Question

1. (4pts) give the equation of the function graphed to the right:

$f(x) =$

Explanation:

Step1: Identify vertex points

The vertex of the absolute value function is at $(-3, 1)$. The general form of a vertex-based absolute value function is $f(x)=a|x-h|+k$, where $(h,k)$ is the vertex. Substituting $h=-3$, $k=1$, we get $f(x)=a|x+3|+1$.

Step2: Find slope $a$ using a point

Use the x-intercept $(-4, 0)$:
Substitute $x=-4$, $f(x)=0$ into the equation:
$0=a|-4+3|+1$
$0=a|{-1}|+1$
$0=a(1)+1$
Solve for $a$: $a=-1$.

Step3: Verify with another point

Use the point $(-2, 0)$:
$f(-2)=-1|{-2+3}|+1=-1(1)+1=0$, which matches the graph.

Answer:

$f(x) = -|x+3| + 1$