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 \((0,0)\). The graph of \( g(x) \) is a translation of \( f(x) \). We need to find the horizontal and vertical shifts.

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

From the graph, the vertex of \( g(x) \) is at \((3,0)\) (wait, no, looking at the grid, the vertex is at \((3,0)\)? Wait, no, the graph's vertex: let's check the x - coordinate where the graph changes direction. Looking at the grid, the vertex is at \((3,0)\)? Wait, no, the graph: when x = 3, y = 0? Wait, no, the grid lines: the vertex is at (3,0)? Wait, no, let's re - examine. The parent function \( f(x)=|x| \) has vertex (0,0). The graph of \( g(x) \) has its vertex at (3,0)? Wait, no, looking at the graph, the vertex is at (3,0)? Wait, no, the x - axis: the vertex is at x = 3, y = 0? Wait, no, the graph: when x = 3, y = 0? Wait, the grid: each square is 1 unit. So the vertex of \( g(x) \) is at \((3,0)\)? Wait, no, the original function \( f(x)=|x| \) is \( y = |x| \). A horizontal translation of \( f(x) \) to the right by \( h \) units and vertical translation by \( k \) units is \( g(x)=|x - h|+k \).
Looking at the graph, the vertex of \( g(x) \) is at \((3,0)\)? Wait, no, wait the graph: let's check the point where the two lines meet. The left line: when x = 0, y = 3? No, wait the graph: the vertex is at (3,0). Wait, no, let's take two points. For the right - hand side of the absolute value graph (where \( x\geq3 \)), let's take a point. When x = 8, y = 5? Wait, no, the graph at x = 8, y = 5? Wait, no, the grid: from the vertex (3,0), when x = 8, the y - value: let's see, the slope of the right - hand line. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Taking (3,0) and (8,5), slope \( m = 1 \). Wait, no, the parent function \( f(x)=|x| \) has a slope of 1 for \( x\geq0 \) and - 1 for \( x<0 \). The graph of \( g(x) \): for \( x\geq3 \), the slope is 1, and for \( x<3 \), the slope is - 1. So the vertex is at (3,0). So the function is \( g(x)=|x - 3|+0 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's check the y - intercept. When x = 0, what is y? From the graph, when x = 0, y = 3? Wait, no, the graph: when x = 0, the y - value is 3? Wait, no, the left - hand line: when x = 0, y = 3? Wait, the equation of the left - hand line: let's take two points on the left - hand line. Let's take (0,3) and (3,0). The slope \( m=\frac{0 - 3}{3 - 0}=\frac{- 3}{3}=-1 \). The equation of the left - hand line is \( y-3=-1(x - 0) \), so \( y=-x + 3 \). For the right - hand line, take (3,0) and (8,5). The slope \( m=\frac{5 - 0}{8 - 3}=\frac{5}{5}=1 \). The equation of the right - hand line is \( y-0 = 1(x - 3) \), so \( y=x - 3 \). So combining these two, the function is \( g(x)=|x - 3| \)? Wait, no, when x = 0, \( g(0)=|0 - 3|=3 \), which matches the y - intercept. When x = 3, \( g(3)=|3 - 3|=0 \), which is the vertex. So the function rule for \( g(x) \) is \( g(x)=|x - 3| \). Wait, but let's re - check the graph. Wait, the original problem: the graph of \( g(x) \), translation of \( f(x)=|x| \). The parent function \( f(x)=|x| \) has vertex (0,0). The graph of \( g(x) \) has vertex (3,0), so it's a horizontal translation to the right by 3 units. So the function is \( g(x)=|x - 3| \).

Answer:

\( g(x)=|x - 3| \)