Stitz-Zeager_College_Algebra_e-book

Horizontal stretch by a factor of 2 solution we build

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Unformatted text preview: −−−−−−− 3 2 multiply each x-coordinate by 2 y = m2 (x) = m1 1 x 2 = 1 x 2 + 3 2 1 We now examine what’s happening to the outputs. From m(x) = −f 2 x + 3 + 1, we see the 2 output from f is being multiplied by −1 (a reflection about the x-axis) and then a 1 is added (a vertical shift up 1). As before, we can determine the correct order by looking at how the point (4, 2) is moved. We have already determined that to make use of the equation f (4) = 2, 1 we need to substitute x = 5. We get m(5) = −f 2 (5) + 3 + 1 = −f (4) + 1 = −2 + 1 = −1. 2 We see that f (4) (the output from f ) is first multiplied by −1 then the 1 is added meaning we first reflect the graph about the x-axis then shift up 1. Theorem 1.4 tells us m3 (x) = −m2 (x) will handle the reflection. y y (5, 2) 2 2 (−1, 1) (−3, 0) −2 −1 (−3, 0) 1 2 3 4 5 1 −2 −1 x −1 1 2 3 4 x 5 (−1, −1) −2 −2 (5, −2) reflect across x-axis y = m2 (x) = 1 x 2 + 3 2 −− − − − −→ −−−−−− multiply each y -coordinate by −1 y = m3 (x) = −m2 (x) = − 1 x 2 + 3 2 1.8 Transformations 99 Finally, to handle the vertical shift, Theorem 1.2 gives m(x) = m3 (x) + 1, and we see that the range of m is (−∞, 1]. y y (−3, 1) 2 (−3, 0) 2 1 (−1, 0) −2 −1 1 2 3...
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