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5-1 Additional PracticeThe Absolute Value Function

Graph each absolute value function.

1. f(x) = 3 |x|

xO

y

2

4

–2 2 4–4

–2

2. g(x) = − 1 __ 2 |x|

xOy

–2

–2

–4

2 4–4

–6

3. h(x) = 4 |x|

xO

y

4

–2 2 4

–2

4. f(x) = −2 |x|

xOy

–2

–2

–4

–6

2 4–4

5. g(x) = 1 __ 4 |x| 6. h(x) = − |x|

xOy

–2

–2

–4

2 4–4

–4

–6

7. How does the value of a determine the range for f(x) = a |x| ?

8. A bus is traveling between two cities that are in neighboring states. The graph of the function d(t) = 60 |t − 1.25| shows the distance of the bus in miles from the state line, where t represents time in hours.

a. What does the graph tell you about the bus ride?

b. What is the average rate of change from 1.5 h to 2 h? What does the average rate of change represent?

xO

y2

–2–4 2 4

–2

–4

–6

y

30

00

60

x0.5 1 1.5 2 2.5

Time (hours)

Dis

tan

ce (

mile

s)

Sample answer: When a > 0, the range is all real numbers greater than or equal to zero. When a < 0, the range is all real numbers less than or equal to zero.

Sample answer: The bus started 75 m away from the state line, crossed the state line in 1.25 h, and then was 75 m away from the state line after another 1.25 h.

60; the speed of the bus

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5-4 Additional PracticeTransformations of Piecewise-Defined Functions

For each function, identify the vertex and axis of symmetry.

1. f(x) = |x| − 5 2. g(x) = |x − 3| 3. h(x) = |x + 7| − 4

Compare each function with f(x) = |x| . Describe the graph of g as a transformation of the graph of f. Then graph the function.

4. g(x) = 2 |x + 3| − 4

xO

y4

2

–2–4–6

–2

–4

5. g(x) = − 1 __ 2 |x − 5| + 3

xO

y4

2

2 4 6 8

–2

–4

6. g(x) = −3 |x − 4| − 1

xO

y4

2

2 4 6 8

–2

–4

7. Explain why the vertex for f(x) = a |x − h| + k is unaffected by a.

8. For three years, a local coffee shop’s rewards program offered 1 point for every whole dollar spent and an additional 5 points for each purchase. Suppose the shop changes the points offered to an additional 8 points for each purchase and 1 point for every whole dollar spent.

a. Write two step function rules, f(x) and g(x), that model each rewards program.

b. Describe the effect of the new points program on the corresponding graphs.

(3, 0); x = 3(0, −5); y-axis (−7, −4); x = −7

translated 3 units left, 4 units down, and vertically stretched by a factor of 2

translated 5 units right, 3 units up, reflected over the x-axis, and vertically compressed by 1 __

2

translated 4 units right, 1 unit down, reflected over the x-axis, and vertically stretched by a factor of 3

Sample answer: Substituting h for x results in the following: f(h) = a |h − h| + k = a |0| + k = 0 + k = k. Regardless of the value of a, f(h) = k.

f(x) = ⌊ x ⌋ + 5; g(x) = ⌊ x ⌋ + 8

The graph of g will be a vertical translation 3 units up from the graph of f.

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5-2 Reteach to Build UnderstandingPiecewise-Defined Functions

1. Match each piecewise-defined function with its graph.

y = { −2x, x < −1

2x, x ≥ −1

y = { 2x, x ≤ −1

−2x, x > −1

y = { −2x, x < 1

2x, x ≥ 1

2. Describe the error Bart made when graphing the piecewise-defined function shown. Graph the function correctly.

f(x) = { 4, x ≤ 2

3x, x > 2

3. Write a piecewise-defined function that describes the graph shown.

y = { , x 5

, x 5

xO

y

2

4

6

–2

–2

2 4

xO

y

2

4

–2–4

–2

2 4

xO

y2

–2–4

–2

–4

–6

2 4

xO

y

2

8

6

–2–4 2 4x

O

y

2

8

6

–2–4 2 4

xO

y

4

8

–4

–4

–8

8–8x − 5

5 ≥<

Bart used an open circle where he should have used a closed circle and a closed circle where he should have used an open circle. The correct graph is shown.

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5-2 Additional PracticePiecewise-Defined Functions

Write each absolute value function as a piecewise–defined function.

1. f(x) = 7 |x| 2. g(x) = − 1 __ 3 |x| 3. h(x) = 9 |x|

Which graph matches each piecewise–defined function?

4. f(x) = { −3x −2, x < 1 −5, x ≥ 1

xO

y

4

2

6

–2–4–6

–2

2

5. g(x) = { 2x + 3, x ≤ 1

−x + 6, x > 1

xO

y

2

–2

–2

–4

2 4 6

6. h(x) = { x + 4, x < −2

2x + 6, x ≥ −2

xO

y

4

–2

–2

2 4

7. A DJ is shopping for cloud storage for his music collection. He finds an online data storage cloud with the following monthly charges per GB (or fraction thereof): $0.29/GB for 0 < x ≤ 25 GB, $0.26/GB for 25 < x ≤ 50 GB, $0.23/GB for 50 < x ≤ 75 GB, and $0.20/GB for x > 75 GB. What is the difference in price for storing just the 75 GB and for storing 75 GB plus a new music collection that is 1 GB or less?

f(x) = { 7x, x ≥ 0 −7x, x < 0

g(x) =

⎧

⎪

⎨ ⎪

⎩ − 1 __

3 x, x ≥ 0

1 __ 3 x, x < 0

h(x) = {

9x, x ≥ 0

−9x, x < 0

Sample Answer: The monthly charge for 75 GB is $17.25. However, by adding the new music collection, the DJ will have just over 75 GB and will pay a monthly charge of $15.20. He will save $2.05 a month by storing the added music collection.

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