that gives the mth number in the nth row.

Answer:

Question 57.
REASONING The explicit rule a_n = a₁ + (n – 1)d defines an arithmetic sequence.
a. Explain why a_{n – 1} = a₁ + [(n – 1) – 1]d.
b. Justify each step in showing that a recursive equation for the sequence is a_n = a_{n – 1} + d.

Answer:

Question 58.
MAKING AN ARGUMENT Your friend claims that the sequence

cannot be represented by a recursive rule. Is your friend correct? Explain.
Answer:

Question 59.
PROBLEM SOLVING Write a recursive rule for the sequence.
3, 7, 15, 31, 63, . . .
Answer:

Maintaining Mathematical Proficiency

Simplify the expression.

Question 60.
5x + 12x
Answer:

Question 61.
9 – 6y – 14
Answer:

Question 62.
2d – 7 – 8d
Answer:

Question 63.
3 – 3m + 11m
Answer:

Write a linear function f with the given values.(Section 4.2)

Question 64.
f(2) = 6, f(-1) = -3
Answer:

Question 65.
f (-2) = 0, f(6) = -4
Answer:

Question 66.
f(-3) = 5, f(-1) = 5
Answer:

Question 67.
f(3) = -1, f(-4) = -15
Answer:

Exponential Functions and Sequences Performance Task: The New Car

6.5–6.7

Core Vocabulary

Core Concepts
Section 6.5
Property of Equality for Exponential Equations, p. 326
Solving Exponential Equations by Graphing, p. 328

Section 6.6
Geometric Sequence, p. 332
Equation for a Geometric Sequence, p. 334

Section 6.7
Recursive Equation for an Arithmetic Sequence, p. 340
Recursive Equation for a Geometric Sequence, p. 340

Mathematical Practices

Question 1.
How did you decide on an appropriate level of precision for your answer in Exercise 49 on page 330?

Question 2.
Explain how writing a function in Exercise 39 part (a) on page 337 created a shortcut for answering part (b).

Question 3.
How did you choose an appropriate tool in Exercise 52 part (b) on page 345?

Performance Task

The New Car

There is so much more to buying a new car than the purchase price. Interest rates, depreciation, and inflation are all factors. So, what is the real cost of your new car?

To explore the answers to this question and more, go to

Exponential Functions and Sequences Chapter Review

Simplify the expression. Write your answer using only positive exponents.

Question 1.
y³ • y^-5

Question 2.
\(\frac{x^{4}}{x^{7}}\)

Question 3.
(x⁰y²)³

Question 4.

Evaluate the expression.

Question 5.
\(\sqrt[3]{8}\)

Question 6.
\(\sqrt[5]{-243}\)

Question 7.
625^{3 / 4}

Question 8.
(-25)^{1 / 2}

Question 9.
f(x) = -4 (\(\sqrt[1]{4}\))^x

Question 10.
f(x) = 3^{x + 2}

Question 11.
f(x) = 2^{x – 4} – 3

Question 12.
Write and graph an exponential function f represented by the table. Then compare the graph to the graph of g(x) = (\(\frac{1}{2}\))^x

Determine whether the table represents an exponential growth function, an exponential decay function, or neither. Explain.

Question 13.

Question 14.

Rewrite the function to determine whether it represents exponential growth or exponential decay. Identify the percent rate of change.

Question 15.
f(t) = 4(1.25)^{t + 3}

Question 16.
y = (1.06)^8t

Question 17.
f(t) = 6(0.84)^{t – 4}

Question 18.
You deposit $750 in a savings account that earns 5% annual interest compounded quarterly.
(a) Write a function that represents the balance after t years.
(b) What is the balance of the account after 4 years?

Question 19.
The value of a TV is $1500. Its value decreases by 14% each year.
(a) Write a function that represents the value y (in dollars) of the TV after t years.
(b) Find the approximate monthly percent decrease in value.
(c) Graph the function from part (a). Use the graph to estimate the value of the TV after 3 years.

Solve the equation.

Question 20.
5^x = 5^{3x – 2}

Question 21.
3^{x – 2} = 1

Question 22.
-4 = 6^{4x – 3}

Question 23.
(\(\frac{1}{3}\))^{2x + 3} = 5

Question 24.
(\(\frac{1}{16}\))^3x = 64^{2(x + 8)}

Question 25.
27^{2x + 2} = 81^{x + 4}

Decide whether the sequence is arithmetic, geometric, or neither. Explain your reasoning. If the sequence is geometric, write the next three terms and graph the sequence.

Question 26.
3, 12, 48, 192, . . .

Question 27.
9, -18, 27, -36, . . .

Question 28.
375, -75, 15, -3, . . .

Write an equation for the nth term of the geometric sequence. Then find a₉.

Question 29.
1, 4, 16, 64, . . .

Question 30.
5, -10, 20, -40, . . .

Question 31.
486, 162, 54, 18, . . .

Write the first six terms of the sequence. Then graph the sequence.

Question 32.
a₁ = 4, a_n = a_{n – 1} + 5

Question 33.
a₁ = -4, a_n = -3a_{n – 1}

Question 34.
a₁ = 32, a_n = \(\frac{1}{4}\)a_{n − 1}

Write a recursive rule for the sequence.

Question 35.
3, 8, 13, 18, 23, . . .

Question 36.
3, 6, 12, 24, 48, . . .

Question 37.
7, 6, 13, 19, 32, . .

Question 38.
The first term of a sequence is 8. Each term of the sequence is 5 times the preceding term. Graph the first four terms of the sequence. Write a recursive rule and an explicit rule for the sequence.

Exponential Functions and Sequences Chapter Test

Evaluate the expression.

Question 1.
–\(\sqrt[4]{16}\)

Question 2.
729\(1 / 6\)

Question 3.
(-32)\(7 / 5\)

Simplify the expression. Write your answer using only positive exponents.

Question 4.
z^-2 • z⁴

Question 5.

Question 6.

Write and graph a function that represents the situation.

Question 7.
Your starting annual salary of $42,500 increases by 3% each year.

Question 8.
You deposit $500 in an account that earns 6.5% annual interest compounded yearly.

Write an explicit rule and a recursive rule for the sequence.

Question 9.

Question 10.

Solve the equation. Check your solution.

Question 11.
2^x = \(\frac{1}{128}\)

Question 12.
256^{x + 2} = 16^{3x – 1}

Question 13.
Graph f(x) = 2(6)^x. Compare the graph to the graph of g(x) = 6^x. Describe the domain and range of f.

Use the equation to complete the statement with the symbol < , > , or =. Do not attempt to solve the equation.

Question 14.
\(\frac{5^{a}}{5^{b}}\)

Question 15.
9^a • 9^-b

Question 16.
The first two terms of a sequence are a₁ = 3 and a₂ = -12. Let a₃ be the third term when the sequence is arithmetic and let b₃ be the third term when the sequence is geometric. Find a₃ – b₃.

Question 17.
At sea level, Earth’s atmosphere exerts a pressure of 1 atmosphere. Atmospheric pressure P (in atmospheres) decreases with altitude. It can be modeled by P =(0.99988)^a, where a is the altitude (in meters).
a. Identify the initial amount, decay factor, and decay rate.
b. Use a graphing calculator to graph the function. Use the graph to estimate the atmospheric pressure at an altitude of 5000 feet.

Question 18.
You follow the training schedule from your coach.
a. Write an explicit rule and a recursive rule for the geometric sequence.
b. On what day do you run approximately 3 kilometers?

Exponential Functions and Sequences Cumulative Assessment

Question 1.
Fill in the exponent of x with a number to simplify the expression.

Question 2.
The graph of the exponential function f is shown. Find f(-7).

Question 3.
Student A claims he can form a linear system from the equations shown that has infinitely many solutions. Student B claims she can form a linear system from the equations shown that has one solution. Student C claims he can form a linear system from the equations shown that has no solution.

a. Select two equations to support Student A’s claim.
b. Select two equations to support Student B’s claim.
c. Select two equations to support Student C’s claim.

Question 4.
Fill in the inequality with < , ≤ , > , or ≥ so that the system of linear inequalities has no solution.

Question 5.
The second term of a sequence is 7. Each term of the sequence is 10 more than the preceding term. Fill in values to write a recursive rule and an explicit rule for the sequence.

Question 6.
A data set consists of the heights y (in feet) of a hot-air balloon t minutes after it begins its descent. An equation of the line of best fit is y = 870 – 14.8t. Which of the following is a correct interpretation of the line of best fit?
A. The initial height of the hot-air balloon is 870 feet. The slope has no meaning in this context.
B. The initial height of the hot-air balloon is 870 feet, and it descends 14.8 feet per minute.
C. The initial height of the hot-air balloon is 870 feet, and it ascends 14.8 feet per minute.
D. The hot-air balloon descends 14.8 feet per minute. The y-intercept has no meaning in this context.

Question 7.
Select all the functions whose x-value is an integer when f(x) = 10.

Question 8.
Place each function into one of the three categories. For exponential functions, state whether the function represents exponential growth, exponential decay, or neither.

Question 9.
How does the graph shown compare to the graph of f(x) = 2^x?

Your practice session without Big Ideas Math Algebra 1 Answers Chapter 10 Radical Functions and Equations is incomplete. As it includes all crucial and important study resources like Questions from Exercises 10.1 to 10.4, along with Chapter Test, Review Tests, Cumulative Practice, Quiz, etc. So, check out the below modules and enhance your math proficiency by referring to the BIM Algebra 1 Solution Key of Ch 10 Radical Functions and Equations.

Big Ideas Math Book Algebra 1 Answer Key Chapter 10 Radical Functions and Equations

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Radical Functions and Equations

• Radical Functions and Equations Maintaining Mathematical Proficiency – Page 541
• Radical Functions and Equations Mathematical Practices – Page 542

Lesson: 1 Graphing Square Root Functions

• Lesson 10.1 Graphing Square Root Functions – Page(543-550)
• Graphing Square Root Functions 10.1 Exercises – Page(548-550)

Lesson: 2 Graphing Cube Roots Functions

• Lesson 10.2 Graphing Cube Roots Functions – Page(551-556)
• Graphing Cube Roots Functions 10.2 Exercises – Page(555-556)
• Radical Functions and Equations Study Skills: Making Note Cards – Page 557
• Study Skills: Making Note Cards

Quiz

• Radical Functions and Equations 10.1–10.2 Quiz – Page 558

Lesson: 3 Solving Radical Equations

• Lesson 10.3 Solving Radical Equations – Page(559-566)
• Solving Radical Equations 10.3 Exercises – Page(564-566)

Lesson: 4 Inverse of a Function

• Lesson 10.4 Inverse of a Function – Page(567-574)
• Inverse of a Function 10.4 Exercises – Page(572-574)

Performance Task

• Radical Functions and Equations Performance Task: Medication and the Mosteller Formula – Page 575
• Radical Functions and Equations Chapter Review – Page (576-578)
• Radical Functions and Equations Chapter Test – Page 579
• Radical Functions and Equations Cumulative Assessment – Page(580-581)

Radical Functions and Equations Maintaining Mathematical Proficiency

Evaluate the expression.
Question 1.
7\(\sqrt{25}\) + 10
Answer:
Given the expression
7\(\sqrt{25}\) + 10
\(\sqrt{25}\) = 5
7(5) + 10 = 35 +10 = 45
Thus 7\(\sqrt{25}\) + 10 = 45

Question 2.
-8 – \(\sqrt{\frac{64}{16}}\)
Answer:
Given the expression
-8 – \(\sqrt{\frac{64}{16}}\)
\(\sqrt{\frac{64}{16}}\) = \(\sqrt{4}\) = 2
– 8 – 2 = -10
Thus -8 – \(\sqrt{\frac{64}{16}}\) = -10

Question 3.
\(5\left(\frac{\sqrt{81}}{3}-7\right)\)
Answer:
Given the expression
\(5\left(\frac{\sqrt{81}}{3}-7\right)\)
\(\sqrt{81}\) = 9
5(9/3 – 7)
5(3 – 7) = 5(-4) = -20
Thus \(5\left(\frac{\sqrt{81}}{3}-7\right)\) = -20

Question 4.
-2(3\(\sqrt{9}\) + 13)
Answer:
Given the expression
-2(3\(\sqrt{9}\) + 13)
= -2(3 (3) + 13)
= -2(9 + 13)
= -2(22)
= -44
Thus -2(3\(\sqrt{9}\) + 13) = -44

Graph f and g. Describe the transformations from the graph of f to the graph of g.
Question 5.
f(x) = x; g(x) = 2x – 2
Answer:
Given,
f(x) = x,
g(x) = 2x – 2
Substitute the value of x in f(x)
g(x) = 2(f(x)) – 2
h(x) = 2x

Question 6.
f(x) = x; g(x) = \(\frac{1}{3}\)x + 5
Answer:

Question 7.
f(x) = x; g(x) = -x + 3
Answer:

Question 8.
ABSTRACT REASONING
Let a and b represent constants, where b ≥ 0. Describe the transformations from the graph of m(x) = ax + b to the graph of n(x) = -2ax – 4b.
Answer:

Radical Functions and Equations Mathematical Practices

Mathematically proficient students distinguish correct reasoning from flawed reasoning.

Monitoring Progress

Question 1.
Which of the following square roots are rational numbers? Explain your reasoning.
\(\sqrt{0}, \sqrt{1}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{9}\)
Answer:

Question 2.
The sequence of steps shown appears to prove that 1 = 0. What is wrong with this argument?

Answer:

Lesson 10.1 Graphing Square Root Functions

Essential Question What are some of the characteristics of the graph of a square root function?

EXPLORATION 1

Graphing Square Root Functions
Work with a partner.

• Make a table of values for each function.
• Use the table to sketch the graph of each function.
• Describe the domain of each function.
• Describe the range of each function.

Answer:

EXPLORATION 2

Writing Square Root Functions
Work with a partner. Write a square root function, y = f (x), that has the given values. Then use the function to complete the table.

Answer:

Communicate Your Answer

Question 3.
What are some of the characteristics of the graph of a square root function?
Answer:
The domain of the square root function f(x)=√x is given in interval form by: [0,+∞)
The range of the square root function f(x)=√x is given in interval form by: [0,+∞)
The x and y intercepts are both at (0,0)
The square root function is an increasing function.

Question 4.
Graph each function. Then compare the graph to the graph of f(x) = \(\sqrt{x}\).
a. g(x) = \(\sqrt{x-1}\)
b. g(x) = \(\sqrt{x-1}\)
c. g(x) = 2\(\sqrt{x}\)
d. g(x) = -2 \(\sqrt{x}\)
Answer:

Monitoring Progress

Describe the domain of the function.
Question 1.
f(x) = 10 \(\sqrt{x}\)
Answer:

Question 2.
y = \(\sqrt{2x}\) + 7
Answer:

Question 3.
h(x) = \(\sqrt{-x+1}\)
Answer:

Graph the function. Describe the range.
Question 4.
g(x) = \(\sqrt{x}\) – 4
Answer:

Question 5.
y = \(\sqrt{2x}\) + 5
Answer:

Question 6.
n(x) = 5\(\sqrt{x}\)
Answer:

Graph the function. Compare the graph to the graph of f(x) = \(\sqrt{x}\) .
Question 7.
h(x) = \(\sqrt{\frac{1}{4} x}\)
Answer:

Question 8.
g(x) = \(\sqrt{x}\) – 6
Answer:

Question 9.
m(x) = -3\(\sqrt{x}\)
Answer:

Question 10.
Let g(x) = \(\frac{1}{2} \sqrt{x+4}+1\). Describe the transformations from the graph of f(x) = \(\sqrt{x}\) to the graph of g. Then graph g.
Answer:

Question 11.
In Example 5, compare the velocities by finding and interpreting their average rates of change over the interval d = 30 to d = 40.
Answer:

Question 12.
WHAT IF?
At what depth does the velocity of the tsunami exceed 100 meters per second?
Answer:

Graphing Square Root Functions 10.1 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
A ________ is a function that contains a radical expression with the independent variable in the radicand.
Answer:
A radical is a function that contains a radical expression with the independent variable in the radicand.

Question 2.
VOCABULARY
Is y = 2x\(\sqrt{5}\) a square root function? Explain.
Answer:
No; y = 2x\(\sqrt{5}\) is a linear function.

Question 3.
WRITING
How do you describe the domain of a square root function?
Answer:
The domain of a square root function is the x-values for which the radicand is greater than or equal to 0.

Question 4.
REASONING
Is the graph of g(x) = 1.25\(\sqrt{x}\) a vertical stretch or a vertical shrink of the graph of f(x) = \(\sqrt{x}\)? Explain.
Answer:
g(x) = 1.25\(\sqrt{x}\)
Graph of g(x) is vertical stretch of graph f(x) = \(\sqrt{x}\)
Since corresponding to each x value of g(x) is more than f(x)

Monitoring Progress and Modeling with Mathematics

In Exercises 5–14, describe the domain of the function.
Question 5.
y = 8\(\sqrt{x}\)
Answer:
x ≥ 0
The domain is the set of real numbers greater than or equal to 0.

Question 6.
y = \(\sqrt{4x}\)
Answer:
4x ≥ 0 that implies x ≥ 0
Therefore the domain is [0, ∞)

Question 7.
y = 4 + \(\sqrt{-x}\)
Answer:

Question 8.
y = \(\sqrt{-\frac{1}{2^{x}}}\) + 1
Answer:
y = \(\sqrt{-\frac{1}{2^{x}}}\) + 1 to be defined
–\(\frac{1}{3}\) x ≥ 0
x  ≤ 0
Therefore the domain is (-∞, 0]

Question 9.
h(x) = \(\sqrt{x-4}\)
Answer:

Question 10.
p(x) = \(\sqrt{x+7}\)
Answer:
The square root will be defined only when the function inside the square root must be non negative.
p(x) = \(\sqrt{x+7}\) to be defined
x + 7 ≥ 0
x ≥ -7
Therefore the domain is [-7, ∞]

Question 11.
f(x) = \(\sqrt{-x+8}\)
Answer:

Question 12.
g(x) = \(\sqrt{-x-1}\)
Answer:
g(x) = \(\sqrt{-x-1}\)
-x – 1 ≥ 0
-(x + 1)≥ 0
x + 1 ≤ 0
x ≤ -1
Therefore the domain is (-∞, -1]

Question 13.
m(x) = 2\(\sqrt{x+4}\)
Answer:

Question 14.
n(x) = \(\frac{1}{2} \sqrt{-x}-2\)
Answer:
n(x) = \(\frac{1}{2} \sqrt{-x}-2\)
-x – 2 ≥ 0
-(x + 2)≥ 0
x + 2 ≤ 0
x ≤ -2
Therefore the domain is (-∞, -2]

In Exercises 15–18, match the function with its graph. Describe the range.
Question 15.
y = \(\sqrt{x-3}\)
Answer:

Question 16.
y = 3\(\sqrt{x}\)
Answer:
y = 3\(\sqrt{x}\) ≥ 0∀ x ∈ R
That is the range is [0, ∞)
Thus the correct answer is option C.

Question 17.
y = \(\sqrt{x}\) – 3
Answer:

Question 18.
y = \(\sqrt{-x+3}\)

Answer:
y = \(\sqrt{-x+3}\)
≥ 0∀ x ∈ R
That is the range is [0, ∞)
Thus option A is eliminated
x-intercept of y is 3.
Also as x will decrease y will increase.
Thus the correct answer is option B.

In Exercises 19–26, graph the function. Describe the range.
Question 19.
y = \(\sqrt{3x}\)
Answer:

Question 20.
y = 4\(\sqrt{-x}\)
Answer:

Question 21.
y = \(\sqrt{x}\) + 5
Answer:

Question 22.
y = -2 + \(\sqrt{x}\)
Answer:

The range is [-2, ∞)

Question 23.
f(x) = – \(\sqrt{x-3}\)
Answer:

Question 24.
g(x) = \(\sqrt{x+4}\)
Answer:

The range is [0, -∞)

Question 25.
h(x) = \(\sqrt{x+2}\) – 2
Answer:

Question 26.
f(x) = –\(\sqrt{x-1}\) + 3
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 1 ≥ 0
x ≥ 1
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (1, 3)

In Exercises 27–34, graph the function. Compare the graph to the graph of f (x) = \(\sqrt{x}\).
Question 27.
g(x) = \(\frac{1}{4} \sqrt{x}\)
Answer:

Question 28.
r(x) = \(\sqrt{2x}\)
Answer:
x = 0 g(0) = 0
x = 1 g(1) = 1.41
x = 2 g(2) = 2
x = 3 g(3) = 2.45
x = 4 g(4) = 2.83

Question 29.
h(x) = \(\sqrt{x+3}\)
Answer:

Question 30.
q(x) = \(\sqrt{x}\) + 8
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x ≥ 0
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (0, 8)

The graph of q(x) is the graph of f(x) that is:
Vertically shifted 8 units upward

Question 31.
p(x) = \(\sqrt{-\frac{1}{3} x}\)x
Answer:

Question 32.
g(x) = -5\(\sqrt{x}\)
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x ≥ 0
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (0, 0)

The graph of q(x) is the graph of f(x) that is:
Reflected in the x-axis
Vertically stretched by a factor of 5.

Question 33.
m(x) = –\(\sqrt{x}\) – 6
Answer:

Question 34.
n(x) = –\(\sqrt{x}\) – 4
Answer:
Step 1: Use the domain to make the table of values
The radicand must be greater than or equal to 0
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (4, 0)

The graph of m(x) is the graph of f(x)
-Reflected in the x-axis
– Horizontally shifted 4 units to the right

Question 35.
ERROR ANALYSIS
Describe and correct the error in graphing the function y = \(\sqrt{x}\) + 1 .

Answer:

Question 36.
ERROR ANALYSIS
Describe and correct the error in comparing the graph of g(x) = \(-\frac{1}{4} \sqrt{x}\) to the graph of f (x) = \(\sqrt{x}\).

Answer:
The graph of g is a reflection in the x-axis and a vertical shrink by a factor of 1/4 not a horizontal stretch.

In Exercises 37–44, describe the transformations from the graph of f (x) = \(\sqrt{x}\) to the graph of h. Then graph h.
Question 37.
h(x) = 4\(\sqrt{x+2}\) – 1
Answer:

Question 38.
h(x) = \(\frac{1}{2} \sqrt{x-6}\)+ 3
Answer:
The graph of h(x) is the graph of f(x)
-Vertically shrinked by a factor of 1/2 →\(\frac{1}{2} \sqrt{x}\)
– Horizontally shifted 6 units to the right \(\frac{1}{2} \sqrt{x-6}\)
– Vertically shifted 3 units upward → \(\frac{1}{2} \sqrt{x-6}\)+ 3
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 6 ≥ 0
x ≥ 6
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (6, -2)

Question 39.
h(x) = 2\(\sqrt{-x}\) – 6
Answer:

Question 40.
h(x) = –\(\sqrt{x-3}\) – 2
Answer:
The graph of h(x) is the graph of f(x)
– Reflected in the x-axis – √x
– Horizontally shifted 3 units to the right → –\(\sqrt{x-3}\)
– Vertically shifted 2 units downward→ –\(\sqrt{x-3}\) – 2
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 3 ≥ 0
x ≥ 3
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (3, -2)

Question 41.
h(x) = \(\frac{1}{3} \sqrt{x+3}\) + 3
Answer:

Question 42.
h(x) = 2\(\sqrt{x-1}\) + 4
Answer:
The graph of h(x) is the graph of f(x)
-Vertically shrinked by a factor of 2 →2\(\sqrt{x}\)
– Horizontally shifted 1 units to the right 2\(\sqrt{x-1}\)
– Vertically shifted 4 units upward → 2\(\sqrt{x-1}\) + 4
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 1 ≥ 0
x ≥ 1
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (1, 4)

Question 43.
h(x) = -2\(\sqrt{x-1}\) + 5
Answer:

Question 44.
h(x) = -5\(\sqrt{x+2}\) – 1
Answer:
The graph of h(x) is the graph of f(x)
-Reflection in the x-axis → –\(\sqrt{x}\)
-Vertically shrinked by a factor of 5 →-5\(\sqrt{x}\)
– Horizontally shifted 2 units to the left -5\(\sqrt{x+2}\)
– Vertically shifted 1 units downward→ -5\(\sqrt{x+2}\) – 1
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x + 2 ≥ 0
x ≤ -2
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (-2, -1)

Question 45.
COMPARING FUNCTIONS
The model S(d ) = \(\sqrt{30df}\) represents the speed S (in miles per hour) of a van before it skids to a stop, where f is the drag factor of the road surface and d is the length (in feet) of the skid marks. The drag factor of Road Surface A is 0.75. The graph shows the speed of the van on Road Surface B. Compare the speeds by finding and interpreting their average rates of change over the interval d = 0 to d = 15.

Answer:

Question 46.
COMPARING FUNCTIONS
The velocity v (in meters per second) of an object in motion is given by v(E ) = \(\sqrt{\frac{2 E}{m}}\), where E is the kinetic energy of the object (in joules) and m is the mass of the object (in kilograms). The mass of Object A is 4 kilograms. The graph shows the velocity of Object B. Compare the velocities of the objects by finding and interpreting the average rates of change over the interval E = 0 to E = 6.

Answer:
To calculate the average rates of change, use points whose E-coordinates are 0 and 6.
Object A: Evaluate S when E = 0 and E = 6
ν(0) = √2(0)/4 = 0
ν(6) = √2(6)/4 = 3
The average rate of change is
s(6) – s(0)/6 – 0 = 0.28
B.
Use the graph to estimate.
(0, 0), (6, 2)
s(6) – s(0)/6 – 0 = (2 – 0)/(6 – 0) = 0.33
From 0 to 6 joules, the velocity of object A increases at an average rate of 0.28 m/s
and the velocity of object B increases at an average rate of 0.33 m/s

Question 47.
OPEN-ENDED
Consider the graph of y = \(\sqrt{x}\).
a. Write a function that is a vertical translation of the graph of y = \(\sqrt{x}\).
b. Write a function that is a reflection of the graph of y = \(\sqrt{x}\).
Answer:

Question 48.
REASONING
Can the domain of a square root function include negative numbers? Can the range include negative numbers? Explain your reasoning.
Answer:
Can the domain of a square root function include negative numbers?
Yes, if the domain includes x < 0
An example would be: y = √-x
where the domain is x ≤ 0
Can the range include negative numbers?
Yes, if the range includes y < 0
An example would be: y = -√x
where the range is y ≤ 0

Question 49.
PROBLEM SOLVING
The nozzle pressure of a fire hose allows firefighters to control the amount of water they spray on a fire. The flow rate f(in gallons per minute) can be modeled by the function f = 12\(\sqrt{p}\), where p is the nozzle pressure (in pounds per square inch).

a. Use a graphing calculator to graph the function. At what pressure does the flow rate exceed 300 gallons per minute?
b. What happens to the average rate of change of the flow rate as the pressure increases?
Answer:

Question 50
PROBLEM SOLVING
The speed s (in meters per second) of a long jumper before jumping can be modeled by the function s = 10.9\(\sqrt{h}\), where h is the maximum height (in meters from the ground) of the jumper.

a. Use a graphing calculator to graph the function. A jumper is running 9.2 meters per second. Estimate the maximum height of the jumper.

Answer:
A jumper is running 9.2 meters per second.

The maximum height of the jumper is approximately 0.71 meters from the ground.

b. Suppose the runway and pit are raised on a platform slightly higher than the ground. How would the graph of the function be transformed?
Answer: The graph of the function will be vertically translated.

Question 51.
MATHEMATICAL CONNECTIONS
The radius r of a circle is given by r = \(\sqrt{\frac{A}{\pi}}\), where A is the area of the circle.

a. Describe the domain of the function. Use a graphing calculator to graph the function.
b. Use the trace feature to approximate the area of a circle with a radius of 5.4 inches.
Answer:

Question 52.
REASONING
Consider the function f(x) = 8a\(\sqrt{x}\).
a. For what value of a will the graph of f be identical to the graph of the parent square root function?
Answer:
The parent function is y = \(\sqrt{x}\)
8a = 1
a = 1/8

b. For what values of a will the graph of f be a vertical stretch of the graph of the parent square root function?
Answer:
A vertical stretch y = k \(\sqrt{x}\)
|k| > 1
8a > 1
a > 1/8

c. For what values of a will the graph of f be a vertical shrink and a reflection of the graph of the parent square root function?
Answer:
A reflection will be y = -k \(\sqrt{x}\) a vertocal shrink y = -k \(\sqrt{x}\)
will occur if 0 < k < 1
0 < -8a < 1
-1/8 < a < 0

Question 53.
REASONING
The graph represents the function f(x) = \(\sqrt{x}\).

a. What is the minimum value of the function?
b. Does the function have a maximum value? Explain.
c. Write a square root function that has a maximum value. Does the function have a minimum value? Explain.
d. Write a square root function that has a minimum value of -4.
Answer:

Question 54.
HOW DO YOU SEE IT?
Match each function with its graph. Explain your reasoning.

Answer:
A. A function is a vertical translation of the parent function 2 units upward so the corresponding graph is graph B
B. The function is a vertical translation of 4 units downwards of the function f(x)so the corresponding graph is graph D
C. The function is a reflection about the y-axis of the function f(x) so the corresponding graph is graph C.
D. The function is a horizontal shrink of the function f(x) so the corresponding graph is graph A.

Question 55.
REASONING
Without graphing, determine which function’s graph rises more steeply, f(x) = 5\(\sqrt{x}\) or g(x) = \(\sqrt{5x}\). Explain your reasoning.
Answer:

Question 56.
THOUGHT PROVOKING
Use a graphical approach to find the solutions of x – 1 = \(\sqrt{5x-9}\). Show your work. Verify your solutions algebraically.
Answer:
Graph both sides as the related functions:
y1 = x – 1
y2 = \(\sqrt{5x-9}\)
y1 has a slope of 1 and y-intercept of -1
y2 has a domain of x ≥ 1.8 so use point plotting

Their point of intersection will be the solution
(2, 1) and (5, 4)
Squaring on both sides
x² – 2x + 1 = 5x – 9
x² – 7x + 1 = – 9
x² – 7x + 10 = 0
(x – 5) (x – 2) = 0
x = 2, 5
Solve for corresponding y-coordinates
x = 2 → y = 2 – 1 = 1
x = 5 → y = 5 – 1 = 4
The solutions are
(2, 1) and (5, 4)

Question 57.
OPEN-ENDED
Write a radical function that has a domain of all real numbers greater than or equal to -5 and a range of all real numbers less than or equal to 3.
Answer:

Maintaining Mathematical Proficiency

Evaluate the expression.(Section 6.2)
Question 58.
\(\sqrt [ 3]{ 343 }\)
Answer:
\(\sqrt [ 3]{ 343 }\) = 7

Question 59.
\(\sqrt [ 3]{ -64 }\)
Answer:

Question 60.
\(-\sqrt[3]{-\frac{1}{27}}\)
Answer:
\(-\sqrt[3]{-\frac{1}{27}}\)
= \(-\sqrt[3]{-\frac{1}{3³}}\)
= -1/3

Factor the polynomial.(Section 7.5)
Question 61.
x² + 7x + 6
Answer:

Question 62.
d² – 11d + 28
Answer:
d² – 11d + 28
d² – 7d – 4d + 28
(d – 7) (d – 4) are the factors
d = 7, 4

Question 63.
y² – 3y – 40
Answer:

Lesson 10.2 Graphing Cube Roots Functions

Essential Question What are some of the characteristics of the graph of a cube root function?

EXPLORATION 1

Graphing Cube Root Functions
Work with a partner.

•  Make a table of values for each function. Use positive and negative values of x.
• Use the table to sketch the graph of each function.• Describe the domain of each function.
• Describe the range of each function.

Answer:

EXPLORATION 2

Writing Cube Root Functions
Work with a partner. Write a cube root function, y = f(x), that has the given values. Then use the function to complete the table.

Answer:

Communicate Your Answer

Question 3.
What are some of the characteristics of the graph of a cube root function?
Answer:

Question 4.
Graph each function. Then compare the graph to the graph of f (x) = \(\sqrt [ 3]{ x }\) .
a. g(x) = \(\sqrt [ 3]{ x-1 }\)
b. g(x) = \(\sqrt [ 3]{ x-1 }\)
c. g(x) = 2 \(\sqrt [ 3]{ x }\)
d. g(x) = -2\(\sqrt [ 3]{ x }\)
Answer:

Monitoring Progress

Graph the function. Compare the graph to the graph of f(x) = \(\sqrt [ 3]{ x }\).
Question 1.
h(x) = \(\sqrt [ 3]{ x }\) + 3
Answer:

Question 2.
m(x) = \(\sqrt [ 3]{ x }\) – 5
Answer:

Question 3.
g(x) = 4\(\sqrt [ 3]{ x }\)
Answer:

Graph the function. Compare the graph to the graph of f(x) = \(\sqrt [ 3]{ x }\).
Question 4.
g(x) =\(\sqrt [3]{ 0.5x+5 }\) + 5
Answer:

Question 5.
h(x) = 4\(\sqrt [3]{ x }\) – 1
Answer:

Question 6.
n(x) = \(\sqrt [ 3]{ 4-x }\)
Answer:

Question 7.
Let g(x) = \(-\frac{1}{2} \sqrt[3]{x+2}\) – 4. Describe the transformations from the graph of f (x) = \(\sqrt [ 3]{ x }\) to the graph of g. Then graph g.
Answer:

Question 8.
In Example 4, compare the average rates of change over the interval x = 2 to x = 10.
Answer:

Question 9.
WHAT IF?
Estimate the age of an elephant whose shoulder height is 175 centimeters.
Answer:

Graphing Cube Roots Functions 10.2 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The __________ of the radical in a cube root function is 3.
Answer:
The index of the radical in a cube root function is 3.

Question 2.
WRITING
Describe the domain and range of the function f(x) = \(\sqrt [3]{ x-4 }\) + 1.
Answer:
Regardless of the transformation, the domain and range of a cubic function are all real numbers.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, match the function with its graph.
Question 3.
y = 3\(\sqrt [3]{ x+2 }\)
Answer:

Question 4.
y = 3\(\sqrt [3]{ x-2 }\)
Answer:
The given function is the parent function
y = \(\sqrt [3]{ x }\) that is horizontally shifted 2 units to the right.
Thus the answer is graph A.

Question 5.
y = 3\(\sqrt [3]{ x+2 }\)
Answer:

Question 6.
y = \(\sqrt [3]{ x }\) – 2
Answer:
The given function is the parent function
y = \(\sqrt [3]{ x }\) that is vertically shifted 2 units to the downward.
Thus the answer is graph B

In Exercises 7–12, graph the function. Compare the graph to the graph of f(x) = \(\sqrt [ 3]{ x }\).
Question 7.
h(x) = \(\sqrt [3]{ x-4 }\)
Answer:

Question 8.
g(x) = \(\sqrt [3]{ x+1 }\)
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of g(x) is the graph of f(x) that is horizontally shifted 1 unit to the left.

Question 9.
m(x) = \(\sqrt [3]{ x+5 }\)
Answer:

Question 10.
q(x) = \(\sqrt [3]{ x }\) – 3
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of g(x) is the graph of f(x) that is vertically shifted 3 units downwards.

Question 11.
p(x) = 6\(\sqrt [3]{ x }\)
Answer:

Question 12.
j(x) = \(\sqrt[3]{\frac{1}{2} x}\)
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of g(x) is the graph of f(x) that is horizontally stretched by a factor of 2.

In Exercises 13–16, compare the graphs. Find the value of h, k, or a.
Question 13.

Answer:

Question 14.

Answer:
The graph of g(x) is the graph of f(x) is
– Vertically shifted 4 units upwards
g(x) = \(\sqrt [3]{ x }\) + 4
Therefore k = 4

Question 15.

Answer:

Question 16.

Answer:
The graph of p(x) is the graph of f(x) is vertically stretched by a factor of 3.
p(x) = 3 \(\sqrt [3]{ x }\)
Therefore a = 3

In Exercises 17–26, graph the function. Compare the graph to the graph of f(x) = \(\sqrt [3]{ x }\).
Question 17.
r(x) = – \(\sqrt [3]{ x-2 }\)
Answer:

Question 18.
h(x) = – 3\(\sqrt [3]{ x+3 }\)
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of h(x) is the graph of f(x)
– Reflected in the x-axis
– Vertically shifted 3 units upward

Question 19.
k(x) = 5\(\sqrt [3]{ x+1 }\)
Answer:

Question 20.
j(x) = 0.5\(\sqrt [3]{ x-4 }\)
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of j(x) is the graph of f(x) is
– Vertically shrinked by a factor of 0.5
– Horizontally shifted 4 units to the right.

Question 21.
g(x) = 4\(\sqrt [3]{ x }\) – 3
Answer:

Question 22.
m(x) = 3\(\sqrt [3]{ x }\) + 7
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of m(x) is the graph of f(x) is
– Vertically stretched by a factor of 3
– Vertically shifted 7 units upward

Question 23.
n(x) = \(\sqrt [3]{ -8x }\) – 1
Answer:

Question 24.
v(x) =\(\sqrt [3]{ 5x }\) + 2
Answer:
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of v(x) is the graph of f(x) is
– Horizontally shrinked by a factor of 1/5
– Vertically shifted 2 units upward

Question 25.
q(x) = \(\sqrt[3]{2(x+3)}\)
Answer:

Question 26.
p(x) = \(\sqrt[3]{3(1-x)}\)
Answer:
p(x) = \(\sqrt[3]{3(1-x)}\)
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

The graph of p(x) is the graph of f(x) is
– Horizontally shrinked by a factor of 1/3
– Reflected in the y-axis
– Horizontally shifted 1 unit to the right

In Exercises 27–32, describe the transformations from the graph of f(x) = \(\sqrt [3]{ x }\) to the graph of the given function. Then graph the given function.
Question 27.
g(x) = \(\sqrt [3]{ x-4 }\) + 2
Answer:

Question 28.
n(x) = \(\sqrt [3]{ x+1 }\) – 3
Answer:
The graph of k(x) is the graph of f(x) is
– Horizontally shifted 1 units to the left → \(\sqrt [3]{ x+1 }\)
– Vertically shifted 3 units downward → \(\sqrt [3]{ x+1 }\) – 3
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

Question 29.
j(x) = -5\(\sqrt [3]{ x+3 }\) + 2
Answer:

Question 30.
k(x) = 6\(\sqrt [3]{ x-9 }\) – 5
Answer:
The graph of k(x) is the graph of f(x) is
– Vertically stretched by a factor of 6 → 6\(\sqrt [3]{ x }\)
– Horizontally shifted 9 units to the right → 6\(\sqrt [3]{ x-9 }\)
– Vertically shifted 3 units downward → 6\(\sqrt [3]{ x-9 }\) – 5
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

Question 31.
v(x) = \(\frac{1}{3} \sqrt[3]{x-1}\) + 7
Answer:

Question 32.
h(x) = -\frac{3}{2} \sqrt[3]{x+4} – 3
Answer:
The graph of v(x) is the graph of f(x) is
– Reflected in the x-axis → \(\sqrt [3]{ x }\)
– Vertically stretched by a factor of 3/2 → -3/2\(\sqrt [3]{ x }\)
– Horizontally shifted 4 units to the left → -\frac{3}{2} \sqrt[3]{x+4}
– Vertically shifted 3 units downward → -\frac{3}{2} \sqrt[3]{x+4} – 3
1. Make a table of values
2. Plot the ordered pairs
3. Draw a smooth curve through the points.

Question 33.
ERROR ANALYSIS
Describe and correct the error in graphing the function f(x) = \(\sqrt [3]{ x-3 }\).

Answer:

Question 34.
ERROR ANALYSIS
Describe and correct the error in graphing the function h(x) = \(\sqrt [3]{ x }\) + 1.

Answer:
The graph that is translated is the square root function when it should be the cubic function since the index is 3.

Question 35.
COMPARING FUNCTIONS
The graph of cube root function q is shown. Compare the average rate of change of q to the average rate of change of f(x) = 3\(\sqrt [3]{ x }\) over the interval x = 0 to x = 6.

Answer:

Question 36.
COMPARING FUNCTIONS
The graphs of two cube root functions are shown. Compare the average rates of change of the two functions over the interval x = -2 to x = 2.

Answer:
For g(x): Use the graph to estimate
Use (-2, -1) and (2, 1)
The average rate of change is
g(2) – g(-2)/2 – (-2) = 0.5
For h(x): Use the graph to estimate
Use (-2, -3) and (2, 3)
h(3) – h(-3)/2 – (-2) = 1.5
The average rate of change of h is 1.5/0.5 = 3.
Thus 3 time greater than the average rate of change of g over the interval x = -2 to x = 2

Question 37.
MODELING WITH MATHEMATICS
For a drag race car that weighs 1600 kilograms, the velocity v (in kilometers per hour) reached by the end of a drag race can be modeled by the function v = 23.8\(\sqrt [3]{ p }\), where p is the car’s power (in horsepower). Use a graphing calculator to graph the function. Estimate the power of a 1600-kilogram car that reaches a velocity of 220 kilometers per hour.
Answer:

Question 38.
MODELING WITH MATHEMATICS
The radius r of a sphere is given by the function r = \(\sqrt[3]{\frac{3}{4 \pi}} V\), where V is the volume of the sphere. Use a graphing calculator to graph the function. Estimate the volume of a spherical beach ball with a radius of 13 inches.
Answer:
Using a graphing utility, trace where the  y-value is 13

Therefore the volume of a sphere with a radius of 13 inches is approximately 9202.77 cu. in

Question 39.
MAKING AN ARGUMENT
Your friend says that all cube root functions are odd functions. Is your friend correct? Explain.
Answer:

Question 40.
HOW DO YOU SEE IT?
The graph represents the cube root function f(x) = \(\sqrt [3]{ x }\).

a. On what interval is f negative? positive?
Answer:
Negative on x ≤ 0
Positive on x ≥ 0
b. On what interval, if any, is f decreasing? increasing?
Answer:
Decreasing: None
Increasing: -∞ < x < ∞
c. Does f have a maximum or minimum value? Explain.
Answer:
The graph has neither maximum nor minimum since it is always increasing based on b.
d. Find the average rate of change of f over the interval x = -1 to x = 1.
Answer:
The average rate of change is
f(1) – f(-1)/1 – (-1)
= 1 – (-1)/1 – (-1) = 1

Question 41.
PROBLEM SOLVING
Write a cube root function that passes through the point (3, 4) and has an average rate of change of -1 over the interval x = -5 to x = 2.
Answer:

Question 42.
THOUGHT PROVOKING
Write the cube root function represented by the graph. Use a graphing calculator to check your answer.

Answer:
Imagine that the coordinate axis is on the center of the cubic function.
This is a vertical stretch by a factor of 2
y = 2 \(\sqrt [3]{ x }\)
Also, from the original graph, it is a translation of 1 unit to the right so tthe function will be
y = 2 \(\sqrt [3]{ x-1 }\)

Maintaining Mathematical Proficiency

Factor the polynomial.(Section 7.6)
Question 43.
3x² + 12x – 36
Answer:

Question 44.
2x² – 11x + 9
Answer:
2x² – 11x + 9
(2x – 9)(x – 1) factor thr polynomial

Question 45.
4x² + 7x – 15
Answer:

Solve the equation using square roots.(Section 9.3)
Question 46.
x² – 36 = 0
Answer:
x² – 36 = 0
x² = 36
squaring on both sides
x = 6 or x = -6

Question 47.
5x² + 20 = 0
Answer:

Question 48.
(x + 4)² – 81
Answer:
(x + 4)² – 81
(x + 4)²  = 81
squaring on both sides
x + 4 = √81
x + 4 = 9 or x + 4 = -9
x = 9 – 4 or x = -9 – 4
x = 5 or x = -13

Question 49.
25(x – 2)² = 9
Answer:

Radical Functions and Equations Study Skills: Making Note Cards

10.1–10.2 What Did YouLearn?

Core Vocabulary
square root function, p. 544
radical function, p. 545
cube root function, p. 552

Core Concepts
Lesson 10.1
Square Root Functions, p. 544
Transformations of Square Root Functions, p. 545
Comparing Square Root Functions Using Average Rates of Change, p. 546

Lesson 10.2
Cube Root Functions, p. 552
Comparing Cube Root Functions
Using Average Rates of Change, p. 554

Mathematical Practices

Question 1.
In Exercise 45 on page 549, what information are you given? What relationships are present? What is your goal?
Answer:

Question 2.
What units of measure did you use in your answer to Exercise 38 on page 556? Explain your reasoning.
Answer:

Study Skills: Making Note Cards

Invest in three different colors of note cards. Use one color for each of the following: vocabulary words, rules, and calculator keystrokes.

• Using the first color of note cards, write a vocabulary word on one side of a card. On the other side, write the definition and an example. If possible, put the definition in your own words.
• Using the second color of note cards, write a rule on one side of a card. On the other side, write an explanation and an example.
• Using the third color of note cards, write a calculation on one side of a card. On the other side, write the keystrokes required to perform the calculation.
  Use the note cards as references while completing your homework. Quiz yourself once a day.
  

Radical Functions and Equations 10.1–10.2 Quiz

Describe the domain of the function.(Lesson 10.1)
Question 1.
y = \(\sqrt{x-3}\)
Answer:
The domain is determined by the radicand which must be greater than or equal to 0
x – 3 ≥ 0
x ≥ 3

Question 2.
f(x) = 15\(\sqrt{x}\)
Answer:
The domain is determined by the radicand which must be greater than or equal to 0
x ≥ 0

Question 3.
y = \(\sqrt{3-x}\)
Answer:
The domain is determined by the radicand which must be greater than or equal to 0
3 – x ≥ 0
– x ≥ -3
x ≤ 3

Graph the function. Describe the range. Compare the graph to the graph of f(x) = \(\sqrt{x}\). (Lesson 10.1)
Question 4.
g(x) = \(\sqrt{x}\) + 5
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x ≥ 0
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (0, 5)

From the graph, we can see that the range is
y ≥ 5
The graph of g(x) is the graph of f(x) is vertically shifted 5 units upward

Question 5.
n(x) = \(\sqrt{x-4}\)
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 4 ≥ 0
x ≥ 4
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (4, 0)

From the graph, we can see that the range is
y ≥ 0
The graph of n(x) is the graph of f(x) is horizontally shifted 4 units to the right.

Question 6.
r(x) = –\(\sqrt{x-1}\) + 1
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 2 ≥ 0
x ≥ 2
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (2, 1)

The graph of r(x) is the graph of f(x) is
– Reflected in the x-axis
– Horizontally shifted 2 units to the right
– Vertically shifted 1 unit upward

Graph the function. Compare the graph to the graph of f(x) = \(\sqrt [3]{ x }\). (Lesson 10.2)
Question 7.
b(x) = \(\sqrt [3]{ x+2 }\)
Answer:
Step 1: Make a table of values
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points

The graph of b(x) is the graph of f(x) is horizontally shifted 2 units to the left.

Question 8.
h(x) = -3\(\sqrt [3]{ x-6 }\)
Answer:
Step 1: Make a table of values
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points

The graph of h(x) is the graph of f(x) is
– Reflected in the x-axis
– Vertically stretched by a factor of 3
– Vertically shifted 6 units downward

Question 9.
q(x) = \(\sqrt [3]{ -4-x }\)
Answer:
Step 1: Make a table of values
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points

The graph of q(x) is the graph of f(x) is
– Reflected in the y-axis
– Horizontally shifted 4 units to the left

Compare the graphs. Find the value of h, k, or a. (Lesson 10.1 and Lesson 10.2)
Question 10.

Answer:
The graph of v(x) is the graph of f(x) is
– Vertically shifted 3 units upwards
v(x) = \(\sqrt [3]{ x }\) + 3
Thus k = 3

Question 11.

Answer:
The graph of g(x) is the graph of f(x) is
– Reflected in the x-axis
– Vertically stretched by a factor of 2
v(x) = -2√x
Thus a = -2

Question 12.

Answer:
The graph of p(x) is the graph of f(x) is
– Horizontally shifted 1 unit to the left
q(x) = \(\sqrt [3]{ x + 1 }\)
Thus h = -1

Describe the transformations from the graph of f to the graph of h. Then graph h. (Section 10.1 and Section 10.2)
Question 13.
f(x) = \(\sqrt{x}\); h(x) = -3 \(\sqrt{x+2}\) + 6
Answer:
The graph of h(x) is the graph of f(x) is
– Reflected in the x-axis → \(\sqrt{x}\)
– Vertically stretched by a factor of 3 → -3\(\sqrt{x}\)
– Horizontally shifted 2 units to the left → -3 \(\sqrt{x+2}\)
– Vertically shifted 6 units upward → -3 \(\sqrt{x+2}\) + 6
Step 1: Use the domain to make a table of values
x + 2 ≥ 0
x ≥ -2
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (-2, 6)

Question 14.
f(x) = \(\sqrt[3]{x}\); h(x) = \(\frac{1}{2} \sqrt[3]{x}-3\)
Answer:
The graph of h(x) is the graph of f(x) is
– Vertically shrinked by a factor of \(\frac{1}{2}\) → \(\frac{1}{2}\) \(\sqrt[3]{x}\)
– Vertically shifted 3 units downward → \(\frac{1}{2}\) \(\sqrt[3]{x}\) – 3
Step 1: Make a table of values.
Step 2: Plot the ordered pairs.
Step 3: Draw a smooth curve through the points.

Question 15.
The time t (in seconds) it takes a dropped object to fall h feet is given by t = \(\frac{1}{4} \sqrt{h}\). (Section 10.1)

a. Use a graphing calculator to graph the function. Describe the domain and range.
b. It takes about 7.4 seconds for a stone dropped from the New River Gorge Bridge in West Virginia to reach the water below. About how high is the bridge above the New River?
Answer:
Using a graphing utility, the domain is h ≥ 0 and the range is t ≥ 0.
Graph the function as y = \(\frac{1}{4} \sqrt{h}\) and the line y = 7.4 corresponding to a time of 7.4 seconds.
Use the intersect feature to obtain (876.16, 7.4)
Hence the bridge is about 876 feet high.

Question 16.
The radius r of a sphere is given by the function r = \(\sqrt[3]{\frac{3}{4 \pi} V}\), where V is the volume of the sphere. Spaceship Earth is a spherical structure at Walt Disney World that has an inner radius of about 25 meters. Use a graphing calculator to graph the function. Estimate the volume of Spaceship Earth. (Section 10.2)
Answer:

Thus the volume of the spaceship Earth with an inner radius of 25 meters is approximately 65449.85 cu. meters

Question 17.
The graph of square root function g is shown. Compare the average rate of change of g to the average rate of change of h(x) = \(\sqrt[3]{\frac{3}{2} x}\)x over the interval x = 0 to x = 3.

Answer:
For g(x): Use the graph to estimate.
Use (0, 0) and (3, 3)
The average rate of change is
g(3) – g(0)/ 3 – 0
= (3 – 0)/(3 -0) = 1
For h(x) = \(\sqrt[3]{\frac{3}{2} x}\)x:
Evaluate f when x = 0 and x = 3
f(0) = \(\sqrt[3]{\frac{3}{2} (0)}\) = 0
f(3) = \(\sqrt[3]{\frac{3}{2} (3)}\) ≈ 1.65
The average rate of change is
f(3) – f(0)/ 3 – 0
= (1.65 – 0)/(3 -0) ≈ 0.55
The average rate of change of g is \(\frac{1}{0.55}\) ≈ 1.8 times greater than the average rate of change of h over the interval x = 0 to x = 3

Lesson 10.3 Solving Radical Equations

Essential Question How can you solve an equation that contains square roots?

EXPLORATION 1

Analyzing a Free-Falling Object
Work with a partner. The table shows the time t (in seconds) that it takes a free-falling object (with no air resistance) to fall d feet.

a. Use the data in the table to sketch the graph of t as a function of d. Use the coordinate plane below.
b. Use your graph to estimate the time it takes the object to fall 240 feet.
c. The relationship between d and t is given by the function t = \(\sqrt{\frac{d}{16}}\).
Use this function to check your estimate in part (b).
d. It takes 5 seconds for the object to hit the ground. How far did it fall? Explain your reasoning.

EXPLORATION 2

Solving a Square Root Equation
Work with a partner. The speed s (in feet per second) of the free-falling object in Exploration 1 is given by the functions
s = \(\sqrt{64d}\).
Find the distance the object has fallen when it reaches each speed.
a. s = 8 ft/sec
b. s = 16 ft/sec
c. s = 24 ft/sec

Communicate Your Answer

Question 3.
How can you solve an equation that contains square roots?
Answer:
If an equation has a square root equal to a negative number, that equation will have no solution. To isolate the radical, subtract 1 from both sides.

Question 4.
Use your answer to Question 3 to solve each equation.
a. 5 = \(\sqrt{x}\) + 20
b. 4 = \(\sqrt{x-18}\)
c. \(\sqrt{x}\) + 2 = 3
d. -3 = -2\(\sqrt{x}\)
Answer:

Monitoring Progress

Solve the equation. Check your solution.
Question 1.
\(\sqrt{x}\) = 6
Answer:
Given,
\(\sqrt{x}\) = 6
Squaring on both sides
\(\sqrt{x}\)² = 6²
x = 36
Check:
\(\sqrt{36}\) = 6
Thus x = 6 is the solution

Question 2.
\(\sqrt{x}\) – 7 = 3
Answer:
Given,
\(\sqrt{x}\) – 7 = 3
\(\sqrt{x}\) = 3 + 7
\(\sqrt{x}\) = 10
Squaring on both sides
x = 10²
x = 100
Check:
\(\sqrt{100}\) – 7 = 3
10 – 7 = 3
Thus x = 100 is the solution

Question 3.
\(\sqrt{y}\) + 15 = 22
Answer:
Given,
\(\sqrt{y}\) + 15 = 22
\(\sqrt{y}\) = 22 – 15
\(\sqrt{y}\) = 7
Squaring on both sides
\(\sqrt{y}\)² = 7² = 49
Check:
\(\sqrt{49}\) + 15 = 22
7 + 15 = 22
Thus y = 49 is the solution

Question 4.
1 – \(\sqrt{c}\) = -2
Answer:
Given,
1 – \(\sqrt{c}\) = -2
1 + 2 = \(\sqrt{c}\)
\(\sqrt{c}\) = 3
Squaring on both sides
c = 9
Check:
1 – \(\sqrt{c}\) = -2
1 – \(\sqrt{9}\) = -2
1 – 3 = -2
Thus c = 9 is the solution

Solve the equation. Check your solution.
Question 5.
\(\sqrt{x+4}\) + 7 = 11
Answer:
Given,
\(\sqrt{x+4}\) + 7 = 11
\(\sqrt{x+4}\) = 11 – 7
\(\sqrt{x+4}\) = 4
Squaring on both sides
x + 4 = 4² = 16
x + 4 = 16
x = 16 – 4
x = 12
\(\sqrt{x+4}\) + 7 = 11
\(\sqrt{12+4}\) + 7 = 11
\(\sqrt{16}\) + 7 = 11
4 + 7 = 11
Thus x = 12 is the solution

Question 6.
15 = 6 + \(\sqrt{3w-9}\)
Answer:
Given,
15 = 6 + \(\sqrt{3w-9}\)
15 – 6 = \(\sqrt{3w-9}\)
9 = \(\sqrt{3w-9}\)
Squaring on both sides
81 = 3w – 9
3w = 81 + 9
3w = 90
w = 90/3
w = 30
check:
15 = 6 + \(\sqrt{3(30)-9}\)
15 = 6 + \(\sqrt{81}\)
15 = 6 + 9
15 = 15
Thus w = 30 is the solution

Question 7.
\(\sqrt{3x+1}\) = \(\sqrt{4x-7}\)
Answer:
Given,
\(\sqrt{3x+1}\) = \(\sqrt{4x-7}\)
Squaring on both sides
3x + 1 = 4x – 7
3x – 4x = -7 – 1
-x = -8
x = 8

Question 8.
\(\sqrt{n}\) = \(\sqrt{5n-1}\)
Answer:
Given,
\(\sqrt{n}\) = \(\sqrt{5n-1}\)
Squaring on both sides
n = 5n – 1
5n – n = 1
4n = 1
n = 1/4

Question 9.
\(\sqrt [3]{ y }\) = 4 = 1
Answer:
Given,
\(\sqrt [3]{ y }\) = 4 – 1
\(\sqrt [3]{ y }\) = 3
cubing on both sides
y = 3³ = 27
y = 27

Question 10.
\(\sqrt [3]{ 3c+7 }\) = 10
Answer:
Given,
\(\sqrt [3]{ 3c+7 }\) = 10
cubing on both sides
3c + 7 = 10³
3c + 7 =  1000
3c = 993
c = 993/3
c = 331

Solve the equation. Check your solution(s).
Question 11.
\(\sqrt{4-3x}\) = x
Answer:
Given,
\(\sqrt{4-3x}\) = x
squaring on both sides
4 – 3x = 1
4 – 1 = 3x
3 = 3x
x = 1 is the solution

Question 12.
\(\sqrt{3m}\) + 10 = 1
Answer:
Given,
\(\sqrt{3m}\) + 10 = 1
\(\sqrt{3m}\) = 1 – 10
\(\sqrt{3m}\) = -9
squaring on both sides
3m = 81
m = 81/3
m = 27

Question 13.
p + 1 = \(\sqrt{7p+15}\)
Answer:
Given,
p + 1 = \(\sqrt{7p+15}\)
squaring on both sides
(p + 1)² = \(\sqrt{7p+15}\)²
p² + 2p + 1 = 7p + 15
p² + 2p + 1 – 7p – 15 = 0
p² -5p -14 = 0
p² -7p + 2p -14 = 0
(p – 7)(p – 2) =
p = 7 or p = 2

Question 14.
What is the length of a pendulum that has a period of 2.5 seconds?
Answer:
A simple pendulum with a period of 1 second will have a length of 0.25 meters or 25 centimeters
2.5 sec = 2.5 × 25 centimeters = 62.5 centimeters

Solving Radical Equations 10.3 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Why should you check every solution of a radical equation?
Answer:
You should check every solution of a radical equation because raising each side of a radical equation to an exponent can introduce extraneous solutions.

Question 2.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

Answer:
x \(\sqrt{3}\) – 5 = 4 does not belong because it is not a radical equation.
\(\sqrt{3}\) is constant.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, solve the equation. Check your solution.
Question 3.
\(\sqrt{x}\) = 9
Answer:

Question 4.
\(\sqrt{x}\) = 4
Answer:
Given,
\(\sqrt{x}\) = 4
Squaring on both sides
\(\sqrt{x}\)² = 4²
x = 16

Question 5.
7 = \(\sqrt{x}\) – 5
Answer:

Question 6.
\(\sqrt{p}\) – 7 = -1
Answer:
\(\sqrt{p}\) – 7 = -1
Add 7 on both sides
\(\sqrt{p}\) – 7 + 7 = -1 + 7
\(\sqrt{p}\) = 6
p = 6² = 36

Question 7.
\(\sqrt{c}\) + 12 = 23
Answer:

Question 8.
\(\sqrt{x}\) + 6 = 8
Answer:
Given,
\(\sqrt{x}\) + 6 = 8
Subtract 6 on both sides
\(\sqrt{x}\) + 6 – 6= 8 – 6
\(\sqrt{x}\) = 2
Squaring on both sides
\(\sqrt{x}\)² = 2²
= 4

Question 9.
4 – \(\sqrt{x}\) = 2
Answer:

Question 10.
-8 = 7 – \(\sqrt{r}\)
Answer:
Given,
-8 = 7 – \(\sqrt{r}\)
Subtract -7 on both sides
-8 – 7 = 7 – \(\sqrt{r}\) – 7
-15 = – \(\sqrt{r}\)
15 = \(\sqrt{r}\)
Squaring on both sides
225 = r

Question 11.
3\(\sqrt{y}\) – 18 = -3
Answer:

Question 12.
2\(\sqrt{q}\) + 5 = 11
Answer:
Given,
2\(\sqrt{q}\) + 5 = 11
Subtract 5 on both sides
2\(\sqrt{q}\) + 5 – 5 = 11 – 5
2\(\sqrt{q}\) = 6
\(\sqrt{q}\) = 6/2 = 3
Squaring on both sides
q = 9

In Exercises 13–20, solve the equation. Check your solution.
Question 13.
\(\sqrt{a-3}\) + 5 = 9
Answer:

Question 14.
\(\sqrt{b+7}\) – 5 = -2
Answer:
Given,
\(\sqrt{b+7}\) – 5 = -2
Add 5 on both sides
\(\sqrt{b+7}\) – 5 + 5 = -2 + 5
\(\sqrt{b+7}\) = 3
Squaring on both sides
\(\sqrt{b+7}\)² = 3² = 9
b + 7 = 9
b = 9 – 7
b = 2

Question 15.
2 \(\sqrt{x+4}\) = 16
Answer:

Question 16.
5\(\sqrt{y-2}\) = 10
Answer:
5\(\sqrt{y-2}\) = 10
Divide both sides by 5
5\(\sqrt{y-2}\)/5 = 10/5
\(\sqrt{y-2}\) = 2
Squaring on both sides
\(\sqrt{y-2}\)² = 2² = 4
y – 2 = 4
Add 2 on both sides
y – 2 + 2 = 4 + 2
y = 6

Question 17.
-1 = \(\sqrt{5r+1}\) – 7
Answer:

Question 18.
2 = \(\sqrt{4s-4}\) – 4
Answer:
Given,
2 = \(\sqrt{4s-4}\) – 4
Add 4 to each side
2 + 4 = \(\sqrt{4s-4}\) – 4 + 4
6 = \(\sqrt{4s-4}\)
Squaring on both sides
6² = \(\sqrt{4s-4}\)²
4s – 4 = 36
4s = 36 + 4
4s = 40
s = 40/4
s = 10

Question 19.
7 + 3\(\sqrt{3p-9}\) = 25
Answer:

Question 20.
19 – 4\(\sqrt{3c-11}\) = 11
Answer:
Given,
19 – 4\(\sqrt{3c-11}\) = 11
Subtract19 on both sides
19 – 4\(\sqrt{3c-11}\) – 19 = 11 – 19
– 4\(\sqrt{3c-11}\) = -8
4\(\sqrt{3c-11}\) = 8
\(\sqrt{3c-11}\) = 8/4
\(\sqrt{3c-11}\) = 2
Squaring on both sides
\(\sqrt{3c-11}\)² = 2² = 4
3c – 11 = 4
3c = 4 + 11
3c = 15
c = 15/3
c = 5

Question 21.
MODELING WITH MATHEMATICS
The Cave of Swallows is a natural open-air pit cave in the state of San Luis Potosí, Mexico. The 1220-foot- deep cave was a popular destination for BASE jumpers. The function t = \(\frac{1}{4} \sqrt{d}\) represents the time t (in seconds) that it takes a BASE jumper to fall d feet. How far does a BASE jumper fall in 3 seconds?

Answer:

Question 22.
MODELING WITH MATHEMATICS
The edge length s of a cube with a surface area of A is given by s = \(\sqrt{\frac{A}{6}}\). What is the surface area of a cube with an edge length of 4 inches?

Answer:
Given that,
The edge length s of a cube with a surface area of A is given by s = \(\sqrt{\frac{A}{6}}\)
s = 4
4 = \(\sqrt{\frac{A}{6}}\)
Squaring on both sides
16 = \(\sqrt{\frac{A}{6}}\)²
\({\frac{A}{6}}\) = 16
A = 16 × 6 = 96

In Exercises 23–26, use the graph to solve the equation.
Question 23.

Answer:

Question 24.

Answer:
The intersection of the two parabolas is the point (5, 4)
Thus the solution of the equation is (5, 4)

Question 25.

Answer:
The graphs intersect at (2, 2). So, the solution of the original equation is x = 2

Question 26.

Answer:
The x value where the 2 lines intersect is x = -1

In Exercises 27–34, solve the equation. Check your solution. (See Example 3.)
Question 27.
\(\sqrt{2x-9}\) = \(\sqrt{x}\)
Answer:

Question 28.
\(\sqrt{y+1}\) = \(\sqrt{4y-8}\)
Answer:
Given,
\(\sqrt{y+1}\) = \(\sqrt{4y-8}\)
Squaring on both sides
\(\sqrt{y+1}\)² = \(\sqrt{4y-8}\)²
y + 1 = 4y – 8
y = 4y – 8 – 1
4y – y = 9
3y = 9
y = 9/3 = 3
y = 3

Question 29.
\(\sqrt{3g+1}\) = \(\sqrt{7g-19}\)
Answer:

Question 30.
\(\sqrt{8h-7}\) = \(\sqrt{6h+7}\)
Answer:
Given,
\(\sqrt{8h-7}\) = \(\sqrt{6h+7}\)
Squaring on both sides
\(\sqrt{8h-7}\)² = \(\sqrt{6h+7}\)²
8h – 7 = 6h + 7
8h – 6h = 7 + 7
2h = 14
h = 7

Question 31.
\(\sqrt{\frac{p}{2}-2}\) = \(\sqrt{p-8}\)
Answer:

Question 32.
\(\sqrt{2v-5}\) = \(\sqrt{\frac{v}{3}+5}\)
Answer:
Given,
\(\sqrt{2v-5}\) = \(\sqrt{\frac{v}{3}+5}\)
Squaring on both sides
\(\sqrt{2v-5}\)² = \(\sqrt{\frac{v}{3}+5}\)²
2v – 5 = 5 + v/3
2v = 10 + v/3
6v = 30 + v
6v – v = 30
5v = 30
v = 6

Question 33.
\(\sqrt{2c+1}\) = \(\sqrt{4c}\) = 0
Answer:

Question 34.
\(\sqrt{5r}\) – \(\sqrt{8r-2}\) = 0
Answer:
Given,
\(\sqrt{5r}\) – \(\sqrt{8r-2}\) = 0
\(\sqrt{5r}\) = \(\sqrt{8r-2}\)
Squaring on both sides
\(\sqrt{5r}\)² = \(\sqrt{8r-2}\)²
5r = 8r – 2
2 = 8r – 5r
2 = 3r
r = 2/3

MATHEMATICAL CONNECTIONS In Exercises 35 and 36, find the value of x.
Question 35.
Perimeter = 22 cm

Answer:

Question 36.

Answer:
\(\sqrt{3x+12}\)(2)(1/2) = \(\sqrt{5x-4}\)
\(\sqrt{3x+12}\) = \(\sqrt{5x-4}\)
Squaring on both sides
3x + 12 = 5x – 4
3x – 5x = -4 – 12
-2x = -16
2x = 16
x = 8

In Exercises 37–44, solve the equation. Check your solution.
Question 37.
\(\sqrt [3]{ x }\) = 4
Answer:

Question 38.
\(\sqrt [3]{ y }\) = 2
Answer:
Given,
\(\sqrt [3]{ y }\) = 2
Cubing on both sides
y = 8

Question 39.
6 = 3\(\sqrt [3]{ 8g }\)
Answer:

Question 40.
\(\sqrt [3]{ r+19 }\) = 3
Answer:
Given,
\(\sqrt [3]{ r+19 }\) = 3
Cubing on both sides
r + 19 = 27
r = 27 – 19
r = 8

Question 41.
\(\sqrt [3]{ 2x+9 }\) = -3
Answer:

Question 42.
-5 = \(\sqrt [3]{ 10x+15 }\)
Answer:
-5 = \(\sqrt [3]{ 10x+15 }\)
Cubing on both sides
(-5)³ = \(\sqrt [3]{ 10x+15 }\)³
-125 = 10x + 15
-125 – 15 = 10x
-140 = 10x
x = -14

Question 43.
\(\sqrt [3]{ y+6 }\) = \(\sqrt [3]{ 5y-2 }\)
Answer:

Question 44.
\(\sqrt [3]{ 7j-2 }\) = \(\sqrt [3]{ j+4 }\)
Answer:
Given,
\(\sqrt [3]{ 7j-2 }\) = \(\sqrt [3]{ j+4 }\)
Cubing on both sides
7j – 2 = j + 4
7j – j = 4 + 2
6j = 6
j = 1

In Exercises 45–48, determine which solution, if any, is an extraneous solution.
Question 45.
\(\sqrt{6x-5}\) = x; x = 5, x = 1
Answer:

Question 46.
\(\sqrt{2y+3}\) = y; y = -1, y = 3
Answer:
Given,
\(\sqrt{2y+3}\) = y
Substitute y = -1 in the above equation
\(\sqrt{2(-1)+3}\) = -1
\(\sqrt{-2+3}\) = -1
\(\sqrt{1}\) ≠ -1
Now for y = 3
\(\sqrt{2(3)+3}\) = 3
\(\sqrt{6+3}\) = 3
\(\sqrt{9}\) = 3
y = -1 does not satisfy the original equation, it is an extraneous solution.
The only solution is y = 3

Question 47.
\(\sqrt{12p+16}\) = -2p; p = -1, p = 4
Answer:

Question 48.
-3g = \(\sqrt{-18-27g}\); g = -2, g = -1
Answer:
Given,
-3g = \(\sqrt{-18-27g}\)
For g = -2
-3(-2) = \(\sqrt{-18-27(-2)}\)
6 = \(\sqrt{36}\)
6 = 6
For g = -1
-3(-1) = \(\sqrt{-18-27(-1)}\)
3 = \(\sqrt{9}\)
3 = 3
There is no extraneous solution.

In Exercises 49–58, solve the equation. Check your solution(s).
Question 49.
y = \(\sqrt{5y-4}\)
Answer:

Question 50.
\(\sqrt{-14x-9x}\) = x
Answer:
Given,
\(\sqrt{-14x-9x}\) = x
Squaring on both sides
-14x – 9x = x²
x² +14 + 9x = 0
(x + 7)(x + 2) = 0
x + 7 = 0 or x + 2 = 0
x = -7 or x = -2

Question 51.
\(\sqrt{1-3a}\) = 2a
Answer:

Question 52.
2q = \(\sqrt{10q-6}\)
Answer:
Given,
2q = \(\sqrt{10q-6}\)
Squaring on both sides
(2q)² = \(\sqrt{10q-6}\)²
4q² = 10q – 6
4q² – 10q + 6 = 0
(2q – 3)(q – 1) = 0
2q – 3 = 0 or q – 1 = 0
2q = 3 or q = 1
q = 3/2 or q = 1

Question 53.
9 + \(\sqrt{5p}\) = 4
Answer:

Question 54.
\(\sqrt{3n}\) – 11 = -5
Answer:
Given,
\(\sqrt{3n}\) – 11 = -5
\(\sqrt{3n}\) = -5 + 11
\(\sqrt{3n}\) = 6
Squaring on both sides
3n = 36
n = 36/3
n = 12

Question 55.
\(\sqrt{2m+2}\) – 3 = 1
Answer:

Question 56.
15 + \(\sqrt{4b-8}\) = 13
Answer:
Given,
15 + \(\sqrt{4b-8}\) = 13
\(\sqrt{4b-8}\) = 13 – 15
\(\sqrt{4b-8}\) = -2
Squaring on both sides
4b – 8 = 4
4b = 4 + 8
4b = 12
b = 12/4
b = 3

Question 57.
r + 4 = \(\sqrt{-4r-19}\)
Answer:

Question 58.
\(\sqrt{3-s}\) = s – 1
Answer:
Given,
\(\sqrt{3-s}\) = s – 1
Squaring on both sides
3 – s = (s – 1)²
3 – s = s² – 2s + 1
s² – 2s + 1 – 3 + s
s² – s – 2 = 0
(s – 2)(s + 1) = 0
s – 2 = 0 or s + 1 = 0
s = 2 or s = -1

ERROR ANALYSIS In Exercises 59 and 60, describe and correct the error in solving the equation.
Question 59.

Answer:

Question 60.

Answer:
Forgot to check the solutioons where x = -6 is an extraneous solution
-6 = \(\sqrt{12-4(-6)}\)
-6 = \(\sqrt{36}\)
– 6 = 6

Question 61.
REASONING
Explain how to use mental math to solve \(\sqrt{2x}\) + 5 = 1.
Answer:

Question 62.
WRITING
Explain how you would solve \(\sqrt [4]{ m+4 }\) – \(\sqrt [4]{ 3m }\) = 0.
Answer:
\(\sqrt [4]{ m+4 }\) – \(\sqrt [4]{ 3m }\) = 0
\(\sqrt [4]{ m+4 }\) = \(\sqrt [4]{ 3m }\)
m + 4 = 3m
m – 3m = -4
-2m = -4
2m = 4
m = 4/2
m = 2

Question 63.
MODELING WITH MATHEMATICS
The formula V = \(\sqrt{PR}\) relates the voltage V (in volts), power P (in watts), and resistance R (in ohms) of an electrical circuit. The hair dryer shown is on a 120-volt circuit. Is the resistance of the hair dryer half as much as the resistance of the same hair dryer on a 240-volt circuit? Explain your reasoning.

Answer:

Question 64.
MODELING WITH MATHEMATICS
The time t (in seconds) it takes a trapeze artist to swing back and forth is represented by the function t = 2π \(\sqrt{\frac{r}{32}}\), where r is the rope length (in feet). It takes the trapeze artist 6 seconds to swing back and forth. Is this rope \(\frac{3}{2}\) as long as the rope used when it takes the trapeze artist 4 seconds to swing back and forth? Explain your reasoning.

Answer:
Solve for r at t = 4 and t = 6
At t = 6 seconds
6 = 2π \(\sqrt{\frac{r}{32}}\)
\(\frac{3}{π}\) = \(\sqrt{\frac{r}{32}}\)
r = \(\frac{288}{π²}\)
At t = 4 seconds
4 = 2π \(\sqrt{\frac{r}{32}}\)
\(\frac{4}{π²}\) = \({\frac{r}{32}}\)
r = \(\frac{128}{π²}\)
Therefore, the length of the rope that takes 6 seconds to swing back and forth is 2.25 times the length of the rope that takes 4 seconds to swing back and forth, not 3/2 or 1.5

REASONING In Exercises 65–68, determine whether the statement is true or false. If it is false, explain why.
Question 65.
If \(\sqrt{a}\) = b, then (\(\sqrt{a}\))² = b².
Answer:

Question 66.
If \(\sqrt{a}\) = \(\sqrt{b}\), then a = b.
Answer:
\(\sqrt{a}\) = \(\sqrt{b}\)
Squaring on both sides
\(\sqrt{a}\)² = \(\sqrt{b}\)²
a = b
Thus the above statement is true.

Question 67.
If a² = b², then a = b.
Answer:

Question 68.
If a² = \(\sqrt{b}\), then a⁴ = (\(\sqrt{b}\))²
Answer:
a² = \(\sqrt{b}\)
squaring on both sides
We get a⁴ = (\(\sqrt{b}\))²

Question 69.
COMPARING METHODS
Consider the equation x + 2 = \(\sqrt{2x-3}\).
a. Solve the equation by graphing. Describe the process.
b. Solve the equation algebraically. Describe the process.
c. Which method do you prefer? Explain your reasoning.
Answer:

Question 70.
HOW DO YOU SEE IT?
The graph shows two radical functions.

a. Write an equation whose solution is the x-coordinate of the point of intersection of the graphs.
Answer:
Using the above graph, equate the 2 y’s since the point of intersection is the solution
y1 = y2
\(\sqrt{2x+3}\) = \(\sqrt{4x-3}\).

b. Use the graph to solve the equation.
Answer:
From the graph, the solution is x = 3, the x-coordinate of the point of intersection.

Question 71.
MATHEMATICAL CONNECTIONS
The slant height s of a cone with a radius of r and a height of h is given by s = \(\sqrt{r^{2}+h^{2}}\). The slant heights of the two cones are equal. Find the radius of each cone.

Answer:

Question 72.
CRITICAL THINKING
How is squaring \(\sqrt{x+2}\) different from squaring \(\sqrt{x}\) + 2?
Answer:
Squaring \(\sqrt{x+2}\) will cancel out the radical symbol and results into x + 2.
Squaring \(\sqrt{x+2}\) is the same as squaring a binomial sum: x + 4 \(\sqrt{x}\) + 4

USING STRUCTURE In Exercises 73–78, solve the equation. Check your solution.
Question 73.
\(\sqrt{m+15}\) = \(\sqrt{m}\) + \(\sqrt{5}\)
Answer:

Question 74.
2 – \(\sqrt{x+1}\) = \(\sqrt{x+2}\)
Answer:
Given,
2 – \(\sqrt{x+1}\) = \(\sqrt{x+2}\)
Squaring on both sides
(2 – \(\sqrt{x+1}\))² = \(\sqrt{x+2}\)²
4 – 4 \(\sqrt{x+1}\) + (x + 1) = x + 2
– 4 \(\sqrt{x+1}\) + x + 1 = x + 2 – 4
– 4 \(\sqrt{x+1}\) = x – 2 – x – 1
– 4 \(\sqrt{x+1}\) = -3
4 \(\sqrt{x+1}\) = 3
\(\sqrt{x+1}\) = 3/4
x + 1 = 9/16
x = -7/16

Question 75.
\(\sqrt{5y+9}\) + \(\sqrt{5y}\) = 9
Answer:

Question 76.
\(\sqrt{2c-8}\) – \(\sqrt{2c}\) – 4 = 0
Answer:
Given,
\(\sqrt{2c-8}\) – \(\sqrt{2c}\) – 4 = 0
\(\sqrt{2c-8}\) = \(\sqrt{2c}\) + 4
Squaring on both sides
\(\sqrt{2c-8}\)² = (\(\sqrt{2c}\) + 4)²
2c – 8 = 2c + 8\(\sqrt{2c}\) + 16
2c – 8 – 2c = 8\(\sqrt{2c}\) + 16
-8 = 8\(\sqrt{2c}\) + 16
Divide by 8 on both sides
-1 = \(\sqrt{2c}\) + 2
-3 = \(\sqrt{2c}\)
Squaring on both sides
9 = 2c
c = 9/2

Question 77.
2\(\sqrt{1+4h}\) – 4\(\sqrt{h}\) – 2 = 0
Answer:

Question 78.
\(\sqrt{20-4z}\) + 2\(\sqrt{-z}\) = 10
Answer:
Given,
\(\sqrt{20-4z}\) + 2\(\sqrt{-z}\) = 10
\(\sqrt{20-4z}\) = 10 – 2\(\sqrt{-z}\)
Squaring on both sides
\(\sqrt{20-4z}\)² = (10 – 2\(\sqrt{-z}\))²
20 – 4z = 100 – 40\(\sqrt{-z}\) – 4z
20 = 100 – 40\(\sqrt{-z}\)
40\(\sqrt{-z}\) = 100 – 20 = 80
40\(\sqrt{-z}\) = 80
\(\sqrt{-z}\) = 2
-z = 4
z = -4

Question 79.
OPEN-ENDED
Write a radical equation that has a solution of x = 5.
Answer:

Question 80.
OPEN-ENDED
Write a radical equation that has x = 3 and x = 4 as solutions.
Answer:
The possible answer is writing the quadratic function (x – 3)(x – 4) under the radical symbol and simplify it.
then equate to 0
\(\sqrt{(x-3)(x-4)}\) = 0

Question 81.
MAKING AN ARGUMENT
Your friend says the equation \(\sqrt{(2 x+5)^{2}}\) = 2x + 5 is always true, because after simplifying the left side of the equation, the result is an equation with infinitely many solutions. Is your friend correct? Explain.
Answer:

Question 82.
THOUGHT PROVOKING
Solve the equation \(\sqrt [3]{ x+1 }\) = \(\sqrt{x-3}\). Show your work and explain your steps.
Answer:
\(\sqrt [3]{ x+1 }\) = \(\sqrt{x-3}\)
let y1 = \(\sqrt [3]{ x+1 }\) and y2 = \(\sqrt{x-3}\)
Let x = 7
\(\sqrt [3]{ 7+1 }\) = \(\sqrt{7-3}\)
\(\sqrt [3]{ 8 }\) = \(\sqrt{4}\)
2 = 2
Therefore the solution is x = 7.

Question 83.
MODELING WITH MATHEMATICS
The frequency f (in cycles per second) of a string of an electric guitar is given by the equation f = \(\frac{1}{2 \ell} \sqrt{\frac{T}{m}}\), where ℓ is the length of the string (in meters), T is the string’s tension (in newtons), and m is the string’s mass per unit length (in kilograms per meter). The high E string of an electric guitar is 0.64 meter long with a mass per unit length of 0.000401 kilogram per meter.

a. How much tension is required to produce a frequency of about 330 cycles per second?
b. Would you need more or less tension to create the same frequency on a string with greater mass per unit length? Explain.
Answer:

Maintaining Mathematical Proficiency

Find the product.(Section 7.2)
Question 84.
(x + 8)(x – 2)
Answer:
(x + 8)(x – 2)
x(x – 2) + 8 (x – 2)
x² -2x + 8x – 16
x² +6x – 16

Question 85.
(3p – 1)(4p + 5)
Answer:

Question 86.
(s + 2)(s² + 3s – 4)
Answer:
Given,
(s + 2)(s² + 3s – 4)
s(s² + 3s – 4) + 2 (s² + 3s – 4)
s³ + 3s² – 4s + 2s² + 6s – 8
s³ + 5s² + 2s – 8

Graph the function. Compare the graph to the graph of f(x) = x².(Section 8.1)
Question 87.
r(x) = 3x²
Answer:

Question 88.
g(x) = \(\frac{3}{4}\)x²
Answer:

The graph of g(x) is the graph of f(x) that is
– Vertically shrinked by a factor of 3/4

Question 89.
h(x) = -5x²
Answer:

Lesson 10.4 Inverse of a Function

Essential Question How are a function and its inverse related?

EXPLORATION 1

Exploring Inverse Functions
Work with a partner. The functions f and g are inverses of each other. Compare the tables of values of the two functions. How are the functions related?

EXPLORATION 2

Exploring Inverse Functions
Work with a partner.
a. Plot the two sets of points represented by the tables in Exploration 1. Use the coordinate plane below.
b. Connect each set of points with a smooth curve.
c. Describe the relationship between the two graphs.
d. Write an equation for each function.

Communicate Your Answer

Question 3.
How are a function and its inverse related?
Answer:

Question 4.
A table of values for a function f is given. Create a table of values for a function g, the inverse of f.

Answer:

Question 5.
Sketch the graphs of f(x) = x + 4 and its inverse in the same coordinate plane. Then write an equation of the inverse of f. Explain your reasoning.

Answer:

Monitoring Progress

Find the inverse of the relation.
Question 1.
(-3, -4), (-2, 0), (-1, 4), (0, 8), (1, 12), (2, 16), (3, 20)
Answer:
Switch the coordinates of each ordered pair.
(-4, -3), (0, -2), (4, -1), (8, 0), (12, 1), (16, 2), (20, 3)

Question 2.

Answer:

Solve y = f(x) for x. Then find the input when the output is 4.
Question 3.
f(x) = x – 6
Answer:
f(x) = y
y = x – 6
x = y + 6
If y = 4
x = 4 + 6
x = 10

Question 4.
f(x) = \(\frac{1}{2}\)x + 3
Answer:
f(x) = y
y = \(\frac{1}{2}\)x + 3
y – 3 = \(\frac{1}{2}\)x
x = 2y – 6
If y = 4
x = 2(4) – 6
x = 8 – 6
x = 4

Question 5.
f(x) = 4x²
Answer:
f(x) = y
y = 4x²
x² = y/4
If y = 4
x² = 4/4 = 1
x = 1

Find the inverse of the function. Then graph the function and its inverse.
Question 6.
f(x) = 6x
Answer:

Question 7.
f(x) = -x + 5
Answer:

Question 8.
f(x) = \(\frac{1}{4}\)x – 1
Answer:

Find the inverse of the function. Then graph the function and its inverse.
Question 9.
f(x) = -x², x ≤ 0
Answer:

Question 10.
f(x) = 4x² + 3, x ≥ 0
Answer:

Question 11.
Is the inverse of f(x) = \(\sqrt{2x-1}\) a function? Find the inverse.
Answer:

Inverse of a Function 10.4 Exercises

Vocabulary and Core ConceptCheck

Question 1.
COMPLETE THE SENTENCE
A relation contains the point (-3, 10). The ____________ contains the point (10, -3).
Answer:
A relation contains the point (-3, 10). The inverse contains the point (10, -3).

Question 2.
DIFFERENT WORDS, SAME QUESTION
Consider the function f represented by the graph. Which is different? Find “both” answers.

Answer:
The blue points A, B, C and D represent the original function f that was given.
The first, third and forth statement then results in the orange points E, F, G and H while the second statement results in the pink points I, J, K and L.
Thus we can say that the second statement is different.

In Exercises 3–8, find the inverse of the relation.
Question 3.
(1, 0), (3, -8), (4, -3), (7, -5), (9, -1)
Answer:

Question 4.
(2, 1), (4, -3), (6, 7), (8, 1), (10, -4)
Answer:
Switch the coordinates of each ordered pair.
(1, 2), (-3, 4), (7, 6), (1, 8), (-4, 10)

Question 5.

Answer:
Switch the coordinates of each ordered pair.

Question 6.

Answer:
Switch the coordinates of each ordered pair.

Question 7.

Answer:
Switch the coordinates of each ordered pair.

Question 8.

Answer:
Switch the coordinates of each ordered pair.
The inverse relation of the above function is given below.

In Exercises 9–14, solve y = f(x) for x. Then find the input when the output is 2.
Question 9.
f(x) = x + 5
Answer:

Question 10.
f(x) = 2x – 3
Answer:
Given,
f(x) = 2x – 3
y = 2x + 3
To find x:
y – 3 = 2x
x = \(\frac{1}{2}\)y + \(\frac{3}{2}\)
If y = 2:
x = \(\frac{1}{2}\)(2) + \(\frac{3}{2}\)
x = 1 + 1.5
x = 2.5

Question 11.
f(x) = \(\frac{1}{4}\)x – 1
Answer:

Question 12.
f(x) = \(\frac{2}{3}\)x + 4
Answer:
Given,
f(x) = \(\frac{2}{3}\)x + 4
y = \(\frac{2}{3}\)x + 4
3y = 2x + 12
3y – 12 = 2x
x = \(\frac{3}{2}\)y – 6
If y = 2
x = \(\frac{3}{2}\)(2) – 6
x = 3 – 6
x = -3

Question 13.
f(x) = 9x²
Answer:

Question 14.
f(x) = \(\frac{1}{2}\)x² – 7
Answer:
Given,
f(x) = \(\frac{1}{2}\)x² – 7
y = \(\frac{1}{2}\)x² – 7
y + 7 = \(\frac{1}{2}\)x²
2y + 14 = x²
x = \(\sqrt{2y+14}\)
If y = 2
x = \(\sqrt{2(2)+14}\)
x = \(\sqrt{18}\)

In Exercises 15 and 16, graph the inverse of the function by reflecting the graph in the line y = x. Describe the domain and range of the inverse.
Question 15.

Answer:

Question 16.

Answer:
The line of the given function passes the two points (-3, -1) and (2, 4)
The function of the line is
y = x + 2
By reflecting the graph in the line y = x, we get the function
y = x – 2
The domain of the inverse is -∞ < x < ∞
The range of the inverse is -∞ < x < ∞

In Exercises 17–22, find the inverse of the function. Then graph the function and its inverse.
Question 17.
f(x) = 4x – 1
Answer:

Question 18.
f(x) = -2x + 5
Answer:
f(x) = -2x + 5
y = -2x + 5
Interchange x and y
x = -2y + 5
x – 5 = -2y
(x – 5)/-2 = y
Finally replace y with f^-1 (x) and interchange the left and right side of the equation:
f^-1 (x) = –\(\frac{1}{2}\)x + \(\frac{5}{2}\)

Question 19.
f(x) = -3x – 2
Answer:

Question 20.
f(x) = 2x + 3
Answer:
Given,
f(x) = 2x + 3
y = 2x + 3
Replace x and y
x = 2y + 3
x – 3 = 2y
y = (x – 3)/2
g(x) = (x – 3)/2
= \(\frac{1}{2}\)x – \(\frac{3}{2}\)

Based on the graphs, we see that the inverse function is symmetric with respect to the line x = y

Question 21.
f(x) =\(\frac{1}{3}\)x + 8
Answer:
Given,
f(x) =\(\frac{1}{3}\)x + 8
Interchange x and y
x = \(\frac{1}{3}\)y + 8
x – 8 = \(\frac{1}{3}\)y
3(x – 8) = y
f^-1 (x) = 3(x – 8) = 3x – 24

Question 22.
f(x) = – \(\frac{3}{2}\)x + \(\frac{7}{2}\)
Answer:
Given,
f(x) = – \(\frac{3}{2}\)x + \(\frac{7}{2}\)
y = – \(\frac{3}{2}\)x + \(\frac{7}{2}\)
x = – \(\frac{3}{2}\)y + \(\frac{7}{2}\)
x – \(\frac{7}{2}\) = –\(\frac{3}{2}\)y
–\(\frac{2}{3}\)(x – \(\frac{7}{2}\)) = y
f^-1 (x) =-\(\frac{2}{3}\)(x – \(\frac{7}{2}\))

In Exercises 23–28, find the inverse of the function. Then graph the function and its inverse.
Question 23.
f(x) = 4x², x ≥ 0
Answer:

Question 24.
f(x) = \(\frac{2}{25}\)x², x ≤ 0
Answer:
Set y = f(x)
y = –\(\frac{1}{25}\)x²
x = –\(\frac{1}{25}\)y²
-25x = y²
±5\(\sqrt{-x}\) = y
Since the domain of f is restricted to x ≤ 0, then the range of the inverse must also be restricted to y ≤ 0
g(x) = -5\(\sqrt{-x}\)
The function and its inverse are reflections of each other about the line y = x.

Question 25.
f(x) = -x² + 10, x ≤ 0
Answer:

Question 26.
f(x) = 2x² + 6, x ≥ 0
Answer:
y = f(x)
y = 2x² + 6
x = 2y² + 6
x – 6 = 2y²
(x – 6)/2 = y²
± \(\sqrt{x-6/2}\) = y
Since the domain of f is restricted to x ≥ 0, then the range of the inverse must also be restricted to y ≥ 0
g(x) = \(\sqrt{x-6/2}\)
The function and its inverse are reflections of each other about the line y = x.

Question 27.
f(x) = \(\frac{1}{9}\)x² + 2, x ≥ 0
Answer:

Question 28.
f(x) = -4x² – 8, x ≤ 0
Answer:
y = f(x)
f(x) = -4x² – 8
y = -4x² – 8
x = -4y² – 8
x + 8 = -4y²
–\(\frac{x}{4}\) – 2 = y²
±√-\(\frac{x}{4}\) – 2 = y
Since the domain of f is restricted to x ≥ 0, then the range of the inverse must also be restricted to y ≥ 0
Thus the inverse is
g(x) =- √-\(\frac{x}{4}\) – 2
The function and its inverse are reflections of each other about the line y = x.

In Exercises 29–32, use the Horizontal Line Test to determine whether the inverse of f is a function.
Question 29.

Answer:

Question 30.

Answer:
The below is the graph of the function:
Because no horizontal line intersects the graph more than once, the inverse is a function.

Question 31.

Answer:

Question 32.

Answer:
The below is the graph of the function:
Because there are horizontal lines that intersect the graph more than once, the inverse is not a function.

In Exercises 33–42, determine whether the inverse of f is a function. Then find the inverse.
Question 33.
f(x) = \(\sqrt{x+3}\)
Answer:

Question 34.
f(x) = \(\sqrt{x-5}\)
Answer:
Graph the function f, because no horizontal line intersects the graph more than once, the inverse of f is a function.

y = f(x)
y = \(\sqrt{x-5}\)
x = \(\sqrt{y-5}\)
Squaring on both sides
x² = y – 5
x² + 5 = y
Since the range of f is restricted to y ≥ 0, then the domain of the inverse must also be restricted to x ≥ 0.
Therefore the inverse is
g(x) = x² + 5, x ≥ 0

Question 35.
f(x) = \(\sqrt{2x-6}\)
Answer:

Question 36.
f(x) = \(\sqrt{4x+1}\)
Answer:
Graph the function f, because no horizontal line intersects the graph more than once, the inverse of f is a function.
y = \(\sqrt{4x+1}\)
x = \(\sqrt{4y+1}\)
Squaring on both sides
x² = 4y + 1
x² – 1 = 4y
y = \(\frac{x²}{4}\) – \(\frac{1}{4}\)
Since the range of f is restricted to y ≥ 0, then the domain of the inverse must also be restricted to x ≥ 0.
Thus the inverse is
g(x) = \(\frac{x²}{4}\) – \(\frac{1}{4}\), where x ≥ 0.

Question 37.
f(x) = 3\(\sqrt{x-8}\)
Answer:

Question 38.
f(x) = –\(\frac{1}{4} \sqrt{5 x+2}\)
Answer:
The below is the graph of the function:
Because no horizontal line intersects the graph more than once, the inverse is a function.

y = f(x)
y = –\(\frac{1}{4} \sqrt{5 x+2}\)
x = –\(\frac{1}{4} \sqrt{5 y+2}\)
Squaring on both sides
x² = \(\frac{1}{16}\)(5y + 2)
16x² = 5y + 2
16x² – 2 = 5y
\(\frac{16x²-2}{5}\) = y
Since the range of f is restricted to y ≥ 0, then the domain of the inverse must also be restricted to x ≥ 0.
Thus the inverse is
g(x) = \(\frac{16x²-2}{5}\), where x ≤ 0

Question 39.
f(x) = –\(\sqrt{3x+5}\) – 2
Answer:

Question 40.
f(x) = 2\(\sqrt{x-7}\) + 6
Answer:
The below is the graph of the function:
Because no horizontal line intersects the graph more than once, the inverse is a function.

y = f(x)
f(x) = 2\(\sqrt{x-7}\) + 6
y = 2\(\sqrt{x-7}\) + 6
x = 2\(\sqrt{y-7}\) + 6
x – 6 = 2\(\sqrt{y-7}\)
Squaring on both sides
(x – 6)² = (2\(\sqrt{y-7}\))²
(x – 6)²/4 = y – 7
(x – 6)²/4 + 7 = y
Since the range of f is restricted to y ≥ 6, then the domain of the inverse must also be restricted to x≥ 6.
g(x) = (x – 6)²/4 + 7, where x ≥ 6

Question 41.
f(x) = 2x²
Answer:

Question 42.
f(x) = |x|
Answer:
Graph the function, Because the horizontal lines intersect the graph more than once, the inverse of f is not a function.
Therefore, it has no inverse.

Question 43.
ERROR ANALYSIS
Describe and correct the error in finding the inverse of the function f(x) = – 3x + 5.

Answer:

Question 44.
ERROR ANALYSIS
Describe and correct the error in finding and graphing the inverse of the function f(x) = \(\sqrt{x-3}\).

Answer:
Since the range of f is restricted to y ≥ 0, then the domain of the inverse must also be restricted to x ≥ 0.
Therefore, only the right half of the graph should be shown.

Question 45.
MODELING WITH MATHEMATICS
The euro is the unit of currency for the European Union. On a certain day, the number E of euros that could be obtained for D U.S. dollars was represented by the formula shown.
E = 0.74683D
Solve the formula for D. Then find the number of U.S. dollars that could be obtained for 250 euros on that day.
Answer:

Question 46.
MODELING WITH MATHEMATICS
A crow is flying at a height of 50 feet when it drops a walnut to break it open. The height h (in feet) of the walnut above ground can be modeled by h = -16t² + 50, where t is the time (in seconds) since the crow dropped the walnut. Solve the equation for t. After how many seconds will the walnut be 15 feet above the ground?

Answer:
Given,
h = -16t² + 50
h – 50 = -16t²
(h – 50)/-16 = t²
t = ±\(\sqrt{50-h}\)/\(\sqrt{16}\)
Since, we cannot have a negative time, then take the positive root and since the height goes from 50ft to 0ft, then the domain is 0 ≤ h ≤ 50.
Thus t = ±\(\sqrt{50-h}\)/\(\sqrt{16}\)
t = \(\sqrt{2.1875}\)
t ≈ 1.48 seconds

MATHEMATICAL CONNECTIONS In Exercises 47 and 48, s is the side length of an equilateral triangle. Solve the formula for s. Then evaluate the new formula for the given value.

Answer:
47.

In Exercises 49–54, find the inverse of the function. Then graph the function and its inverse.
Question 49.
f(x) = 2x³
Answer:

Question 50.
f(x) = x³ – 4
Answer:
y = f(x)
y = x³ – 4
x = = y³ – 4
x + 4 = y³
\(\sqrt [3]{ x+4}\) = y
g(x) = \(\sqrt [3]{ x+4}\)
The function and its inverse are reflections of each other about the line y = x

Question 51.
f(x) = (x – 5)³
Answer:

Question 52.
f(x) = 8(x + 2)³
Answer:
y = f(x)
y = 8(x + 2)³
x = 8(y + 2)³
\(\sqrt [3]{ x }\) = 2(y + 2)
\(\sqrt [3]{ x }\)/2 = y + 2
\(\sqrt [3]{ x }\)/2 – 2 = y
g(x) = \(\sqrt [3]{ x }\)/2 – 2
The function and its inverse are reflections of each other about the line y = x

Question 53.
f(x) = 4\(\sqrt [3]{ x }\)
Answer:

Question 54.
f(x) = –\(\sqrt [3]{ x-1 }\)
Answer:
Given,
f(x) = –\(\sqrt [3]{ x-1 }\)
y = f(x)
y = –\(\sqrt [3]{ x-1 }\)
x = –\(\sqrt [3]{ y-1 }\)
cubing on both sides
x³ = -(y – 1)
-x³ = y – 1
-x³ + 1 = y
g(x) = -x³ + 1
The function and its inverse are reflections of each other about the line y = x

Question 55.
MAKING AN ARGUMENT
Your friend says that the inverse of the function f(x) = 3 is a function because all linear functions pass the Horizontal Line Test. Is your friend correct? Explain.
Answer:

Question 56.
HOW DO YOU SEE IT?
Pair the graph of each function with the graph of its inverse.

Answer:
The function and its inverse are reflections of each other about the line y = x
Therefore the pairs are A and D, B and F, C and E

Question 57.
WRITING
Describe changes you could make to the function f(x) = x² – 5 so that its inverse is a function. Describe the domain and range of the new function and its inverse.
Answer:

Question 58.
CRITICAL THINKING
Can an even function with at least two values in its domain have an inverse that is a function? Explain.
Answer:
No. Since the graphs of all even functions are reflections of the y-axis.
Thus it will immediately fail the horizontal line test. It can only have an inverse if its domain is restricted.

Question 59.
OPEN-ENDED
Write a function such that the graph of its inverse is a line with a slope of 4.
Answer:

Question 60.
CRITICAL THINKING
Consider the function g(x) = -x.
a. Graph g(x) = -x and explain why it is its own inverse.
Answer: We note that the graph of g is perpendicular to the line y = x and thus reflecting the graph of g will result in the same graph and thus the inverse function of g is g itself.
b. Graph other linear functions that are their own inverses. Write equations of the lines you graph.
Answer:
We note that every function parallel to the line y = -x is also its own inverse because the graph remains perpendicular to the line y = x. However also note that y = x will be its own inverse.

c. Use your results from part (b) to write a general equation that describes the family of linear functions that are their own inverses.
Answer:
Thus y = -x + c and y = x with c a constant are all functions that have itself as inverse.

Question 61.
REASONING
Show that the inverse of any linear function f(x) = mx + b, where m ≠ 0, is also a linear function. Write the slope and y-intercept of the graph of the inverse in terms of m and b.
Answer:

Question 62.
THOUGHT PROVOKING
The graphs of f(x) = x³ – 3x and its inverse are shown. Find the greatest interval -a ≤ x ≤ a for which the inverse of f is a function. Write an equation of the inverse function.

Answer:
Restricting the domain to -1 ≤ x ≤ 1 will make the graph of f(x) pass the horizontal line test.
Find the inverse:
y = f(x)
y = x³ – 3x
x = y³ – 3x
Since we cannot isolate y, then we cannot solve for the inverse function.

Question 63.
REASONING
Is the inverse of f(x) = 2|x + 1|a function? Are there any values of a, h, and k for which the inverse of f(x) = a |x – h| + k is a function? Explain your reasoning.
Answer:

Maintaining Mathematical Proficiency

Find the sum or difference.(Section 7.1)
Question 64.
(2x – 9) – (6x + 5)
Answer:
Given,
(2x – 9) – (6x + 5)
2x – 9 – 6x – 5
add or subtract the like terms
2x – 6x – 9 – 5
-4x – 14

Question 65.
(8y + 1) + (-y – 12)
Answer:

Question 66.
(t² – 4t – 4) + (7t² + 12t + 3)
Answer:
Given,
(t² – 4t – 4) + (7t² + 12t + 3)
t² – 4t – 4 + 7t² + 12t + 3
8t² + 8t -1

Question 67.
(-3d² + 10d – 8) – (7d² – d – 6)
Answer:

Graph the function. Compare the graph to the graph of f(x) = x². (Section 8.2)
Question 68.
g(x) = x² + 6
Answer:

the graph of g(x) is the graph of f(x) that is vertically shifted 6 units upward

Question 69.
h(x) = -x² – 2
Answer:

Question 70.
p(x) = -4x² + 5
Answer:

The graph of p(x) is the graph of f(x):
– Reflected in the x-axis
– Veertically stretched by a factor of 4
– Vertically shifted 5 units upward

Question 71.
q(x) = \(\frac{1}{3}\)x² – 1
Answer:

Radical Functions and Equations Performance Task: Medication and the Mosteller Formula

10.3–10.4What Did YouLearn?

Core Vocabulary
radical equation, p. 560
inverse relation, p. 568
inverse function, p. 569

Core Concepts
Lesson 10.3
Squaring Each Side of an Equation, p. 560
Identifying Extraneous Solutions, p. 562

Lesson 10.4
Inverse Relation, p. 568
Finding Inverses of Functions Algebraically, p. 570
Finding Inverses of Nonlinear Functions, p. 570
Horizontal Line Test, p. 571

Mathematical Practices
Question 1.
Could you also solve Exercises 37–44 on page 565 by graphing? Explain.
Answer:

Question 2.
What external resources could you use to check the reasonableness of your answer in Exercise 45 on page 573?
Answer:

Performance Task: Medication and the Mosteller Formula

When taking medication, it is critical to take the correct dosage. For children in particular, body surface area (BSA) is a key component in calculating that dosage. The Mosteller Formula is commonly used to approximate body surface area. How will you use this formula to calculate BSA for the optimum dosage?
To explore the answers to this question and more, go to

Radical Functions and Equations Chapter Review

10.1 Graphing Square Root Functions (pp. 543–550)

Graph the function. Describe the domain and range. Compare the graph to the graph of f(x) = \(\sqrt{x}\).
Question 1.
g(x) = \(\sqrt{x}\) + 7
Answer:
Step 1: Use the domain to make a table of values
The radicand must be greater than or equal to 0
x ≥ 0
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (0, 7)

From the graph, we can see that the range is
y ≥ 1
The graph of g(x) is the graph of f(x) is vertically shifted 7 units upwards.

Question 2.
h(x) = \(\sqrt{x-6}\)
Answer:
Step 1: Use the domain to make a table of values
The radicand must be greater than or equal to 0
x – 6 ≥ 0
x ≥ 6
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (6, 0)

From the graph, we can see that the range is
y ≥ 0
The graph of h(x) is the graph of f(x) is horizontally shifted 6 units to the right.

Question 3.
r(x) = –\(\sqrt{x+3}\) – 1
Answer:
Step 1: Use the domain to make a table of values
The radicand must be greater than or equal to 0
x + 3 ≥ 0
x ≥ -3
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (-3, -1)

From the graph, we can see that the range is
y ≤ -1
The graph of r(x) is the graph of f(x) is
– Reflected in the x-axis
– Horizontally shifted 3 units to the left
– Vertically shifted 1 unit downward

Question 4.
Let g(x) = \(\frac{1}{4} \sqrt{x-6}\) + 2. Describe the transformations from the graph of f(x) = \(\sqrt{x}\) to the graph of g. Then graph g.
Answer:
The graph of g(x) is the graph of f(x) is
– Vertically shrinked by a factor of 1/4
– Horizontally shifted 6 units to the right
– Vertically shifted 2 units upward
Step 1: Use the domain to make a table of values
The radicand must be greater than or equal to 0
x – 6 ≥ 0
x ≥ 6
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (6, 2)

10.2 Graphing Cube Root Functions (pp. 551–556)

Graph the function. Compare the graph to the graph of f(x) = \(\sqrt [3]{ x }\).
Question 5.
g(x) = \(\sqrt [3]{ x }\) + 4
Answer:
Step 1: Make a table of values.
Step 2: Plot the ordered pairs.
Step 3: Draw a smooth curve through the points.

The graph of g(x) is the graph of f(x) is vertically shifted 4 units upward.

Question 6.
h(x) = -8\(\sqrt [3]{ x }\)
Answer:
Step 1: Make a table of values.
Step 2: Plot the ordered pairs.
Step 3: Draw a smooth curve through the points.

The graph of h(x) is the graph of f(x) is
– Reflected in the x-axis
– Horizontally stretched by a factor of 8
– Vertically shifted 6 unit downward

Question 7.
s(x) = \(\sqrt[3]{-2(x-3)}\)
Answer:
Step 1: Make a table of values.
Step 2: Plot the ordered pairs.
Step 3: Draw a smooth curve through the points.

The graph of s(x) is the graph of f(x) is
– Reflected in the y-axis
– Horizontally stretched by a factor of 1/2
– Horizontally shifted 3 unit to the right

Question 8.
Let g(x) = -3\(\sqrt [3]{ x+2 }\) – 1. Describe the transformations from the graph of f(x) = \(\sqrt [3]{ x }\) to the graph of g. Then graph g.
Answer:
Step 1: Make a table of values.
Step 2: Plot the ordered pairs.
Step 3: Draw a smooth curve through the points.

Question 9.
The graph of cube root function r is shown. Compare the average rate of change of r to the average rate of change of p(x) = \(\sqrt[3]{\frac{1}{2} x}\) over the interval x = 0 to x = 8.
Answer:
Use (0, 0) and (8, 5)
The average rate of change is
r(8) – r(0)/8-0 = (5 -0)/(8 – 0) = 0.625
For p(x) = \(\sqrt[3]{\frac{1}{2} x}\)
Evaluate g when x = 0 and x = 8
p(0) = \(\sqrt[3]{\frac{1}{2} (0)}\) = 0
p(8) = \(\sqrt[3]{\frac{1}{2} (8)}\) = \(\sqrt [3]{ 4 }\)
The average rate of change is
p(8) – p(0)/8-0 = (\(\sqrt [3]{ 4 }\)  -0)/(8 – 0) = 0.2
The average rate of change of r is 0.625/0.2 = 3.1 times greater than the average rate of change of p over the interval x = 0 to x = 8.

10.3 Solving Radical Equations (pp. 559-566)

Solve the equation. Check your solution(s).
Question 10.
8 + \(\sqrt{x}\) = 18
Answer:
Given,
8 + \(\sqrt{x}\) = 18
\(\sqrt{x}\) = 18 – 8
\(\sqrt{x}\) = 10
Squaring on both sides
x = 10² = 100
Check:
8 + \(\sqrt{x}\) = 18
8 + \(\sqrt{100}\) = 18
8 + 10 =18
Thus x = 100 is the solution

Question 11.
\(\sqrt [3]{ x-1 }\) = 3
Answer:
Given,
\(\sqrt [3]{ x-1 }\) = 3
Cubing on both sidees
x – 1 = 3³ = 27
x – 1 = 27
x = 27 + 1
x = 28
Check:
\(\sqrt [3]{ x-1 }\) = 3
\(\sqrt [3]{ 28-1 }\) = 3
\(\sqrt [3]{ 27 }\) = 3
3 = 3
Thus x = 28 is the solution

Question 12.
\(\sqrt{5x-9}\) = \(\sqrt{4x}\)
Answer:
Given,
\(\sqrt{5x-9}\) = \(\sqrt{4x}\)
squaring on both sides
5x – 9 = 4x
5x – 4x = 9
x = 9

Question 13.
x = \(\sqrt{3x+4}\)
Answer:
Given,
x = \(\sqrt{3x+4}\)
squaring on both sides
x² = 3x + 4
x² – 3x + 4 = 0
(x – 1) (x – 4) = 0
x = 1 or x = 4

Question 14.
8\(\sqrt{x-5}\) + 34 = 58
Answer:
Given,
8\(\sqrt{x-5}\) + 34 = 58
8\(\sqrt{x-5}\) = 58 – 34
8\(\sqrt{x-5}\) = 24
\(\sqrt{x-5}\) = 24/8
\(\sqrt{x-5}\) = 3
squaring on both sides
x – 5 = 9
x = 9 + 5
x = 14

Question 15.
\(\sqrt{5x}\) + 6 = 5
Answer:
Given,
\(\sqrt{5x}\) + 6 = 5
\(\sqrt{5x}\) = 5 – 6
\(\sqrt{5x}\) = -1
squaring on both sides
5x = 1
x = 1/5

Question 16.
The radius r of a cylinder is represented by the function r = \(\sqrt{\frac{V}{\pi h}}\), where V is the volume and h is the height of the cylinder. What is the volume of the cylindrical can?

Answer:
r = \(\sqrt{\frac{V}{\pi h}}\)
r = 2 in
2 = \(\sqrt{\frac{V}{\pi 4}}\)
4 = \(\frac{V}{4π}\)
16π = V
V ≈ 50.3 cu. in

10.4 Inverse of a Function (pp. 567–574)

Find the inverse of the relation.
Question 17.
(1, -10), (3, -4), (5, 4), (7, 14), (9, 26)
Answer:
Switch the coordinates of each ordered pairs
(1, -10), (3, -4), (5, 4), (7, 14), (9, 26)
(-10, 1), (-4, 3), (4, 5), (14, 7), (26, 9)

Question 18.

Answer:
Switch the coordinates of each ordered pairs

Find the inverse of the function. Then graph the function and its inverse.
Question 19.
f(x) = -5x + 10
Answer:
y = f(x)
y = -5x + 10
x = -5y + 10
x – 10 = -5y
-x/5 + 2 = y
g(x) = -x/5 + 2
The function and its inverse are reflections of each other about the line y = x

Question 20.
f(x) = 3x² – 1, x ≥ 0
Answer:
y = f(x)
y = 3x² – 1
x = 3y² – 1
x + 1 = 3y²
(x + 1)/3 = y²
y = ± \(\sqrt{\frac{x+1}{\3}}\)
Since the domain of f is restricted to x ≥ 0, then the range of the inverse must also be restricted to y ≥ 0.
Therefore, the inverse is
g(x) = \(\sqrt{x}\)/2
The function and its inverse are reflections of each other about the line y = x

Question 21.
f(x) = \(\frac{1}{2} \sqrt{2 x+6}\)
Answer:
y = f(x)
y = \(\frac{1}{2} \sqrt{2 x+6}\)
x = \(\frac{1}{2} \sqrt{2 y+6}\)
2x = \(\sqrt{2 y+6}\)
Squaring on both sides
4x² = 2y + 6
4x² – 6 = 2y
Taking 2 as a common number
2x² – 3 =y
y = 2x² – 3
Since the range of f is restricted to y ≥ 0, then the domain of the inverse must also be restricted to x ≥ 0.
Therefore the inverse is
g(x) = 2x² – 3
The function and its inverse are reflections of each other about the line y = x

Question 22.
Consider the function f(x) = x² + 4. Use the Horizontal Line Test to determine whether the inverse of f is a function.
Answer:
The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once.

Since the horizontal lines intersect the graph of f more than once, then the inverse of f is not a function.

Question 23.
In bowling, a handicap is an adjustment to a bowler’s score to even out differences in ability levels. In a particular league, you can find a bowler’s handicap h by using the formula h = 0.8(210 – a), where a is the bowler’s average. Solve the formula for a. Then find a bowler’s average when the bowler’s handicap is 28.

Answer:
Given,
h = 0.8(210 – a)
h/0.8 = 210 – a
h/0.8 – 210 = -a
a = -h/0.8 + 210
substitute h = 28 and then find a
a = -28/0.8 + 210
a = 175

Radical Functions and Equations Chapter Test

Find the inverse of the function.
Question 1.
f(x) = 5x – 8
Answer:
f(x) = 5x – 8
y = 5x – 8
y + 8 = 5x
x = (y + 8)/5
Thus f^-1(x) = (y + 8)/5

Question 2.
f(x) = 2\(\sqrt{x+3}\) – 1
Answer:
f(x) = 2\(\sqrt{x+3}\) – 1
y = 2\(\sqrt{x+3}\) – 1
y + 1 = 2\(\sqrt{x+3}\)
(y + 1)/2 = \(\sqrt{x+3}\)
[(y + 1)/2]² = x + 3
[(y + 1)/2]² – 3 = x
Thus f^-1(x) = [(x + 1)/2]² – 3

Question 3.
f(x) = –\(\frac{1}{3}\)x² + 4, x ≥ 0
Answer:
y = f(x)
f(x) = –\(\frac{1}{3}\)x² + 4
y = –\(\frac{1}{3}\)x² + 4
x = –\(\frac{1}{3}\)y² + 4
x – 4 = –\(\frac{1}{3}\)y²
-3x + 12 = y²
y = \(\sqrt{-3x+12}\)
Since the domain of f is restricted to x ≤ 0, then the range of the inverse must also be restricted to y ≤ 0
g(x) = \(\sqrt{-3x+12}\)
The function and its inverse are reflections of each other about the line y = x.

Graph the function f. Describe the domain and range. Compare the graph of f to the graph of g.
Question 4.
f(x) = –\(\sqrt{x+6}\); g(x) = \(\sqrt{x}\)
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x + 6 ≥ 0
x ≥ -6
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (-6, 0)

From the graph, we can see that the range is y ≥ 0
The graph of f(x) is the graph of g(x) is
– Reflected in the x-axis
– Horizontally shifted 6 units to the left

Question 5.
f(x) = \(\sqrt{x-3}\) + 2; g(x) = \(\sqrt{x}\)
Answer:
Step 1: Use the domain to make a table of values.
The radicand must be greater than or equal to 0
x – 3 ≥ 0
x ≥ 3
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points starting at (3, 2)

From the graph, we can see that the range is y ≥ 2
The graph of f(x) is the graph of g(x) is
– Horizontally shifted 3 units to the right
– Vertically shifted 2 units upward

Question 6.
f(x) = \(\sqrt [3]{ x }\) – 5; g(x) = \(\sqrt [3]{ x }\)
Answer:
The domain and range of a cubic function are all real numbers.
Step 1: Use the domain to make a table of values.
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points.

The graph of f(x) is the graph of g(x)
– Vertically shifted 5 units downward

Question 7.
f(x) = -2\(\sqrt [3]{ x+1 }\); g(x) = \(\sqrt [3]{ x }\)
Answer:
The domain and range of a cubic function are all real numbers.
Step 1: Use the domain to make a table of values.
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points.

The graph of f(x) is the graph of g(x)
– Reflected in the x-axis
– Vertically stretched by a factor of 2
– Horizontally shifted 1 unit to the right

Solve the equation. Check your solution(s).
Question 8.
9 – \(\sqrt{x}\) = 3
Answer:
Given,
9 – \(\sqrt{x}\) = 3
9 – 3 = \(\sqrt{x}\)
6 = \(\sqrt{x}\)
squaring on both sides
36 = x
check:
9 – \(\sqrt{x}\) = 3
9 – \(\sqrt{36}\) = 3
9 – 6 = 3
Thus x = 36 is the solution

Question 9.
\(\sqrt{2x-7}\) – 3 = 6
Answer:
Given,
\(\sqrt{2x-7}\) – 3 = 6
\(\sqrt{2x-7}\) = 6 + 3
\(\sqrt{2x-7}\) = 9
squaring on both sides
2x – 7 = 9²
2x – 7 = 81
2x = 81 + 7
2x = 88
x = 44
Check:
\(\sqrt{2(44)-7}\) – 3 = 6
\(\sqrt{81}\) – 3 = 6
9 – 3 = 6
6 = 6
Thus x = 44 is the solution

Question 10.
\(\sqrt{8x-21}\) = \(\sqrt{18-5x}\)
Answer:
Given,
\(\sqrt{8x-21}\) = \(\sqrt{18-5x}\)
squaring on both sides
8x – 21 = 18 – 5x
8x + 5x = 18 + 21
13x = 39
x = 3
Check:
\(\sqrt{8x-21}\) = \(\sqrt{18-5x}\)
\(\sqrt{8(3)-21}\) = \(\sqrt{18-5(3)}\)
\(\sqrt{3}\) = \(\sqrt{3}\)
Thus x = 3 is the solution

Question 11.
x + 5 = \(\sqrt{7x+53}\)
Answer:
Given,
x + 5 = \(\sqrt{7x+53}\)
(x + 5)² = 7x + 53
x² + 10x + 25 = 7x + 53
x² + 10x + 25 – 7x – 53 = 0
x² + 3x – 28 = 0
(x + 7)(x – 4) = 0
x + 7 = 0 or x – 4 = 0
x = -7 or x = 4

Question 12.
When solving the equation x – 5 = \(\sqrt{ax+b}\), you obtain x = 2 and x = 8. Explain why at least one of these solutions must be extraneous.
Answer:
x = 2 is an extraneous solution because the left side will be negative but the right side will be positive.

Describe the transformations from the graph of f(x) = \(\sqrt [3]{ x }\) to the graph of the given function. Then graph the given function.
Question 13.
h(x) = 4\(\sqrt [3]{ x-1 }\) + 5
Answer:
Step 1: Use the domain to make a table of values.
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points.

Question 14.
w(x) = –\(\sqrt [3]{ x+7 }\) – 2
Answer:
Step 1: Use the domain to make a table of values.
Step 2: Plot the ordered pairs
Step 3: Draw a smooth curve through the points.

Question 15.
The velocity v (in meters per second) of a roller coaster at the bottom of a hill is given by v = \(\sqrt{19.6h}\), where h is the height (in meters) of the hill.
(a) Use a graphing calculator to graph the function. Describe the domain and range.
Answer:

Domain: h ≥ 0
Range: v ≥ 0

(b) How tall must the hill be for the velocity of the roller coaster at the bottom of the hill to be at least 28 meters per second?
Answer:
v = 28 meters
28 = \(\sqrt{19.6h}\)
Squaring on both sides
784 = 19.6h
h = 40 meters

(c) What happens to the average rate of change of the velocity as the height of the hill increases?
Answer: The average rate of change of the velocity decreases as the height of the hill increases.

Question 16.
The speed s (in meters per second) of sound through air is given by s = 2\(\sqrt{T+273}\), where T is the temperature (in degrees Celsius).

a. What is the temperature when the speed of sound through air is 340 meters per second?
Answer:
s = 340
340 = 20\(\sqrt{T+273}\)
17 = \(\sqrt{T+273}\)
289 = T + 273
T = 16(16°C)
b. How long does it take you to hear the wolf howl when the temperature is -17°C?
Answer:
s = 20\(\sqrt{T+273}\)
s = \(\sqrt{-17+273}\)
s = \(\sqrt{256}\)
s = 20(16)
s = 320
Divide the distance by the speed to find the time
t = d/s = 1760/320
t = 5.5 seconds

Question 17.
How can you restrict the domain of the function f(x) = (x – 3)² so that the inverse of f is a function?
Answer:
We can restrict the domain by taking half of the parabola which occurs at the vertex.
In this case, the vertex is at (3, 0)
Therefore we can restrict the domain as x ≤ 3 or x ≥ 3 for the inverse of f to be a function.

Question 18.
Write a radical function that has a domain of all real numbers less than or equal to 0 and a range of all real numbers greater than or equal to 9.
Answer:
From the statement, the domain is x ≤ 0 which means the function contains the radical \(\sqrt{x}\) and is reflected in the y-axis: \(\sqrt{-x}\)
The range is y ≥ 9 means that this radical is vertically shifted 9 units upward so that the function will be
f(x) = \(\sqrt{-x}\) + 9

Radical Functions and Equations Cumulative Assessment

Question 1.
Fill in the function so that it is represented by the graph.

Answer:
The graph of f(x) is the graph of the parent function y = \(\sqrt{x}\) that is
– Reflected in the x-axis → –\(\sqrt{x}\)
– Horizontally shifted 1 unit to the right → –\(\sqrt{x-1}\)
– Vertically shifted 2 units upward → –\(\sqrt{x-1}\) + 2
Thus the function is –\(\sqrt{x-1}\) + 2

Question 2.
Consider the equation y = mx + b. Fill in values for m and b so that each statement is true.
a. When m = ______ and b = ______, the graph of the equation passes through the point (-1, 4).
Answer:
Write any slope say m = 1 and solve for b by using the slope intercept form
y = mx + b
Now substitute slope and given point
4 = 1(-1) + b
4 = -1 + b
Therefore m = 1, b = 5
b. When m = ______ and b = ______, the graph of the equation has a positive slope and passes through the point (-2, -5).
Answer:
Write any slope say m = 1 and solve for b by using the slope intercept form
y = mx + b
Now substitute slope and given point
-5 = 1(-2) + b
-5 = -2 + b
b = -3
Therefore m = 1, b = -3
c. When m = ______ and b = ______, the graph of the equation is perpendicular to the graph of y = 4x – 3 and passes through the point (1, 6).
Answer:
The lines are perpendicular if their slopes are negative reciprocal of each other.
Thus m = -1/4
Solve for b by using the slope intercept form
y = mx + b
Now substitute slope and given point
6 = 1(-1/4) + b
6 = -1/4 + b
b = 25/4
Therefore m = -1/4, b = 25/4

Question 3.
Which graph represents the inverse of the function f(x) = 2x + 4?

Answer:
y = f(x)
y = 2x + 4
x = 2y + 4
x – 4 = 2y
x/2 – 2 = y
The graph where m = 1/2 and b = -2 is option b

Question 4.
Consider the equation x = \(\sqrt{ax+b}\). Student A claims this equation has one real solution. Student B claims this equation has two real solutions. Use the numbers to answer parts (a)–(c).

a. Choose values for a and b to create an equation that supports Student A’s claim.
Given,
x = \(\sqrt{ax+b}\)
Squaring on both sides
x² = \(\sqrt{ax+b}\)²
x² = ax + b
x² – ax – b = 0
There is one real solution if b² – 4ac = 0
(-a)² – 4(1)(-b) = 0
a² + 4b = 0
The possible answer is a = 4, b = -4
b. Choose values for a and b to create an equation that supports Student B’s claim.
Answer:
There are 2 real solutions if b² – 4ac > 0
(-a)² – 4(1)(-b) > 0
a² + 4b > 0
The possible answer is a = 2, b = 1
c. Choose values for a and b to create an equation that does not support either student’s claim.
Answer:
There are no real solutions if b² – 4ac < 0
(-a)² – 4(1)(-b) < 0
a² + 4b < 0
The possible answer is a = 2, b = -2

Question 5.
Which equation represents the nth term of the sequence 3, 12, 48, 192, . . .?
A. a_n = 3(4)^n-1
B. a_n = 3(9)^n-1
C. a_n = 9n – 6
D. a_n = 9n + 3
Answer:
This is a geometric sequence
a1 = 3 and the common ratio is r = 4
Use the formula for the nth term
a_n = a1(r)^r-1
The equation will be
a_n = 3(4)^n-1
Thus the correct answer is option A.

Question 6.
Consider the function f(x) = \(\frac{1}{2} \sqrt[3]{x+3}\). The graph represents function g. Select all the statements that are true.

Answer:
The x-intercept of f is -3 while g is 0. (-3 < 0)
The graph of both f and g are always increasing
The average rate of change for both functions decreases as x increases.
For g(x). Estimate using the graph. Use (0, 0) and (8, 4)
The average rate of change us
g(8) – g(0)/8-0
= (4-0)/(8-0) = 0.50
For f(x) = [/latex]\sqrt[3]{x+3}[/latex]. Evaluate f when x = 0 and x = 8
f(0) = 1/2 [/latex]\sqrt[3]{0+3}[/latex] = 0.72
f(8) = 1/2 [/latex]\sqrt[3]{8+3}[/latex] = 1.11
The average rate of change us
f(8) – f(0)/8-0
= (1.11-0.72)/(8-0) = 0.05
The average rate of change of g is 0.5/0.05 = 10 times greater than the average rate of change of f over the interval x = 0 and x = 8

Question 7.
Place each function into one of the three categories.

Answer:
Recall the use of discriminant b² – 4ac to find the number of solutions
b² – 4ac < 0
b² – 4ac = 0
b² – 4ac > 0
f(x) = 3x² + 4x + 2
f(x) = -x² + 2x
f(x) = 4x² – 8x + 4
f(x) = x² – 3x – 21
f(x) = 7x²
f(x) = -6x² – 5

Question 8.
You are making a tabletop with a tiled center and a uniform mosaic border.
a. Write the polynomial in standard form that represents the perimeter of the tabletop.
b. Write the polynomial in standard form that represents the area of the tabletop.
c. The perimeter of the tabletop is less than 80 inches, and the area of tabletop is at least 252 square inches. Select all the possible values of x.

Answer:
From the figure, the dimensions of the tabletop are
L = 2x + 16 and W = 2x + 12
a. The perimeter is
P = 2L + 2W
P = 2(2x + 16 ) + 2(2x + 12)
P = 8x + 56
b. The area is
A = L . W
A = (2x + 16 ) × (2x + 12)
A = 4x² + 56x + 192
c. From the given,
P < 80
8x + 56 < 82
8x < 24
x < 3
A ≥ 252
4x² + 56x + 192 ≥ 252
4x² + 56x – 60 ≥ 0
(x + 15)(x – 1) ≥ 0