Precalculus Sixth Edition Chapter 1 Functions and Graphs

1. 6 Transformations of Functions Copyright © 2018, 2014, 2010 Pearson Education, Inc. All

Objectives • Recognize graphs of common functions. • Use vertical shifts to graph functions.

Graphs of Common Functions (1 of 8) Seven functions that are frequently encountered in

Graphs of Common Functions (2 of 8) f(x) = c is known as the

Graphs of Common Functions (3 of 8) f(x) = x is known as the

Graphs of Common Functions (4 of 8) This function is even. Copyright © 2018,

Graphs of Common Functions (5 of 8) This function is even. Copyright © 2018,

Graphs of Common Functions (6 of 8) This function is neither even nor odd.

Graphs of Common Functions (7 of 8) This function is odd. Copyright © 2018,

Graphs of Common Functions (8 of 8) This function is odd. Copyright © 2018,

Vertical Shifts Let f be a function and c a positive real number. The

Example: Vertical Shift Solution: The graph will shift vertically up by 3 units. Copyright

Horizontal Shifts Let f be a function and c a positive real number. The

Example: Horizontal Shift Solution: The graph will shift to the right 4 units. Copyright

Reflections of Graphs Reflection about the x-Axis The graph of y = −f(x) is

Example: Reflection about the y-Axis Solution: The graph will reflect across the x-axis. Copyright

Vertically Stretching and Shrinking Graphs Let f be a function and c a positive

Example: Vertically Shrinking a Graph Solution: Copyright © 2018, 2014, 2010 Pearson Education, Inc.

Horizontally Stretching and Shrinking Graphs Let f be a function and c a positive

Example: Horizontally Stretching and Shrinking a Graph Solution: Copyright © 2018, 2014, 2010 Pearson

Example: Graphing Using a Sequence of Transformations Solution: Copyright © 2018, 2014, 2010 Pearson

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Objectives • Recognize graphs of common functions. • Use vertical shifts to graph functions. • Use horizontal shifts to graph functions. • Use reflections to graph functions. • Use vertical stretching and shrinking to graph functions. • Use horizontal stretching and shrinking to graph functions. • Graph functions involving a sequence of transformations. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Graphs of Common Functions (1 of 8) Seven functions that are frequently encountered in algebra are: It is essential to know the characteristics of the graphs of these functions. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Graphs of Common Functions (2 of 8) f(x) = c is known as the constant function. The range of this function is: c This function is even. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Vertical Shifts Let f be a function and c a positive real number. The graph of y = f(x) + c is the graph of y = f(x) shifted c units vertically upward. The graph of y = f(x) − c is the graph of y = f(x) shifted c units vertically downward. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Horizontal Shifts Let f be a function and c a positive real number. The graph of y = f(x + c) is the graph of y = f(x) shifted to the left c units. The graph of y = f(x − c) is the graph of y = f(x) shifted to the right c units. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Reflections of Graphs Reflection about the x-Axis The graph of y = −f(x) is the graph of y = f(x) reflected about the x-axis. Reflection about the y-Axis The graph of y = f(−x) is the graph of y = f(x) reflected about the y-axis. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Vertically Stretching and Shrinking Graphs Let f be a function and c a positive real number. If c > 1, the graph of y = cf(x) is the graph of y = f(x) vertically stretched by multiplying each of its y-coordinates by c. If 0 < c < 1, the graph of y = cf(x) is the graph of y = f(x) vertically shrunk by multiplying each of its y-coordinates by c. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Horizontally Stretching and Shrinking Graphs Let f be a function and c a positive real number. If c > 1, the graph of y = f(cx) is the graph of y = f(x) horizontally shrunk by dividing each of its x-coordinates by c. If 0 < c <1, the graph of y = f(cx) is the graph of y = f(x) horizontally stretched by dividing each of its x-coordinates by c. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved