Question

1. starting with the graph of $y = e^x$, write the equation of the graph that results from

(a) shifting 2 units downward.
(b) shifting 2 units to the right.
(c) reflecting about the $x$-axis.
(d) reflecting about the $y$-axis.
(e) reflecting about the $x$-axis and then about the $y$-axis.

Explanation:

Step1: Shift down 2 units

Subtract 2 from $e^x$
$y = e^x - 2$

Step2: Shift right 2 units

Replace $x$ with $x-2$
$y = e^{(x-2)}$

Step3: Reflect over x-axis

Multiply $e^x$ by $-1$
$y = -e^x$

Step4: Reflect over y-axis

Replace $x$ with $-x$
$y = e^{-x}$

Step5: Reflect over x then y-axis

First reflect over x-axis, then replace $x$ with $-x$
$y = -e^{-x}$

Answer:

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