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.
g(x) =

Explanation:

Step1: Identify vertex of $g(x)$

The vertex of the translated absolute value graph is at $(6, -9)$. For the form $a|x-h|+k$, the vertex is $(h,k)$, so $h=6$, $k=-9$.

Step2: Determine stretch factor $a$

The slope of the right segment is 1 (for each 1 unit right, the graph goes up 1 unit), so $a=1$.

Step3: Substitute values into form

Substitute $a=1$, $h=6$, $k=-9$ into $a|x-h|+k$:
$g(x) = 1|x - 6| + (-9) = |x - 6| - 9$

Answer:

$g(x) = |x - 6| - 9$