281 2.7 Graphing Techniques Reflecting Forming the mirror image of a graph across a line is called reflecting the graph across the line. EXAMPLE 2 Reflecting Graphs across Axes Graph each function. (a) g1x2 = -2x (b) h1x2 = 2-x SOLUTION (a) The tables of values for g1x2 = -2x and ƒ1x2 = 2x are shown with their graphs in Figure 74. As the tables suggest, every y-value of the graph of g1x2 = -2x is the negative of the corresponding y-value of ƒ1x2 = 2x. This has the effect of reflecting the graph across the x-axis. –4 –2 2 4 –2 2 x y 0 f(x) = !x g(x) = –!x Figure 74 x ƒ1x2 =!x g1x2 = −!x 0 0 0 1 1 -1 4 2 -2 (b) The domain of h1x2 = 2-x is 1-∞, 04, while the domain of ƒ1x2 = 2x is 30, ∞2. Choosing x-values for h1x2 that are negatives of those used for ƒ1x2, we see that corresponding y-values are the same. The graph of h is a reflection of the graph of ƒ across the y-axis. See Figure 75. x y 4 4 –4 0 f(x) = √x h(x) = √–x Figure 75 x ƒ1x2 =!x h1x2 =!−x -4 undefined 2 -1 undefined 1 0 0 0 1 1 undefined 4 2 undefined S Now Try Exercises 27 and 33. The graphs in Example 2 suggest the following generalizations. Reflecting across an Axis The graph of y = −ƒ1x2 is the same as the graph of y = ƒ1x2 reflected across the x-axis. (If a point 1x, y2 lies on the graph of y = ƒ1x2, then 1x, -y2 lies on this reflection.) The graph of y =ƒ1 −x2 is the same as the graph of y = ƒ1x2 reflected across the y-axis. (If a point 1x, y2 lies on the graph of y = ƒ1x2, then 1-x, y2 lies on this reflection.)
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