Question

part a. describe the transformation of $f$ represented by $g$.
$f(x)=e^{-x}$, $g(x)=e^{-5x}+2$
the graph of $g$ is a $\boldsymbol{square}$ by a factor of $\boldsymbol{square}$, followed by a translation $\boldsymbol{square}$ $\boldsymbol{square}$ units of the graph of $f$.
options:
2, $\frac{1}{4}$, 5, $\frac{1}{5}$
up, down, left, right
vertical shrink, vertical stretch, horizontal shrink

Explanation:

Step1: Identify horizontal transformation

For $f(x)=e^{-x}$, $g(x)=e^{-5x}+2 = f(5x)+2$. A horizontal shrink by factor $\frac{1}{5}$ transforms $f(x)$ to $f(5x)$.

Step2: Identify vertical translation

Adding 2 to $f(5x)$ gives a vertical translation up 2 units.

Answer:

The graph of $g$ is a horizontal shrink by a factor of $\boldsymbol{\frac{1}{5}}$, followed by a translation up $\boldsymbol{2}$ units of the graph of $f$.