1. 3.2 Transformations of the Graphs of Functions
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2. 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.

3. 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.

4. Example 1

5. Example 2

6. Example 3

7. 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.

8. 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.

9. 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.

10. Example 4

11. Example 5

12. Example 6

13. Example 7

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