• How do graph piecewise functions?

2.5 Use Piecewise Functions

Example 1 Evaluate a piecewise functionEvaluate a piecewise function

Evaluate the function when x = 3.

2 if,14

2 if,1

xx

xxxg

SolutionBecause ______, use _______ equation.

23 second _______xg 14 x

Substitute ___ for x.3 _________ g 3 134 Simplify.____ 11

2.5 Use Piecewise FunctionsCheckpoint. Evaluate the function when Checkpoint. Evaluate the function when xx = = 4 and 4 and xx = 2. = 2.

0 if,14

10 if,23

1.xx

xxxf

04

4f 243 14

02

2f 124

1

2

3

2.5 Use Piecewise Functions

Example 2 Graph a piecewise functionGraph a piecewise function

Graph 1 if

11 if

1 if

34

112

x

x

x

x

x

xf

Solution

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

1. To the _____ of x = 1, graph y = 2x + 1. Use

an _____ dot at (1, ___ ) because the equation

y = 2x + 1 __________ apply when x = 1.

left

3opendoes not

2. From x = 1 to x = 1, inclusive, graph y = ¼ x.

Use _____ dots at both ( 1, ___ ) and ( 1, ___ )

because the equation y = ¼ x applies to both x =

1 and x = 1.

solid ¼ ¼

2.5 Use Piecewise Functions

Example 2 Graph a piecewise functionGraph a piecewise function

Graph 1 if

11 if

1 if

34

112

x

x

x

x

x

xf

Solution

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

3. To the right of x = 1, graph y = . Use an _____

dot at (1, ___ ) because the equation y =

__________ apply when x = 1.3

open

does not

4. Examine the graph. Because there are gaps in

the graph at x = _____ and x = ___, these are

the x-coordinates for which there are points of

_____________.

1 1

discontinuity

2.5 Use Piecewise FunctionsCheckpoint. Complete the following exercise.Checkpoint. Complete the following exercise.

2 if

20 if

0 if

1

12

1

1.

x

x

x

x

x

x

xf

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

Discontinuity at x = 0 and x = 2

2.5 Use Piecewise Functions

Example 3 Write a piecewise functionWrite a piecewise function

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

For x between ___ and ___, including x = 1, the graph is the line segment given by y = 1.

1 2

xf

,1 21 if x

For x between ___ and ___, including x = 2, the graph is the line segment given by y = 2.

2 3

,2 32 if x

For x between ___ and ___, including x = 3, the graph is the line segment given by y = 3.

3 4

,3 43 if x

So, a _____________ _________ for the graph is as follows:

piecewise

function

The intervals over which the function is ___________ are ____________,

____________, ______________.

constant 21 x32 x 43 x

2.5 Use Piecewise Functions

Example 4 Write and analyze a piecewise functionWrite and analyze a piecewise function

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.

213 xxf

1. Graph the function. Find and label 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. 2,1 2

2 4,3

7,2

2

2.5 Use Piecewise Functions

Example 4 Write and analyze a piecewise functionWrite and analyze a piecewise function

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.

213 xxf

2. Find linear equations that represent each piece

of the graph.

2,1 2

2 4,3

7,2

Left of vertex:

____

m 24 13

3

___3__ xy 4 3

_____ xy 3 5xy 34 9

2.5 Use Piecewise Functions

Example 4 Write and analyze a piecewise functionWrite and analyze a piecewise function

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.

213 xxf

2. Find linear equations that represent each piece

of the graph.

2,1 2

2 4,3

7,2

Right of vertex:

____

m 27 12

3

__3__ xy 7 2

____ xy 3 1xy 37 6

2.5 Use Piecewise Functions

Example 4 Write and analyze a piecewise functionWrite and analyze a piecewise function

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.

213 xxf

So the function may be written as

2,1 2

2 4,3

7,2

1 if

1 if ,13

,53

x

x

x

xxf

The extrema is a ____________ located at the vertex ( 1, 2 ). The rate of change of the function is ____ when x < 1 and ___ when x > 1.3

minimum

3

2.5 Use Piecewise FunctionsCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.4. Write a piecewise function for the step function shown.

Describe any intervals over which the function is constant.

xf

,2 10 if x,0 31 if x,2 43 if x

Constant intervals:

10 x31 x43 x

2.5 Use Piecewise FunctionsCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.5. Write the function as a piecewise

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

14 xxf

1,4

2,6 2,1

xf 4 if x,5x4 if x,3x

minimum: 1,4 rate of change:

4 when 1 x4 when 1 x

2.5 Use Piecewise Functions

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