Questions: 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 ax-h+k, where a, h, and k are integers or simplified fractions.

Solution Steps

The vertex of the absolute value function $f(x) = |x|$ is at $(0, 0)$. The graph shows the vertex of $g(x)$ is at $(0, -5)$.

The graph of $f(x) = |x|$ passes through the points $(1,1)$ and $(-1,1)$. We observe on the given graph that when $x=5$, $y=0$ and when $x=-5$, $y=0$. This indicates that starting from the vertex, the graph rises one unit for every one unit increase in $x$. Thus, there is no vertical stretch or compression. So, $a=1$.

Since the vertex of $g(x)$ is $(0, -5)$, we have $h=0$ and $k=-5$. Since $a=1$, the function rule for $g(x)$ is $g(x) = |x - 0| - 5$, which simplifies to $g(x) = |x| - 5$.

Final Answer