3.2 Transformations of the Graphs of Functions Copyright © Cengage Learning. All rights reserved.

Vertical Translations If f is a function and k is a positive number, then The graph of y = f(x) + k is identical to the graph of y = f(x) except that it is translated k units upward. The graph of y = f(x) – k is identical to the graph of y = f(x) except that it is translated k units downward.

Horizontal Translations If f is a function and k is a positive number, then The graph of y = f(x – k) is identical to the graph of y = f(x) except that it is translated k units to the right. The graph of y = f(x + k) is identical to the graph of y = f(x) except that it is translated k units to the left.

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Reflections If f is a function, then The graph of y = –f(x) is identical to the graph of y = f(x) except that it is reflected about the x-axis. The graph of y = f(–x) is identical to the graph of y = f(x) except that it is reflected about the y-axis.

Vertical Stretching If f is a function and k > 1, then The graph of y = kf(x) can be obtained by stretching the graph of y = f(x) vertically by multiplying each value of f(x) by k.

Vertical Shrinking If f is a function and 0 < k < 1, then The graph of y = kf(x) can be obtained by shrinking the graph of y = f(x) vertically by multiplying each value of f(x) by k.

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Summarizing the Ideas If f is a function and k represents a positive number then