Question

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

answer attempt 1 out of 2
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Explanation:

Step1: Identify vertex of the function

The vertex of the absolute value graph is at $(-4, -10)$. The general form of an absolute value function is $f(x)=a|x-h|+k$, where $(h,k)$ is the vertex. So $h=-4$, $k=-10$, so $f(x)=a|x+4|-10$.

Step2: Find the slope $a$

From the graph, when $x=0$, $f(0)=5$. Substitute into the function:
$5=a|0+4|-10$
$5=4a-10$
$4a=15$
$a=\frac{15}{4}$
So the function is $f(x)=\frac{15}{4}|x+4|-10$.

Step3: Set $f(x)=20$ and solve

$\frac{15}{4}|x+4|-10=20$
$\frac{15}{4}|x+4|=30$
$|x+4|=30\times\frac{4}{15}$
$|x+4|=8$

Step4: Solve absolute value equation

Case 1: $x+4=8$
$x=8-4=4$
Case 2: $x+4=-8$
$x=-8-4=-12$

Answer:

$x=4$ and $x=-12$