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Unformatted text preview: e horizontal asymptote of the graph. Solution. 1. To find T (0), we replace every occurrence of the independent variable t with 0 to obtain T (0) = 70 + 90e−0.1(0) = 160. This means that the coffee was served at 160◦ F. 2. To graph y = T (t) using transformations, we start with the basic function, f (t) = et . As we have already remarked, e ≈ 2.718 > 1 so the graph of f is an increasing exponential with y -intercept (0, 1) and horizontal asymptote y = 0. The points −1, e−1 ≈ (−1, 0.37) and (1, e) ≈ (1, 2.72) are also on the graph. Since the formula T (t) looks rather complicated, we rewrite T (t) in the form presented in Theorem 1.7 and use that result to track the changes to our three points and the horizontal asymptote. We have T (t) = 90e−0.1t + 70 = 90f (−0.1t) + 70. Multiplication of the input to f , t, by −0.1 results in a horizontal expansion by a factor of 10 as well as a reflection about the y -axis. We divide each of the x values of our points by −0.1 (which amounts to multiplying them by −10) to obtain...
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