Use Piecewise Functions 2.5 Use Piecewise Functions Example 1 Evaluate a piecewise function Evaluate the function when x = 3. Solution Because ______, use _______ equation. second Substitute ___ for x. Simplify.

Use Piecewise Functions 2.5 Use Piecewise Functions Checkpoint. Evaluate the function when x = -4 and x = 2.

Use Piecewise Functions 2.5 Use Piecewise Functions Example 2 Graph a piecewise function 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. solid - ¼ ¼

Use Piecewise Functions 2.5 Use Piecewise Functions Example 2 Graph a piecewise function 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

Use Piecewise Functions 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. Discontinuity at x = 0 and x = 2

Use Piecewise Functions 2.5 Use Piecewise Functions Example 3 Write 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. For x between ___ and ___, including x = 2, the graph is the line segment given by y = 2. So, a _____________ _________ for the graph is as follows: piecewise 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

Use Piecewise Functions 2.5 Use Piecewise Functions Example 4 Write 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. 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.

Use Piecewise Functions 2.5 Use Piecewise Functions Example 4 Write 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. Find linear equations that represent each piece of the graph. Left of vertex:

Use Piecewise Functions 2.5 Use Piecewise Functions Example 4 Write 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. Find linear equations that represent each piece of the graph. Right of vertex:

Use Piecewise Functions 2.5 Use Piecewise Functions Example 4 Write 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. 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

Use Piecewise Functions 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:

Use Piecewise Functions 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: