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Translations and Special Functions Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

Reflecting the Graph Across the x and y-Axis

Multiplying the x -values by −1 will reverse the graph horizontally. This is called reflecting the graph across the y-axis . Multiplying the y-values by −1 will reverse the graph vertically. This is called reflecting the graph across the x-axis .

Fig. 5.14

Fig. 5.15

When a function is even, reflecting the graph across the y -axis does not change the graph. When a function is odd, reflecting the graph across the y -axis is the same as reflecting it across the x -axis.

We can use function notation to summarize these transformations.

y = af ( x + h ) + k

• If h is positive, the graph is shifted to the left h units.
• If h is negative, the graph is shifted to the right h units.
• If k is positive, the graph is shifted up k units.
• If k is negative, the graph is shifted down k units.
• If a > 1, the graph is vertically stretched. The larger a is, the greater the stretch.
• If 0 < a < 1, the graph is vertically compressed. The closer to 0 a is, the greater the compression.
• The graph of − f ( x ) is reflected across the x -axis.
• The graph of f (− x ) is reflected across the y -axis.

The graphs below are various transformations of the graph of y = | x |.

Fig. 5.16

Fig. 5.17

Fig. 5.18

Fig. 5.19

Examples

The graph of y = f ( x ) is given in Figure 5.20. Sketch the transformations. We will sketch the graph by moving the points (−4, 5), (−1, −1), (1, 3), and (4, 0).

Fig. 5.20

• y = f ( x + 1) − 3

Table 5.1

 Original point Left 1 x − 1 Down 3 y − 3 Plot this point (−4, 5) −4 − 1 = −5 5 − 3 = 2 (−5, 2) (−1,−1) −1 − 1 = − 2 −1 − 3 = −4 (−2,−4) (1, 3) 1 − 1 = 0 3 − 3 = 0 (0, 0) (4, 0) 4 − 1 = 3 0 − 3 = −3 (3,−3)

Fig. 5.21

• y = − f ( x )

Table 5.2

 Original point x does not change x Opposite of y      − y Plot this point (−4, 5) −4 −5 (−4,−5) (−1,−1) −1 −(−1) = 1 (−1, 1) (1, 3) 1 −3 (1,−3) (4, 0) 4 −0 = 0 (4, 0)

Fig. 5.22

• y = 2 f ( x − 3)

Table 5.3

 Original point Right 3 x + 3 Stretched 2 y Plot this point (−4, 5) −4 + 3 = −1 2(5) = 10 (−1, 10) (−1,−1) −1 + 3 =2 2(−1) = −2 (2,−2) (1, 3) 1 + 3 = 4 2(3) = 6 (4, 6) (4, 0) 4 + 3 = 7 2(0) = 0 (7, 0)

Fig. 5.23

• Table 5.4

 Original point Opposite of x − x Compressed and up 2 y + 2 Plot this point ( − 4, 5) − (−4) = 4 (5) + 2 = (4, ) ( − 1, − 1) −(−1)= 1 (−1) + 2 = (1, ) (1, 3) − 1 (3) + 2 = (− 1, ) (4, 0) − 4 (0) + 2 = 2 (−4, 2)

Fig. 5.24

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