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Essential Questions

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Monday, February 22

Essential Questions

- How do graph piecewise functions?

2.5

Use Piecewise Functions

Evaluate a piecewise function

Example 1

Evaluate the function when x = 3.

Solution

Because ______, use _______ equation.

second

Substitute ___ for x.

Simplify.

2.5

Use Piecewise Functions

Checkpoint. Evaluate the function when x = -4 and x = 2.

2.5

Use Piecewise Functions

Graph a piecewise function

Example 2

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

Graph

Solution

- 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

open

does not

- 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.

- ¼

¼

solid

2.5

Use Piecewise Functions

Graph a piecewise function

Example 2

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

Graph

Solution

- To the right of x = 1, graph y = 3. Use an _____ dot at (1, ___ ) because the equation y = 3 __________ apply when x = 1.

open

does not

- 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 _____________.

discontinuity

2.5

Use Piecewise Functions

Checkpoint. Complete the following exercise.

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

and x = 2

Discontinuity at x = 0

2.5

Use Piecewise Functions

Write a piecewise function

Example 3

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.

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

piecewise

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

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

function

The intervals over which the function is ___________ are ____________,

____________, ______________.

constant

2.5

Use Piecewise Functions

Write and analyze a piecewise function

Example 4

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.

- 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.

2.5

Use Piecewise Functions

Write and analyze a piecewise function

Example 4

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.

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

Left of vertex:

2.5

Use Piecewise Functions

Write and analyze a piecewise function

Example 4

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.

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

Right of vertex:

2.5

Use Piecewise Functions

Write and analyze a piecewise function

Example 4

So the function may be written as

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.

minimum

2.5

Use Piecewise Functions

Checkpoint. Complete the following exercises.

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

Constant intervals:

2.5

Use Piecewise Functions

Checkpoint. Complete the following exercises.

- 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.

minimum:

rate of change:

2.5

Use Piecewise Functions

Pg. 52, 2.5 #1-14