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Chapter 1 | Functions and Graphs
Figure 1.27 (a) The graph of y =− f ( x ) is the graph of y = f ( x ) reflected about the x -axis. (b) The graph of y = f (− x ) is the graph of y = f ( x ) reflected about the y -axis.
If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function f ( x ), the graph of the related function y = cf ⎛ ⎝ a ( x + b ) ⎞ ⎠ + d can be obtained from the graph of y = f ( x ) by performing the transformations in the following order. 1. Horizontal shift of the graph of y = f ( x ). If b >0, shift left. If b <0, shift right. 2. Horizontal scaling of the graph of y = f ( x + b ) by a factor of | a |. If a <0, reflect the graph about the y -axis. 3. Vertical scaling of the graph of y = f ( a ( x + b )) by a factor of | c |. If c <0, reflect the graph about the x -axis. 4. Vertical shift of the graph of y = cf ( a ( x + b )). If d >0, shift up. If d <0, shift down. We can summarize the different transformations and their related effects on the graph of a function in the following table.
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