the graph shows $g(x)$, which is a translation of $f(x) = |x|$. write the function rule for $g(x)$. \nwrite…

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

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

Answer

Explanation:

Step1: Identify vertex of $f(x)=|x|$

The vertex of $f(x)=|x|$ is at $(0,0)$.

Step2: Find vertex of $g(x)$

From the graph, the vertex of $g(x)$ is at $(0,-2)$.

Step3: Determine translation parameters

For the form $a|x-h|+k$, the vertex is $(h,k)$. Here, $h=0$, $k=-2$. The slope $a$ remains 1 (no stretch/compression, same direction as $|x|$).

Step4: Substitute into the form

Substitute $a=1$, $h=0$, $k=-2$ into $a|x-h|+k$. $\begin{align*} g(x)&=1|x-0|+(-2)\ &=|x|-2 \end{align*}$

Answer:

$g(x)=|x|-2$