Question

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

Explanation:

Step1: Recall translation of absolute function

The parent function is \( f(x) = |x| \), which has its vertex at \((0,0)\). For a vertical translation of \( f(x) \) by \( k \) units down, the function becomes \( g(x)=|x|+k \) (if \( k<0 \), it's a downward shift).

Step2: Identify the vertex of \( g(x) \)

From the graph, the vertex of \( g(x) \) is at \((0, - 10)\). This means the parent function \( f(x)=|x| \) (vertex at \((0,0)\)) is shifted down by 10 units.

Step3: Write the function rule

Using the vertical translation formula, if we shift \( f(x)=|x| \) down by 10 units, the function \( g(x) \) is \( g(x)=|x|-10 \).

Answer:

\( g(x)=|x|-10 \)