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3-1 Additional PracticeRelations and Functions

What is the domain and range of each function?

1. 2.

Is each relation a function? If so, state whether it is one-to-one or many-to-one.

3. {(–4, 7), (–3, 5), (1, 4), (3, –8), (5, –11)} 4. {(–4, 8), (–2, 4), (0, 1), (2, 4), (4, 8)}

5. −2013

−3134

6. y

4

8

Ox

−4

−8

84−4−8

7. Explain how the vertical line test proves that a relation is not a function.

8. Fiona buys different amounts of gas at $2.25. She has a graph which shows the different amounts she should pay. What constraints are there on the domain of the function?

3

6

5

7

7

7

8 11

9 14

x

y

−3

9

−1

5

2

4

5

−5

7

−7

x

y

domain: {3, 5, 7, 8, 11} range: {6, 7, 9, 14}

function; one-to-one

not a function

Sample answer: A vertical line intersects a relation at the same domain value. If the vertical line contains multiple range values for the same domain value, the relation is not a function.

Sample answer: The domain of the function consists of values greater than 0.

domain: {–3, –1, 2, 5, 7} range: {9, 5, 4, –5, –7}

function; many-to-one

function; many-to-one

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3-2 Additional PracticeLinear Functions

What is the value of f(−3) for each function?

1. f(x) = 4x − 9 2. f(x) = − 1 __ 3 x + 13 3. f(x) = −2x − 11

Draw the graph of each linear function.

4. f(x) = 3x − 6 5. f(x) = −2(x + 4) 6. f(x) = 1 __ 2 x + 5

y2

Ox

−2

−4

−6

42−2−4

yO

x

−2

−4

−6

−8

42−2−4

y

Ox

2

4

6

42−2−4

Use the data in each table to write a linear function using function notation.

7.

−3

1

7

0.2

−0.6

1.4

yx 8.

−5

−2

4

−1

−10

17

yx 9.

−5

−2

8

2

8

−18

yx

10. A function, f(x) = 4x + 5, has a domain 0 ≤ x ≤ 50. What is its range?

11. For a basic subscription, a cable television provider charges an activation fee of $60, plus $125 per month. What linear function represents the total cost of a basic cable subscription for t months? What is the total cost for two years of service?

−21 14 −5

f(x) = 0.2x f(x) = 3x + 5 f(x) = −2x − 2

5 ≤ y ≤ 205

f(t) = 125t + 60; $3,060

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3-4 Additional PracticeArithmetic Sequences

Tell whether or not each sequence is an arithmetic sequence. If it is an arithmetic sequence, give the common difference.

1. 4, 8, 12, 16, … 2. −11, 5, 0, 6, … 3. 12, 23, 34, 45, …

Write a recursive formula and an explicit formula for each arithmetic sequence.

4. 9, 15, 21, 27, … 5. 1.5, 2.25, 3, 3.75, … 6. 7, 0, −7, −14, …

Recursive: Recursive: Recursive:

Explicit: Explicit: Explicit:

Write an explicit formula for each recursive formula and a recursive formula for each explicit formula.

7. a 1 = 5 8. a 1 = −8 9. a n = 15 + 4n

a n = a n − 1 + 3 a n = a n − 1 − 3

10. You are given the first four terms of an arithmetic sequence. Why might you use a recursive formula? Why might you use an explicit formula? Under what conditions might a recursive formula be preferred over the explicit formula? Under what conditions might an explicit formula be preferred over the recursive formula?

11. You open a savings account with a $400 deposit. Each month after that, you deposit $25. Write an explicit rule to represent the amount of money you deposit into your savings account. How much money will you have in the account on month 12?

arithmetic; 4 not arithmetic arithmetic; 11

Sample answer: If you need to know the 7th, 8th, and 9th terms, the recursive form might be preferred as you need to apply the common difference successively to the next few terms. If you want to know terms, such as those beyond 100, then the explicit would be preferred.

a n = 375 + 25n; $675

a 1 = 9

a n = a n − 1 + 6

a n = 3 + 6n

a 1 = 1.5

a n = a n − 1 + 0.75

a n = 0.75 + 0.75n

a 1 = 7

a n = a n − 1 −7

a n = 14 − 7n

a n = 2 + 3n a n = −5 −3n a 1 = 19

a n = a n − 1 + 4

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3-5 Additional PracticeScatter Plots and Lines of Fit

What is the association between the x- and y-values for each graph?

1.

14

18

2 106x

y

100

0

2.

2

4

6

0 4 62

8

x

y

0

Describe the type of correlation each scatter plot shows. Draw a trend line that models each data set and find the equation of that trend line.

3.

1 3 4 5 62x

y

500

60

70

80

Tem

per

atu

re

Time (hours)0

4.

20 4 5 6 7 83x

y

20

4

6

8Sa

les

(mill

ion

s)

Cost in Dollars

5. For the trend line in Item 3, what would the expected temperature be after 2 hours? Explain what this means in the context of the data.

6. For the trend line in Item 4, what would the expected sales be if the cost was set at $6.50? Explain what this means in the context of the data.

7. Would you expect the trend line for the temperature to continue in the same direction indefinitely? Explain.

positive correlation Sample answer: y = 7x + 43

negative correlation Sample answer: y = −x + 10

Sample answer: The expected temperature is 57°. The line shows an increase of about 7° per hour.

Sample answer: The expected sales would be about 3,500,000. The sales decrease by about 1,000,000 units per dollar increase in cost.

Sample answer: No, the temperature would drop as the sun sets. Also, sooner or later new seasons would occur and the overall temperature would either be warmer or colder.

positive association

negative association

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3-6 Additional PracticeAnalyzing Lines of Best Fit

Describe the type of correlation indicated by each correlation coefficient.

1. r = 0.875 2. r = −0.976 3. r = 0.043

For each situation described given a linear model, is there a correlation? If so, is there a causal correlation? Explain.

4. the number of minutes studied for a test and the final test grade

5. the attendance at a baseball game and number of runs the home team scores in the game

Use the table for Items 6 and 7.

0

900

1

1,900

3

3,200

5

4,500

6

5,150

7

6,750

9

7,550

10

8,100

12

9,525

13

10,300

Years Since2002

Gross Revenue(Millions)

6. What is the equation of the line of best fit for the above data? Round the slope and y-intercept to the nearest hundredth. Interpret the slope and y-intercept.

7. Use the line of best fit to determine the predicted gross revenue for 2010 and 2018.

8. Explain why a pattern in a residual plot can suggest that a linear model may not be a good fit for a set of data.

strong positive correlation

strong negative correlation

weak correlation

Sample answer: correlation; causal. The amount of time spent studying for a test will likely affect the test grade positively.

Sample answer: no correlation; A team may have high or low attendance but the number of runs cannot be predicted based on attendance.

y = 713.71x + 1077.05; The slope represents the approximate increase in revenue per year, and the y-intercept represents the initial revenue from the first year.

2010: $6,786.73 2018: $12,496.41

Sample answer: A residual plot that has a definite pattern suggests that the deviations between actual and predicted values will generally not be close to one another even though they may occasionally be close together. Since more data values will be far away from the predicted values, a linear model may not be appropriate.

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