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: 17

Answer

(a) $$y=e^x-2$$ (b) $$y=e^{x-2}$$ (c) $$y=-e^x$$ (d) $$y=e^{-x}$$ (e) $$y=-e^{-x}$$

Work Step by Step

(a) Shifting the graph $y=e^x$ 2 units downward means that for every value of $x$, there is a new $y'$ such that $y'=y-2$. So, $y'=y-2=e^x-2$ Hence, the equation of the graph $y$ resulting from the downward shift would be (where $y'$ is taken as the new value of $y$): $y=e^x-2$ (b) Shifting the graph $y=e^x$ 2 units to the right means that for every value of $y$, there is a new $x'$ such that $x'=x+2$. Thus, substituting $x'$ for $x$ in the equation of the graph $y$: $y=e^x=e^{x'-2}$ Hence, the new equation of the graph resulting from a rightward shift would be (where $x'$ is taken as the new value of $x$): $y=e^{x-2}$ (c) Reflecting the graph $y=e^x$ about the $x$-axis means that for every value of $x$, there is a new $y'$ such that $y'=-y$ (You can think of it as flipping the sign of every value of $y$). Thus, substituting $y'$ for $y$ in the equation of the graph $y$: $y'=-y=-e^{x}$ Hence, the new equation of the graph resulting from a reflection about the $x$-axis would be (where $y'$ is taken as the new value of $y$): $y=-e^{x}$ (d) Reflecting the graph $y=e^x$ about the $y$-axis means that for every value of $y$, there is a new $x'$ such that $x'=-x$ (You can think of it as flipping the sign of every value of $x$). Thus, substituting $x'$ for $x$ in the equation of the graph $y$: $y=e^x=e^{-x'}$ Hence, the new equation of the graph resulting from a reflection about the $y$-axis would be (where $x'$ is taken as the new value of $x$): $y=e^{-x}$ (e) From part c, reflecting the graph $y=e^x$ about the $x$-axis yields the new equation $y=-e^x$. Similar to part d, further reflecting this new equation of the graph about the $y$-axis means that for every value of $y$, there is a new $x'$ such that $x'=-x$. Thus, substituting $x'$ for $x$ in the equation of the graph $y$: $y=-e^x=-e^{-x'}$ Hence, the new equation of the graph resulting from a reflection about the $x$-axis and then about the $y$-axis would be (where $x'$ is taken as the new value of $x$): $y=-e^{-x}$
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