Using Problem Solving Strategies and Models

22 Georgia Notetaking Guide, Mathematics 2 Copyright © McDougal Littell/Houghton Mifflin Company.

2.1 Using Problem Solving Strategies and Models

GeorgiaPerformanceStandard(s)

MM2P1d, MM2P3a

Your Notes

Goal p Solve problems using verbal models.

VOCABULARY

Verbal model

A bus travels at an average rate of 55 miles per hour. The distance between Chicago and San Francisco is 2130 miles. How long would it take for the bus to travel from Chicago to San Francisco?

SolutionUse the distance formula for distance traveled as a verbal model.

Distance(miles)

5Rate

(mi/h)p Time

(hours)

5 p t

An equation for this situation is 5 t. Solve for t.

5 t Write equation.

ø t Divide each side by .

The amount of time it would take to travel from Chicago to San Francisco is about hours.

CHECK You can use unit analysis to check your answer.

miles øhour

milesp hours

Example 1 Use a formula

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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 2 23

Your Notes

1. In Example 1, how fast is the bus traveling if it takes 22 hours to travel from San Francisco to Colorado Springs, a distance of 1335 miles?

Checkpoint Complete the following exercise.

The table shows the height h of a jet airplane t minutes after beginning its descent. Find the height of the airplane after 9 minutes.

Time (min), t 0 1 2 3 4

Height (ft), h 35,000 32,000 29,000 26,000 23,000

SolutionThe height decreases by 3000 feet per minute.

35,000 32,000 29,000 26,000 23,000

23000 23000 23000 23000

You can use this pattern to write a verbal model for the height.

Height(feet)

5Initial height

(feet)2

Rate of descent(feet/min)

p Time(min)

h 5 2 p t

An equation for the height is h 5 2 t.

So, the height after 9 minutes is

h 5 2 ( ) 5 feet.

Example 2 Look for a pattern

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Your Notes

You want to paint five 1 foot wide stripes on the wall. There should be an equal amount of space between the ends of the wall and the stripes and between each pair of stripes. The wall is 14 feet long. How far apart should the stripes be?

Begin by drawing and labeling x 1 x 1 x 1

14 ft

x 1 x 1 x

a diagram, as shown at the right.

From the diagram, you can write and solve an equation to find x.

x 1 1 1 x 1 1 1 x 1 1 1 x1 1 1 x 1 1 1 x 5 14 Write equation.

x 1 5 14 Combine like terms.

x 5 Subtract from each side.

x 5 Divide each side by .

The stripes should be painted feet apart.

Example 3 Draw a diagram

2. If a jet airplane descends at the rate given in the table, what is its height after 8 minutes?

Time (min), t 0 1 2 3 4

Height (ft), h 36,000 32,800 29,600 26,400 23,200

3. In Example 3, how far apart do the stripes need to be painted if you are only going to put 4 stripes on the wall?

Checkpoint Complete the following exercises.

Homework

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Name ——————————————————————— Date ————————————

LESSON

2.1 PracticeUse the formula d 5 rt for distance traveled to solve for the missing variable.

1. d 5 ? , r 5 35 miles per hour, t 5 2 hours

2. d 5 120 miles, r 5 60 miles per hour, t 5 ?

3. d 5 140 miles, r 5 ? , t 5 3.5 hours

4. d 5 125 miles, r 5 50 miles per hour, t 5 ?

Use the formula A 5 bh for the area of a parallelogram to solve for the missing variable.

5. A 5 ? , b 5 2 feet, h 5 2 feet

6. A 5 8 square feet, b 5 ? , h 5 2 feet

7. A 5 200 square meters, b 5 40 meters, h 5 ?

8. A 5 ? , b 5 21 centimeters, h 5 7 centimeters



Look for a pattern in the table. Then write an equation that represents the pattern.

9. x 0 1 2 3

y 1 3 5 7

10. x 0 1 2 3

y 6 10 14 18

11. x 0 1 2 3

y 17 27 37 47

12. x 0 1 2 3

y 150 125 100 75

13. Amusement Park Price You have $480 to purchase tickets for admission to an amusement park for the members of your rollercoaster club. An “all day” ticket costs $45.50. How many tickets can you purchase?

14. Amusement Park Trip Your travel arrangements to an amusement park include a round trip driving distance of 216 miles. The planned travel time is 4 hours. What must your average speed be to make the trip in the allotted time?

15. Rollercoaster Club Membership Each membership to the rollercoaster club costs $85 per year. In addition to the dues, each member pays $35 per trip taken during the season. If you have $295, how many trips can you take?

LESSON

2.1 Practice continued

Name ——————————————————————— Date ————————————



2.2 Solve Absolute Value Equations and Inequalities


MM2A1c

Your Notes

Goal p Solve absolute value equations and inequalities.

VOCABULARY

Extraneous solution

Solve ⏐x 2 3⏐5 6. Graph the solution.

⏐x 2 3⏐5 6 Write originalequation.

x 2 3 5 or x 2 3 5 Write equivalentequations.

x 5 or x 5 Solve for x.

x 5 or x 5 Simplify.

The solutions are and . These are the values of x that are units away from on a number line.

4 6 8 10202224

Example 1 Solve a simple absolute value equation

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Your NotesCheckpoint Solve the equation. Then graph the solution.

1. ⏐x⏐5 5

4 6 82022242628

2. ⏐x 2 5⏐5 2

4210 3 8 9 1065 7

Solve ⏐4x 1 10⏐5 6x. Check for extraneous solutions.

⏐4x 1 10⏐5 6x Write original equation.

4x 1 10 5 or 4x 1 10 5 Expressioncan equal or .

10 5 or 10 5 Subtract from each side.

5 x or 5 x Solve for x.

Check the apparent solutions to see if either is extraneous.

CHECK

⏐4x 1 10⏐5 6x ⏐4x 1 10⏐5 6x

⏐4( ) 1 10⏐0 6( ) ⏐4( ) 1 10⏐0 6( )

⏐ ⏐ 0 ⏐ ⏐ 0

5 ✓ Þ

The solution is . Reject because it is an solution.

Example 2 Check for extraneous solutions

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Your Notes

Solve ⏐2x 1 5⏐> 3. Then graph the solution.

The absolute value inequality is equivalent to 2x 1 5 < or 2x 1 5 > .

First Inequality Second Inequality

2x 1 5 < Write inequalities. 2x 1 5 >

2x < Subtract 2x > from each side.

x < Divide each side x > by .

The solutions are all real numbers less than or greater than .

4 6 82022242628

Example 3 Solve an inequality of the form ⏐ax 1 b⏐> c

Solve ⏐x 2 1.5⏐≤ 4.5. Then graph the solution.

x 2 1.5 ≤ 4.5 Write inequality.

≤ x 2 1.5 ≤ Write equivalent compound inequality.

≤ x ≤ Add to each expression.

The solution is between and , inclusive.

4 6 82022242628

Example 4 Solve an inequality of the form ⏐ax 1 b⏐≤ c

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Your Notes

3. ⏐2x 1 5⏐ 5 11 4. ⏐3x 1 18⏐ 5 6x

Checkpoint Solve the equation. Check for extraneoussolutions.

Checkpoint Solve the inequality. Then graph the solution.

5. ⏐x 2 2⏐≥ 7

4 6 8 10 122026 24 2228

6. ⏐4x 2 1⏐< 9

4210 321222324Homework

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Name ——————————————————————— Date ————————————

LESSON

2.2 PracticeRewrite the absolute value equation as two linear equations.

1. ⏐x 2 3⏐ 5 2 2. ⏐x 1 4⏐ 5 2 3. ⏐2x 2 7⏐ 5 3

4. ⏐3 2 x⏐ 5 6 5. ⏐2 2 4x⏐ 5 5 6. ⏐5x 1 2⏐ 5 3

7. ⏐3x 1 3⏐ 5 7 8. ⏐3 2 2x⏐5 7 9. ⏐ 1 }

2 x 2 2⏐ 5 14

10. ⏐7x 2 3⏐ 5 4 11. ⏐2.4 2 1.2x⏐ 5 10.9 12. ⏐2x 2 1 }

2 ⏐5 6

Solve the equation.

13. ⏐x⏐ 5 7 14. ⏐x⏐ 5 17 15. ⏐x⏐ 5 0 16. ⏐3x⏐ 5 9

17. ⏐x 1 4⏐ 5 8 18. ⏐2x 2 4⏐ 5 10 19. ⏐7 2 2x⏐ 5 5 20. ⏐3x 2 5⏐ 5 4



Rewrite the absolute value inequality as a compound inequality.

21. ⏐x 1 4⏐ < 7 22. ⏐23x⏐ < 9 23. ⏐x 2 2⏐ ≥ 5

24. ⏐x 1 5⏐ > 6 25. ⏐4x 1 2⏐ ≤ 7 26. ⏐3 2 x⏐ > 4

27. ⏐2x 1 2⏐ ≥ 12 28. ⏐5 2 x⏐ ≤ 2 29. ⏐1.5x 1 2.3⏐ ≥ 3.3

30. ⏐ 2 }

3 x 1 5⏐ ≤ 1 31. ⏐0.3 2 1.2x⏐ < 1.7 32. ⏐

4 } 5 2

1 }

3 x⏐ >

3 } 5

Solve the inequality.

33. ⏐x⏐ < 8 34. ⏐x⏐ ≥ 1 35. ⏐x 2 2⏐ ≤ 3

36. ⏐x 1 1⏐ > 3 37. ⏐2x 2 3⏐ < 9 38. ⏐4 2 x⏐ ≤ 12

39. ⏐2x 2 4⏐ > 1 40. ⏐4 2 3x⏐ ≤ 2 41. ⏐11 2 3x⏐ ≥ 2

42. Travel You plan to visit both your sister and your friend next Saturday. Your sister's house is 10 miles west of your house. Your friend's house is 10 miles east of your house. All three houses lie on the same straight road. Write an absolute value inequality that represents all the possible distances you may be from your house from the time you leave until the time you return home.

43. Tire Pressure A tire manufacturer suggests that the tire pressure for a certain tire style should be within 2 PSI of the recommended operating pressure of 32 PSI. Write an absolute value inequality that represents the range of pressures the tire should sustain under normal conditions.

LESSON


Name ——————————————————————— Date ————————————



2.3 Represent Relations and Functions


MM2A5a

Your Notes

Goal p Represent relations and graph linear functions.

VOCABULARY

Relation

Domain

Range

Function

Equation in two variables

Independent variable

Dependent variable

Solution of an equation in two variables

Linear function

One-to-one

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Your Notes

Consider the relation given by the ordered pairs (1, 0), (0, 22), (22, 3), and (3, 1).

a. Identify the domain and range.

b. Represent the relation using a graph.

c. Use the vertical line test to tell whether the relation is a function.

d. If the relation is a function, tell whether it is a one-to-one function.

Solution

a. The domain consists of all x-coordinates: ,, , and . The range consists of all

y-coordinates , , , and .

b. See the graph below.

x

y

1

1

c. The relation a function because no single vertical line intersects more than one point of the graph.

d. The function a one-to-one function because no two values in the domain have the same value in the range.

Example 1 Represent relations and identify functions

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Your Notes

Identify the domain and range of the given relation. Represent the relation using a graph. Use the vertical line test to tell whether the relation is a function. If it is a function, tell whether it is a one-to-one function.

1. (2, 2), (1, 23), (0, 22), (3, 3)

x

y

1

1


Example 2 Graph an equation in two variables

Graph the equation y 5 22x 2 2.

x

y

1

1SolutionStep 1 Construct a table of values.

x 22 21 0 1 2

y

Step 2 Plot the points. Notice that they all lie on a .

Step 3 the points with a line.

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Your Notes

Tell whether the function is linear. Then evaluate the function when x 5 23.

a. f(x) 5 6x 1 10 b. g(x) 5 2x2 1 4x 2 1

Solution

a. The function f is because it has the form f(x) 5 mx 1 b.

f(x) 5 6x 1 10 Write original function.

f( ) 5 6( ) 1 10 Substitute for x.

5 Simplify.

b. The function g is because it has an x2-term.

g(x) 5 2x2 1 4x 2 1 Write original function.

g( ) 5 2( )2 1 4( ) 2 1 Substitutefor x.

5 Simplify.

Example 3 Classify and evaluate functions

Graph the equation.

2. y 5 x 1 2 3. y 5 22x 1 1

x

y

1

1

x

y

1

1

4. Tell whether the function f(x) 5 3x2 2 4x 1 1 is linear. Then evaluate the function when x 5 1 and x 5 2.


Homework

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Name ——————————————————————— Date ————————————

LESSON

2.3 PracticeIdentify the domain and range of the given relation. Then tell whether the relation is a function. If the relation is a function, tell whether it is a one-to-one function.

1. (23, 1), (21, 0), (2, 4) 2. (22, 1), (0, 0), (2, 3)

3. (1, 2), (3, 1), (5, 2) 4. (22, 1), (3, 5), (22, 4)

5. (0, 2), (21, 22), (3, 21) 6. (21, 2), (2, 0), (2, 2)

Graph the relation and tell whether the relation is a function.

7. x 0 1 2 3 4

y 3 1 2 4 2

8. x 22 21 0 0 1

y 23 21 1 3 5

x

y

1

1

2

2

x

y

9. x 23 22 21 0 1

y 4 2 4 6 4

10. x 24 22 0 2 4

y 3 0 23 0 3

x

y

1

1

2

2

x

y



Use the vertical line test to determine whether the relation is a function.

11.

x

y

1

1

12.

x

y

1

1

13.

x

y

1

1

Graph the equation.

14. y 5 x 1 1 15. y 5 2x 2 2 16. y 5 2x 1 3

x

y

1

1

x

y

1

1

x

y

1

1

17. y 5 23x 2 1 18. y 5 4x 2 3 19. y 5 22x 1 2

x

y

1

1

x

y

1

1

x

y

1

1

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


20. y 5 5x 21. y 5 3 22. y 5 1 }

2 x 2 1

x

y

1

1

x

y

1

1

x

y

1

1

23. World Population The table shows the estimated world

0 1960 1980 2000 20200

1

2

3

4

5

6

7

8

YearP

op

ula

tio

n (

billio

ns)population in billions for the given year. Use a coordinate

plane to graph the data. Identify the domain and range of the relation. Is the relation a one-to-one function? Explain.

Year 1960 1970 1980 1990 2000

Population 3.0 3.7 4.5 5.3 6.1



2.4 Use Absolute Value Functions and Transformations


MM2A1b

Your Notes

Goal p Graph and write absolute value functions.

VOCABULARY

Absolute value function

Vertex of an absolute value graph

Transformation

Translation

Reflection

Axis of symmetry

Zeros

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Your Notes

Graph y 5 22⏐x 1 3⏐1 2. Identify the intercepts, zeros, and intervals of increase and decrease. Compare the graph with the graph of y 5⏐x⏐.

1. Identify and plot the vertex,

x

y

1

1

(h, k) 5 ( , ).

2. Plot another point on the graph such as ( , ).Use symmetry to plot a third point, ( , ).

3. Connect the points with a graph.

4. Examine the graph. The x-intercepts are and , which are also the of the function, and the y-intercept is . The function is increasing when and decreasing when .

5. Compare with y 5⏐x⏐. The graph of y 5 22⏐x 1 3⏐1 2 is the graph of y 5⏐x⏐first stretched by a factor of , then reflected in the , and finally translated units and units.

Example 1 Graph a function of the form y 5 a⏐x 2 h⏐1 k

1. Graph the function y 5 21}2⏐x 2 1⏐2 2. Compare

the graph with the graph of y 5 ⏐x⏐.

x

y

1

1


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Your Notes

Write a function for the graph shown.

SolutionThe vertex of the graph is

x

y

1

1

(1, 22)

(3, 2)( , ). So, the equation has the form y 5 a⏐x 2 ⏐1 .Substitute the coordinates of the point ( , ) into the equation and solve for a.

5 a⏐ ⏐1 Substitute for x and for y.

5 a Solve for a.

An equation for the graph is y 5 .

Example 2 Write an absolute value function

2.

x

y

1

1

(21, 3)

(23, 7)

Checkpoint Write a function for the graph shown.

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Your Notes

The graph of a function y 5 f(x)

x

y

1

1

(0, 4)

(1, 1)

(4, 3)

is shown. Sketch the graph of the given function.

a. y 51}2 p f(x)

b. y 5 2f(x 2 1) 1 2

Solution

a. The graph of y 51}2 p f(x) is

x

y

1

1

the graph of y 5 f(x) shrunk

by a factor of .

To draw the graph, find y

for each labeled point (x, y) on the graph of f(x). Plot points

1x, y2 and connect them.

b. The graph of y 5 2f(x 2 1) 1 2

x

y

1

1

is the graph of y 5 f(x) in the x-axis, then

translated unit and units. To draw the

graph, first reflect the labeledpoints and connect their images. Then translate and

Example 3 Apply transformations to a graph

3. y 5 21}4 p f (x)

x

y

1

1

Checkpoint Use the graph of y 5 f(x) in Example 3 to graph the given function.

Homework

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Match the function with its graph.

1. f (x) 5 ⏐x 1 3⏐ 2. f (x) 5 ⏐x 2 3⏐

3. f (x) 5 ⏐x⏐ − 3 4. f (x) 5 ⏐x⏐ 1 3

5. f (x) 5 3⏐x⏐ 6. f (x) 5 1 }

3 ⏐x⏐

A.

x

y3

1

B.

x

y

1

1

C.

x

y1

1

D.

x

y

1

1

E.

x

y

1

1

F.

x

y

1

1

LESSON

2.4 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


For the function, (a) tell whether the graph opens up or down, (b) identify the vertex, and (c) tell whether the function is wider, narrower, or the same width as the graph of y 5 ⏐x⏐.

7. y 5 22⏐x⏐ 8. f(x) 5 ⏐x 2 3⏐ 9. y 5 ⏐x 1 1⏐ 1 4

10. f (x) 5 5⏐x 2 2⏐ 2 4 11. y 5 28⏐x 1 5⏐ 2 4 12. f (x) 5 4⏐x⏐ 2 4

13. y 5 1 }

2 ⏐x 2 2⏐ 1 2 14. f (x) 5 2

2 } 3 ⏐x 1 6⏐ 2 1 15. y 5 2

3 } 2 ⏐x 2 1⏐ 1 2

Write an equation of the graph shown. Identify the intercepts, zeros, and intervals of increase and decrease.

16.

x

y

1

1

17.

x

y

2

2

18.

x

y

1

1



Let f (x) = x. Sketch the graphs of f (x) and y on the same coordinate plane.

19. y 5 f (x) 1 1 20. y 5 f (x 2 2) 21. y 5 22f (x)

x

y

1

1

x

y

1

1

x

y

1

1

In Exercises 22–24, use the following information.

A-Frame House The roof line of an A-frame house follows the path given by y 5 22⏐x⏐ 1 20. Each unit on the coordinate plane represents one foot.

22. Graph the function for 210 ≤ x ≤ 10.

x

y

2

2

23. Find the vertex of the graph.

24. Assuming the x-axis is parallel to the ground, what does the y-value of the vertex tell you about the house?

LESSON


Name ——————————————————————— Date ————————————



2.5 Use Piecewise FunctionsGoal p Evaluate, graph, and write piecewise functions.Georgia

PerformanceStandard(s)

MM2A1a, MM2A1b

Your Notes

VOCABULARY

Piecewise function

Points of discontinuity

Step function

Extrema

Average rate of change

Evaluate the function when x 5 3.

g(x) 5 h x 1 1, if x ≤ 24x 2 1, if x > 2

Solution

g(x) 5 Because ,use equation.

g( ) 5 Substitute for x.

5 Simplify.

Example 1 Evaluate a piecewise function

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Your Notes

1. f(x) 5 h3x 2 2, x ≤ 01}4 x 1 1, x > 0

Checkpoint Evaluate the function when x 5 24 and x 5 2.

Graph f (x) 5 h22x 1 1, if x < 211}4 x, if 21 ≤ x ≤ 1

3, if x > 1

.

Find the x-coordinates for which there are points of discontinuity.

Solution

1. To the of x 5 21, graph

x

y

1

1

y 5 22x 1 1. Use an dot at (21, ) because the equationy 5 22x 1 1 apply when x 5 21.

2. From x 5 21 to x 5 1, inclusive,

graph y 51}4 x. Use dots

at both 121, 2 and 11, 2 because the equation

y 51}4 x applies to both x 5 21 and x 5 1.

3. To the right of x 5 1, graph y 5 3. Use an dot at (1, ) because the equation y 5 3 apply when x 5 1.

4. Examine the graph. Because there are gaps in the graph at x 5 and x 5 , these are the x-coordinates for which there are points of

.

Example 2 Graph a piecewise function

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Your Notes

2. Graph the following function and find the x-coordinates for which there are points of discontinuity.

f(x) 5 h 1}2 x 1 1, if x < 0

x 2 1, if 0 ≤ x < 2 x, if x ≥ 2 x

y

1

1


Write a piecewise function for the

x

y

1

1

step function shown. Describe any intervals over which the function is constant.

For x between and , including x 5 1, the graph is the line segment given by y 5 1. For x between and , including x 5 2, the graph is the line segment given by y 5 2. For x between and , including x 5 3, the graph is the line segment given by y 5 3. So, a for the graph is as follows:

The intervals over which the function is are , , .

Example 3 Write a piecewise function

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Your Notes

Write the function f(x) 5 3⏐x 1 1⏐ 2 2 as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.

1. Graph the function. Find and label

x

y

2

2

the vertex, one point to the left of the vertex, and one point to the right of the vertex. The graph shows one minimum value of

, located at the vertex, and no maximum.

2. Find linear equations that represent each piece of the graph.

Left of vertex:

m 5 5

y 2 5 23(x 2 ( ))

y 5 x 2

Right of vertex:

m 5 5

y 2 5 3(x 2 )

y 5 x 1

So, the function may be written as

f(x) 5 h23x 2 5, if x < 21

3x 1 1, if x ≥ 21.

The extremum is a located at the vertex (21, 22). The rate of change of the function is when x < 21 and when x > 21.

Example 4 Write and analyze a piecewise function

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Your Notes

Homework

3. Graph the following function and find the x-coordinates for which there are points of discontinuity.

f(x) 5 h 1}2 x 1 1, if x < 0

x 2 1, if 0 ≤ x < 2 x, if x ≥ 2 x

y

1

1

4. Write a piecewise function for the step function shown. Describe any intervals over which the function is constant.

x

y

1

1

5. Write the function f (x) 5 ⏐x 1 4⏐ 2 1 as a piecewise function. Find any extrema as well as the rate of change to the left and to the right of the vertex.


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Evaluate the piecewise function when (a) x 5 21, (b) x 5 0, and (c) x 5 2.

1. f(x) 5 h x 1 3, if x < 0 2x 2 1, if x ≥ 0

2. h(x) 5 h 2x, if x < 21 3x 1 1, if x ≥ 21

3. g(x) 5 h 1 2 2x, if x ≤ 22 x, if 22 < x ≤ 0 3 2 x, if x > 0

4. f (x) 5 h x, if x ≤ 23 2x, if 23 < x ≤ 1 x 1 2, if x > 1

Graph the piecewise function. Find the x-coordinates for which there are points of discontinuity.

5. f(x) 5 h x 1 2, if x < 0 x 2 3, if x ≥ 0

6. h(x) 5 h 3 2 x, if x < 2 1 1 x, if x ≥ 2

x

y

1

1

x

y

1

1

7. f (x) 5 h x, if x ≥ 1 2x, if x < 1

8. g(x) 5 h 1, if x ≤ 2 x, if 2 < x ≤ 3 x 1 3, if x ≥ 3

x

y

1

1

x

y

2

2

LESSON

2.5 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Write a piecewise function for the step function shown. Describe any intervals over which the function is constant.

9.

x

y

1

1

10.

x

y

1

1

11.

x

y

1

1

Write the function as a piecewise function. Find any extrema as well as the rate of change of the function to the left and to the right of the vertex.

12. f (x) 5 ⏐x⏐ 1 2 13. f (x) 5 ⏐x 2 2 ⏐ 1 1 14. f (x) 5 3⏐x 2 2 ⏐ 2 4



Words to ReviewGive an example of the vocabulary word.

Verbal model

Relation

Range

Equation in two variables

Dependent variable

Linear function

Extraneous solution

Domain

Function

Independent variable

Solution of an equation in two variables

One-to-one function

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Absolute value function

Transformation

Reflection

Points of discontinuity

Extrema

Vertex of an absolute value graph

Translation

Piecewise function

Step function

Average rate of change

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