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 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

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.

2.2 Example of a Vertical Shift with the Calculator Give the equation of each function graphed. Solution: Figure 21b pg 2-30

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 to the right a distance of c units. The graph of is obtained by shifting the graph of to the left a distance of c units. Figure 22 pg 2-31

2.2 Example of a Horizontal Shift Give the equation of each function in the graphs below. Solution: Figure 23b pg 2-34

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|

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.

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. Use the equation to estimate U.S. sales of Toyota vehicles in 2007. Solution Year Vehicles 2000 1.6 2001 1.7 2002 1.8 2003 1.9 2004 2.0 The data can be modeled by the equation y = 0.1x + 1.6, where x = 0 corresponds to 2000, x = 1 to 2001, and so on. a. Because 2000 corresponds to 0, the graph of y = 0.1x + 1.6 would have to be shifted 2000 units to the right. Thus, the equation becomes y = 0.1(x – 2000) + 1.6. b. y = 0.1(2007 – 2000) + 1.6 = 2.3 The model estimates U.S. sales of Toyota vehicles in 2007 to be about 2.3 million.