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 translation of the absolute - value function \( y=|x - h|+k \), the vertex of the function \( y = |x - h|+k \) is at \( (h,k) \).
Looking at the graph of \( g(x) \), the vertex of the V - shaped graph (since it's an absolute - value function translation) is at \( (4,0) \). So, \( h = 4 \) and \( k = 0 \) in the form \( y=|x - h|+k \).

Step2: Write the function rule

Substitute \( h = 4 \) and \( k = 0 \) into the formula \( y=|x - h|+k \). We get \( g(x)=|x - 4|+0 \), which simplifies to \( g(x)=|x - 4| \). We can also verify this by checking a point on the graph. For example, when \( x = 0 \), \( g(0)=|0 - 4|=4 \), and from the graph, when \( x = 0 \), the \( y \) - value is 4, which matches. When \( x = 4 \), \( g(4)=|4 - 4| = 0 \), which is the vertex, and when \( x = 8 \), \( g(8)=|8 - 4|=4 \), and from the graph, when \( x = 8 \), the \( y \) - value is 4, which also matches.

Answer:

\( g(x)=|x - 4| \)