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 a function \( y = f(x) \), the rule is \( y = f(x)+k \), where \( k \) is the vertical shift (positive for up, negative for down).

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

From the graph, the vertex of \( g(x) \) is at \( (0,8) \). Comparing to the vertex of \( f(x) \) which is \( (0,0) \), there is a vertical shift of \( 8 \) units up.

Step3: Write the function rule for \( g(x) \)

Using the vertical translation rule, since we shift \( f(x)=|x| \) up by \( 8 \) units, the function \( g(x) \) is \( g(x)=|x| + 8 \).

Answer:

\( g(x)=|x| + 8 \)