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Stitz-Zeager_College_Algebra_e-book

# Example 211 find the slope of the line containing the

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Unformatted text preview: f 2 Solution. We build up to a formula for g (x) using intermediate functions as we’ve seen in previous examples. We let g1 take care of our ﬁrst step. Theorem 1.2 tells us g1 (x) = f (x)+2 = x2 +2. Next, we reﬂect the graph of g1 about the x-axis using Theorem 1.4: g2 (x) = −g1 (x) = − x2 + 2 = −x2 − 2. We shift the graph to the right 1 unit, according to Theorem 1.3, by setting g3 (x) = g2 (x − 1) = −(x − 1)2 − 2 = −x2 + 2x − 3. Finally, we induce a horizontal stretch by a factor of 2 2 1 1 1 using Theorem 1.6 to get g (x) = g3 2 x = − 2 x + 2 2 x − 3 which yields g (x) = − 1 x2 + x − 3. 4 We use the calculator to graph the stages below to conﬁrm our result. shift up 2 units −− − − − −→ −−−−−− add 2 to each y -coordinate y = f (x) = x2 y = g1 (x) = f (x) + 2 = x2 + 2 reﬂect across x-axis −− − − − −→ −−−−−− multiply each y -coordinate by −1 y = g1 (x) = 11 x2 +2 You really should do this once in your life. y = g2 (x) = −g1 (x) = −x2 − 2 1.8 Transformations 103 shift right 1...
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