SECTION 2.5 Graphing Techniques: Transformations 113 Table 9 lists some points on the graphs of Y f x x 1 2 ( ) = = and Y h x 2 ( ) = = f x x 4 4 2 ( ) − = − using a TI-84 Plus CE. Notice that each y-coordinate of h is 4 units less than the corresponding y-coordinate of f. To obtain the graph of h from the graph of f , subtract 4 from each y-coordinate on the graph of f .The graph of h is identical to that of f, except that it is shifted down 4 units. See Figure 55. Vertical Shift Down Use the graph of f x x2 ( ) = to obtain the graph of h x x 4. 2 ( ) = − Find the domain and range of h. Solution EXAMPLE 1 Table 9 Figure 55 24 26 6 6 Y2 5 x2 2 4 Y1 5 x2 y 4 (2, 4) (22, 4) (0, 0) y 5 x2 x 4 25 (2, 0) (22, 0) (0, 24) Down 4 units Down 4 units y 5 x2 2 4 The domain of h is the set of all real numbers. The range of h is 4, [ ) − ∞. Now Work PROBLEM 41 Exploration On the same screen, graph each of the following functions: Y x Y x Y x , 3 , 2 1 2 2 2 3 2 ( ) ( ) = = − = + What do you observe? Result Figure 56 illustrates the graphs and a table of values using Desmos. You should have observed the following pattern. With the graph of Y x 1 2 = on the screen (in black), the graph of Y x 3 2 2 ( ) = − shown in red is identical to that of Y x , 1 2 = except it is shifted horizontally to the right 3 units. The graph of Y x 2 3 2 ( ) = + shown in blue is identical to that of Y x , 1 2 = except it is shifted horizontally to the left 2 units. From the table, we see the x-coordinates on Y x 3 2 2 ( ) = − are 3 units larger than they are for Y x 1 2 = for any given y-coordinate. For example, when Y 0 1 = , then x 0 = , and when Y 0 2 = , then x 3 = . Also, we see the x-coordinates on Y x 2 3 2 ( ) = + are 2 units smaller than they are for Y x 1 2 = for any given y-coordinate. For example, when Y 0 1 = , then x 0 = , and when Y 0 3 = , then x 2 =− . Figure 56 Horizontal shift
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