Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.4 - Exponential Functions - 1.4 Exercises - Page 53: 15

Answer

Refer to the blue graph below.
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Work Step by Step

The parent function of the given function is $y=e^x$. RECALL: (i) The graph of $y=f(-x)$ involves a reflection about the $y$ axis of the graph of $y=f(x)$. (ii) The graph of $y=\dfrac{1}{c} \cdot f(x)$ involves a vertical shrink by a factor of $c$ of the parent function $y=f(x)$. (iii) The graph of $y=-f(x)$ involves a reflection about the $x$-axis of the parent function $y=f(x)$. (iv) The graph of $y=f(x)+c$ involves a vertical shift of $c$ units upward of the parent function $y=f(x)$. The given function is equivalent to $y=-\frac{1}{2}e^{-x}+1$. This function involves the following transformations of the parent function $y=e^x$: (1) reflection about the $y$-axis because of the $-x$ exponent; (2) vertical shrink by a factor of $2$ because of the $\dfrac{1}{2}$ coefficient; (3) reflection about the $x$-axis because of the $-$ sign before the coefficient; and (4) a vertical shift of $1$ unit because of the addition of $1$. Thus, to graph the given function, perform the following steps: (1) Graph the parent function $y=e^x$ to obtain the black graph below. (2) Reflect the graph pf $y=e^x$ about the $y$-axis to obtain $y=e^{-x}$ (the green graph below). (3) Shrink vertically by a factor of $2$ the graph of $y=e^{-x}$ to obtain the graph of $y=\frac{1}{2}e^{-x}$ (the red graph below). (4) Reflect the graph of $y=\frac{1}{2}e^{-x}$ about the $x$-axis to obtain the graph of $y=-\frac{1}{2}e^{-x}$ (the purple graph below). (5) Shift the graph of $y=-\frac{1}{2}e^{-x}$ one unit upward to obtain the graph of $y=-\frac{1}{2}e^{-x}+1$ (the blue graph below).
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