Transformations of Graphs 3.4 Graph functions using vertical and horizontal translations Graph function using stretching and shrinking Graph function using reflections Combine transformations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Vertical Shifts A graph is shifted up or down. The shape of the graph is not changed—only its position. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Horizontal Shifts A graph is shifted left or right. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2006 Pearson Education, Inc Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally. Shift the graph of y = x2 to the left 3 units and downward 2 units. y = x2 y = (x + 3)2 y = (x + 3)2  2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find an equation that shifts the graph of f(x) = x2  2x + 3 left 4 units and down 3 units. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Stretching and Shrinking Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Horizontal Stretching and Shrinking Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the graph of y = f(x) to sketch the graph of each equation. a) y = 2f(x) b) Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution continued b) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Reflections of Graphs Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example For the function f(x) = x2 + x  2 graph its reflection across the x-axis and across the y-axis. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Combining Transformations Transformations of graphs can be combined to create new graphs. For example the graph of y = 3(x + 3)2 + 1 can be obtained by performing four transformations on the graph of y = x2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Combining Transformations continued y = 3(x + 3)2 + 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Describe how the graph of the equation can be obtained by transforming the graph of y = |x|. Then graph the equation. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley