Question

20 multiple choice 10 points
write the equation of a function $g(x)$ whose graph is the graph of $f(x) = |x|$ translated 5 units to the left, 7 units up, and vertically compressed by a factor of $\frac{1}{3}$.
$\frac{1}{3}|x - 5| + 7$
$3|x + 5| + 7$
$3|x - 5| + 7$
$\frac{1}{3}|x + 5| + 7$

Explanation:

Step1: Horizontal translation left 5 units

Replace $x$ with $x+5$: $f_1(x) = |x+5|$

Step2: Vertical compression by $\frac{1}{3}$

Multiply by $\frac{1}{3}$: $f_2(x) = \frac{1}{3}|x+5|$

Step3: Vertical translation up 7 units

Add 7 to the function: $g(x) = \frac{1}{3}|x+5| + 7$

Answer:

$\boldsymbol{\frac{1}{3}|x + 5| + 7}$ (corresponding to the last option)