Question

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

Explanation:

Step1: Identify vertex of \( g(x) \)

The vertex of \( f(x) = |x| \) is at \( (0,0) \). The graph of \( g(x) \) also has its vertex at \( (0,0) \), so \( h = 0 \) and \( k = 0 \).

Step2: Determine the value of \( a \)

For \( f(x)=|x| \), the slope of the right - hand side (where \( x\geq0 \)) is \( 1 \). Let's take a point on \( g(x) \) for \( x\geq0 \), say \( (6,6) \). Substitute into the form \( g(x)=a|x - h|+k \), with \( h = 0 \) and \( k = 0 \), we get \( 6=a|6 - 0|+0 \), which simplifies to \( 6 = 6a \). Solving for \( a \), we divide both sides by \( 6 \), so \( a = 1 \).

Step3: Write the function rule

Using the form \( g(x)=a|x - h|+k \), with \( a = 1 \), \( h = 0 \), and \( k = 0 \), we have \( g(x)=|x - 0|+0=|x| \). Wait, but let's check another point. For \( x=- 6 \), \( g(-6)=6 \), and \( |-6| = 6 \), which matches. So the transformation here is a vertical stretch or compression? Wait, no, actually, the original \( f(x)=|x| \) passes through \( (6,6) \) as well. Wait, maybe I made a mistake. Wait, the graph of \( g(x) \) seems to have the same shape as \( f(x)=|x| \). Let's confirm the general form. The parent function is \( f(x)=|x| \), and if there is no horizontal or vertical shift (since the vertex is at \( (0,0) \)) and the slope is \( 1 \) (since for \( x>0 \), the line goes from \( (0,0) \) to \( (6,6) \), slope \( m=\frac{6 - 0}{6 - 0}=1 \)), so \( a = 1 \), \( h = 0 \), \( k = 0 \). So \( g(x)=1\times|x - 0|+0=|x| \). But let's check the graph again. The graph of \( g(x) \) has the same vertex and the same slope as \( f(x)=|x| \), so the transformation is just the identity transformation for the absolute - value function in terms of stretch/compression and shift. So the function rule is \( g(x)=|x| \) or in the form \( a|x - h|+k \), it is \( g(x)=1|x - 0|+0 \).

Answer:

\( g(x)=|x| \) (or \( g(x)=1|x - 0|+0 \))