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

Explanation:

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

The parent function ( f(x) = |x| ) has a vertex at ( (0, 0) ). For ( g(x) ), the vertex (the point of the V - shape) is at ( (5, 0) ) (from the graph, the minimum point is at ( x = 5 ), ( y = 0 )). In the form ( g(x)=a|x - h|+k ), the vertex is ( (h, k) ), so ( h = 5 ) and ( k = 0 ).

Step2: Determine the value of ( a )

We can use another point on the graph to find ( a ). Let's use the y - intercept. When ( x = 0 ), from the graph, ( y = 5 )? Wait, no, looking at the graph, when ( x = 0 ), the point is ( (0, 5) )? Wait, no, the graph passes through ( (0, 5) )? Wait, no, let's re - examine the graph. Wait, the line on the left side: when ( x = 0 ), the y - coordinate is 5? Wait, no, looking at the grid, when ( x = 0 ), the point is at ( y = 5 )? Wait, no, the graph: let's take two points. The vertex is ( (5, 0) ), and when ( x = 0 ), let's see the line. The left - hand line: from ( (5, 0) ) to ( (0, 5) )? Wait, the slope between ( (5, 0) ) and ( (0, 5) ) is ( \frac{5 - 0}{0 - 5}=\frac{5}{-5}=- 1 ). Wait, or take ( x = 0 ), ( y = 5 )? Wait, no, the graph: when ( x = 0 ), the y - value is 5? Wait, the original parent function ( f(x)=|x| ) has a slope of 1 for ( x>0 ) and - 1 for ( x < 0 ). For the translated function ( g(x)=a|x - h|+k ), the slope of the right - hand side (for ( x>h )) is ( a ), and the slope of the left - hand side (for ( x < h )) is ( - a ). The vertex is at ( (5, 0) ). Let's take a point on the right - hand side, say ( (8, 3) ) (since from ( (5, 0) ), moving 3 units right and 3 units up). So when ( x = 8 ), ( y = 3 ). Plug into ( g(x)=a|x - 5|+0 ) (since ( k = 0 )): ( 3=a|8 - 5| ), so ( 3 = 3a ), so ( a = 1 )? Wait, no, that can't be. Wait, maybe I made a mistake in the vertex. Wait, looking at the graph again, the vertex is at ( (5, 0) ), and when ( x = 0 ), the y - value is 5. So substituting ( x = 0 ), ( y = 5 ) into ( g(x)=a|x - 5|+0 ): ( 5=a|0 - 5| ), so ( 5 = 5a ), so ( a = 1 ). Wait, but the left - hand side: the slope should be - a. If ( a = 1 ), then for ( x<5 ), ( g(x)=- (x - 5)=-x + 5 ), which when ( x = 0 ), gives ( y = 5 ), which matches the graph. And for ( x>5 ), ( g(x)=x - 5 ), which when ( x = 8 ), gives ( y = 3 ), which matches the graph (since from ( (5, 0) ) to ( (8, 3) ), the slope is ( \frac{3 - 0}{8 - 5}=1 )). Wait, so ( a = 1 ), ( h = 5 ), ( k = 0 ).

Wait, no, wait the graph: let's check the y - intercept again. The graph crosses the y - axis at ( (0, 5) )? Wait, the original graph: looking at the grid, when ( x = 0 ), the y - coordinate is 5? Wait, the user's graph: the left line goes from the top left, crosses the y - axis at (0,5)? Wait, no, in the provided graph, the left line: when x = 0, y is 5? Wait, the vertex is at (5, 0). So the function is ( g(x)=|x - 5| ), because the translation is 5 units to the right (since ( h = 5 ), ( k = 0 ), ( a = 1 )). Wait, let's verify with the vertex form. The vertex form of a translated absolute value function is ( g(x)=a|x - h|+k ), where ( (h,k) ) is the vertex. The vertex of ( g(x) ) is at ( (5,0) ), so ( h = 5 ), ( k = 0 ). The slope ( a ) is 1 because the graph has the same "steepness" as ( f(x)=|x| ) (no vertical stretch or compression, and the direction is the same as the parent function for the right - hand side, positive slope, left - hand side negative slope, so ( a = 1 )). So ( g(x)=|x - 5|+0 ), which simplifies to ( g(x)=|x - 5| ). Wait, but when ( x = 0 ), ( g(0)=|0 - 5| = 5 ), which matches the y - intercept (the point ( (0,5) ) on the graph). And when ( x = 5 ), ( g(5)=|5 - 5|=0 ), which is the vertex. And when ( x = 8 ), ( g(8)=|8 - 5| = 3 ), which matches the point ( (8,3) ) on the graph.

Wait, maybe I made a mistake earlier. Let's re - do:

1. Vertex of ( f(x)=|x| ) is ( (0,0) ). Vertex of ( g(x) ) is ( (5,0) ), so ( h = 5 ), ( k = 0 ) in ( g(x)=a|x - h|+k ).

2. To find ( a ), we can use the fact that the graph of ( g(x) ) has the same shape as ( f(x)=|x| ), so ( a = 1 ) (since there is no vertical stretch or compression, and the direction is the same as the parent function for the right - hand branch).

So the function rule for ( g(x) ) is ( g(x)=|x - 5|+0 ), or ( g(x)=|x - 5| ). Wait, but let's check with another point. When ( x = 5 ), ( g(5)=0 ), which is correct. When ( x = 10 ), ( g(10)=|10 - 5| = 5 ), and from the graph, when ( x = 10 ), the y - coordinate is 5, which matches.

Answer:

( g(x)=|x - 5| ) (or in the form ( a|x - h|+k ), it is ( 1|x - 5|+0 ))