Question

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

answer attempt 2 out of 2
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Explanation:

Step1: Identify vertex of the function

The vertex of the absolute value graph is at $(h,k)=(-2,-10)$.

Step2: Write absolute value function form

The standard form is $f(x)=a|x-h|+k$, so substitute $h=-2, k=-10$:
$f(x)=a|x+2|-10$

Step3: Solve for $a$ using a point

Use the point $(0,0)$ (from the graph, when $x=0, f(x)=0$):
$0=a|0+2|-10$
$0=2a-10$
$2a=10$
$a=5$
So the function is $f(x)=5|x+2|-10$.

Step4: Set $f(x)=20$ and solve

$5|x+2|-10=20$
Add 10 to both sides:
$5|x+2|=30$
Divide by 5:
$|x+2|=6$
Split into two cases:
Case1: $x+2=6$
$x=6-2=4$
Case2: $x+2=-6$
$x=-6-2=-8$

Answer:

$x=4$ and $x=-8$