Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.5 Transformations of Functions.

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Presentation transcript:

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Transformations of Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Graphs of Common Functions Seven functions that are frequently encountered in algebra are: It is essential to know the characteristics of the graphs of these functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Graphs of Common Functions (continued) is known as the constant function. The domain of this function is: The range of this function is: c The function is constant on This function is even.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Graphs of Common Functions (continued) is known as the identity function. The domain of this function is: The range of this function is: The function is increasing on This function is odd.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Graphs of Common Functions (continued) is the absolute value function. The domain of this function is: The range of this function is: The function is decreasing on and increasing on This function is even.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Graphs of Common Functions (continued) is the standard quadratic function. The domain of this function is: The range of this function is: The function is decreasing on and increasing on This function is even.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Graphs of Common Functions is the square root function. The domain of this function is: The range of this function is: The function is increasing on This function is neither even nor odd.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Graphs of Common Functions (continued) is the standard cubic function. The domain of this function is: The range of this function is: The function is increasing on This function is odd.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Graphs of Common Functions (continued) is the cube root function. The domain of this function is: The range of this function is: The function is increasing on This function is odd.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 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 © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Vertical Shift Use the graph of to obtain the graph of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 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 © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Horizontal Shift Use the graph of to obtain the graph of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 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 © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Reflection about the y-Axis Use the graph of to obtain the graph of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 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 © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Vertically Shrinking a Graph Use the graph of to obtain the graph of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 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 © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Horizontally Stretching and Shrinking a Graph Use the graph of to obtain the graph of