Single Variable Calculus, Chapter 7, 7.2-1, Section 7.2-1, Problem 14
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Use the graph of $y = e^x$ to find the equation of the graph that results from
a.) Reflecting about the line $y = 4$.
To acquire the equation from $y = e^x$, we first multiply it by $-1$ to reflect the graph from $x$-axis then we need to find the appropriate number that will shift our graph upwards. To figure it out, let's look at the $y$-intercepts. The graph crosses $y$-axis at $1$ from $y = e^x$. in our current form $y = -e^x$, the graph crosses $y$-axis at $-1$. If we want to reflect about the line $y = 4$, then we want 4 to be in the middle between our $y$-intercepts. Thus, we add $8$ to our function so that..
$y = -e^x + 8$
b.) Reflecting about the line $x = 2$.
To achieve this, we first multiply the exponent of $y = e^x$ from $y$-axis. Then we need to find the appropriate number that will shift our graph to the right.
Thus,
$ \begin{equation} \begin{aligned} e^x =& e^{-(x - n)} \\ \\ x =& -x + n \\ \\ n =& 2x \end{aligned} \end{equation} $
So if $x = 2$, then
$n = 2(2) = 4$
Therefore,
$y = e^{-x - 4}$
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