the graph shows g(x), which is a transformation of f(x) = |x|. write the function rule for g(x).\nwrite your…

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 ))