Question

the graph shows g(x), which is a translation of f(x) = |x|. write the function rule for g(x).
write your answer in the form a|x - h| + k, where a, h, and k are integers or simplified fractions.

Explanation:

Step1: Identify vertex of $g(x)$

The vertex of $g(x)$ is at $(-6, 0)$.

Step2: Determine vertical stretch factor $a$

The slope of the right segment is $1$, so $a=1$.

Step3: Substitute into vertex form

Vertex form: $g(x)=a|x-h|+k$, where $(h,k)$ is vertex. Substitute $a=1$, $h=-6$, $k=0$:
$g(x)=1|x-(-6)|+0 = |x+6|$

Step4: Verify with y-intercept

When $x=0$, $g(0)=|0+6|=6$, which matches the graph.

Answer:

$g(x) = |x + 6|$