114 CHAPTER 2 Functions and Their Graphs Now Work PROBLEM 45 The domain of f is all real numbers, or , ( ) −∞ ∞ . The range of f is 5, [ ) − ∞ . Horizontal Shifts • If the argument x of a function f is replaced by x h h , 0, − > the graph of the new function y f x h ( ) = − is the graph of f shifted horizontally right h units. • If the argument x of a function f is replaced by x h h , 0, + > the graph of the new function y f x h ( ) = + is the graph of f shifted horizontally left h units. In Words For y f x h h , 0 ( ) = − > , add h to each x -coordinate on the grapha of ( ) = y f x to shift the graph right h units. For y f x h h , 0, ( ) = + > subtract h from each x -coordinate on the graph of ( ) = y f x to shift the graph left h units. Combining Vertical and Horizontal Shifts Graph the function ( ) = + − f x x 3 5. Find the domain and range of f . Solution EXAMPLE 2 We graph f in steps. First, note that f is basically an absolute value function, so begin with the graph of = y x as shown in Figure 57(a). Next, to get the graph of = + y x 3 , shift the graph of = y x horizontally 3 units to the left. See Figure 57(b). Finally, to get the graph of = + − y x 3 5, shift the graph of = + y x 3 vertically down 5 units. See Figure 57(c). Note the points plotted on each graph. Using key points can be helpful in keeping track of the transformation that has taken place. We are led to the following conclusions: NOTE Vertical shifts result when adding or subtracting a real number k after performing the operation suggested by the basic function, while horizontal shifts result when adding or subtracting a real number h to or from x before performing the operation suggested by the basic function. For example, the graph of ( ) = + f x x 3 is obtained by shifting the graph of = y x up 3 units, because we evaluate the square root function first and then add 3. The graph of ( ) = + g x x 3 is obtained by shifting the graph of = y x left 3 units, because we first add 3 to x before we evaluate the square root function. j Figure 57 x y (0, 0) 5 Replace x by x 1 3; Horizontal shift left 3 units. (2, 2) (22, 2) y 5 0 x 0 5 (a) x y (23, 25) 2 (21, 23) (25, 23) y 5 0 x 1 30 2 5 5 (c) x y (23, 0) 2 Subtract 5; Vertical shift down 5 units. (21, 2) (25, 2) y 5 0 x 1 30 5 (b) Check: Graph ( ) = = + − Y f x x 3 5 1 and compare the graph to Figure 57(c). In Example 2, if the vertical shift had been done first, followed by the horizontal shift, the final graph would have been the same. (Try it for yourself.) Now Work PROBLEMS 47 AND 71
RkJQdWJsaXNoZXIy NjM5ODQ=