Single Variable Calculus, Chapter 7, 7.2-1, Section 7.2-1, Problem 14

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,

$y = e^{-x - 4}$