Question

1. the graph of an absolute value function $f(x)$ is shown. find the value of $x$ when $f(x)=20$.

Explanation:

Step1: Identify vertex of the function

The vertex of the absolute value graph is at $(-3, -10)$. The general form of an absolute value function is $f(x)=a|x-h|+k$, where $(h,k)$ is the vertex. Substituting $h=-3$, $k=-10$, we get $f(x)=a|x+3|-10$.

Step2: Find the value of $a$

Use the point on the right line: when $x=0$, $f(0)=5$. Substitute into the function:
$5=a|0+3|-10$
$5=3a-10$
$3a=15$
$a=5$
So the function is $f(x)=5|x+3|-10$.

Step3: Solve for $x$ when $f(x)=20$

Set $5|x+3|-10=20$
First, isolate the absolute value term:
$5|x+3|=20+10$
$5|x+3|=30$
$|x+3|=6$
This gives two cases:
Case 1: $x+3=6$
$x=6-3=3$
Case 2: $x+3=-6$
$x=-6-3=-9$

Answer:

$x=3$ and $x=-9$