Warm Up

Mastery Objectives Identify, graph, and describe parent functions. Identify and graph transformations of parent functions

Linear and Polynomial Parent Functions

Square Root and Reciprocal Parent Functions

Absolute Value Parent Function

Greatest Integer Function

Example Describe the following characteristics of the graph of the parent function domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

A.D:, R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as, and as,. The graph is decreasing on the interval and increasing on the interval. B.D:, R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as,. As,. The graph is decreasing on the interval and increasing on the interval. C.D:, R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as,. As,. The graph is decreasing on the interval and increasing on the interval. D.D:, R: ; no intercepts. The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as,. As,. The graph is decreasing on the interval and increasing on the interval. Describe the following characteristics of the graph of the parent function f (x) = x 2 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

Vertical and Horizontal Translations

Example 2 Use the graph of f (x) = x 3 to graph the function g (x) = x 3 – 2. Answer:

Example 2 Use the parent graph of f (x) = x 3 to graph the function g (x) = (x – 1) 3. Answer:

Example 2 Use the parent graph of f (x) = x 3 to graph the function g (x) = (x – 1) 3 – 2. Answer:

Example 2 Use the graph of f (x) = x 2 to graph the function g (x) = (x – 2) 2 – 1. A. B. C. D.

Reflections in the Coordinate Axes

Example 3 Describe how the graphs of and g (x) are related. Then write an equation for g (x). Answer: The graph is translated 1 unit up;

Example 3 Describe how the graphs of and g (x) are related. Then write an equation for g (x). Answer: The graph is translated 1 unit to the left and reflected in the x-axis;

Vertical and Horizontal Dilations

Example 4 Identify the parent function f (x) of, and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes. The graph of g(x) is the same as the graph of the reciprocal function expanded vertically because and 1 < 3.

Example 4 Answer: ; g (x) is represented by the expansion of f (x) by a factor of 3.

Transformations with Absolute Value

Warm Up

Example 4 Identify the parent function f (x) of g (x) = –|4x|, and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes. The graph of g (x) is the same as the graph of the absolute value function f (x) = |x| compressed horizontally and then reflected in the x-axis because g (x) = –4(|x|) = –|4x| = –f (4x), and 1 < 4.

Example 4 Answer:f (x) = |x| ; g (x) is represented by the compression of f (x) by a factor of 4 and reflection in the x-axis.

Example A.f (x) = x 3 ; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of. B.f (x) = x 3 ; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the x-axis. C.f (x) = x 3 ; g(x) is represented by the reflection of the graph of f (x) in the x-axis. D.f (x) = x 3 ; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the y-axis. Identify the parent function f (x) of g (x) = – (0.5x) 3 and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes.

Peicewise Example Graph.

Example 5 Graph the function. A. B. C. D.

Example 7 Use the graph of f (x) = x 2 – 4x + 3 to graph the function g(x) = |f (x)|. The graph of f (x) is below the x-axis on the interval (1, 3), so reflect that portion of the graph in the x-axis and leave the rest unchanged.

Example 7 Answer:

Example 7 Use the graph of f (x) = x 2 – 4x + 3 to graph the function h (x) = f (|x|).

Example 7 Replace the graph of f (x) to the left of the y-axis with a reflection of the graph to the right of the y-axis Answer:

Example 7 Use the graph of f (x) shown to graph g(x) = |f (x)| and h (x) = f (|x|). A. B. C. D.

Homework:  Study for the quiz on  P 52: 1,2,4,7,10,15,24,31,32,41,42,46