1. Copyright © 2007 Pearson Education, Inc. Slide 2-1

2. Copyright © 2007 Pearson Education, Inc. Slide 2-2 Chapter 2: Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs 2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

3. Copyright © 2007 Pearson Education, Inc. Slide 2-3 2.2 Vertical Translations of Graphs Vertical Shifting of the Graph of a Function If the graph of is obtained by shifting the graph of upward a distance of c units. The graph of is obtained by shifting the graph of downward a distance of c units.

4. Copyright © 2007 Pearson Education, Inc. Slide 2-4 2.2 Example of a Vertical Shift with the Calculator Give the equation of each function in the graphs below. Solution: a. b. Figure 21b pg 2-30

5. Copyright © 2007 Pearson Education, Inc. Slide 2-5 2.2 Horizontal Translations of Graphs Horizontal Shifting of the Graph of a Function If the graph of is obtained by shifting the graph of right a distance of c units. The graph of is obtained by shifting the graph of left a distance of c units. Figure 22 pg 2-31

6. Copyright © 2007 Pearson Education, Inc. Slide 2-6 2.2 Example of a Horizontal Shift Give the equation of each function in the graphs below. Solution: a. b. Figure 23b pg 2-34

7. Copyright © 2007 Pearson Education, Inc. Slide 2-7 2.2 Example of Vertical and Horizontal Shifts Describe how the graph of would be obtained by translating the graph of Sketch the graphs on the same xy- plane. Solution is shifted 2 units left and 6 units down. The graph changes from decreasing to increasing at the point (  2,  6). y = |x+ 2|  6 y = |x|

8. Copyright © 2007 Pearson Education, Inc. Slide 2-8 2.2 Effects of Shifts on Domain and Range The domains and ranges of functions may or may not be affected by translations. –If the domain (or range) is a horizontal (or vertical) shift will not affect the domain (or range). –If the domain (or range) is not a horizontal (or vertical) shift will affect the domain (or range). Example Determine the domain and range of the following shifted graph.

9. Copyright © 2007 Pearson Education, Inc. Slide 2-9 2.2 Applying a Horizontal Shift to an Equation Model The table lists U.S. sales of Toyota vehicles in millions. a. Find the corresponding equation that allows direct input of the year. b. Predict the sale in 2005. Solution YearJuveniles 19981.4 19991.5 20001.6 20011.7 20021.8 The data can be modeled by the equation y =.1x + 1.4, where x = 0 corresponds to 1998, x = 1 to 1999, and so on. a. Because 1998 corresponds to 0, the graph of y =.1x + 1.4 would have to be shifted 1998 units to the right. Thus, the equation becomes y =.1(x – 1998) + 1.4. b. y =.1(2005 – 1998) + 1.4 = 2.1 Toyota vehicle sales in 2005 is predicted to be about 2.1 million.