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 $x$):
$y=-e^{-x}$
