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 \( f(x) \). Let's check the vertex of \( g(x) \). From the graph, the vertex of \( g(x) \) is at \((0, -1)\)? Wait, no, looking at the graph, the vertex is at \((0, -1)\)? Wait, no, the graph touches the y-axis at \((0, -1)\)? Wait, no, the grid: the y-axis has -2, -4, etc. Wait, the vertex of \( g(x) \) is at \((0, -1)\)? Wait, no, let's re-examine. The parent function \( f(x)=|x| \) has vertex at (0,0). The graph of \( g(x) \) has its vertex at (0, -1)? Wait, no, the graph in the picture: the vertex is at (0, -1)? Wait, no, the y-axis: the grid lines. Let's see, the original \( f(x)=|x| \) has vertex at (0,0). The graph of \( g(x) \) is shifted down by 1 unit? Wait, no, looking at the graph, the vertex is at (0, -1)? Wait, no, the point where the two lines meet: at x=0, y=-1? Wait, the graph shows that at x=0, the y-coordinate is -1? Wait, no, the grid: the y-axis has -2, -4, etc. Wait, maybe I made a mistake. Wait, the graph of \( g(x) \): let's check a point. For \( f(x)=|x| \), when x=1, f(1)=1. For \( g(x) \), when x=1, what's the y-value? Looking at the graph, when x=1, y=0? No, wait, the graph: let's see the vertex. Wait, the graph of \( g(x) \) has its vertex at (0, -1)? Wait, no, the original \( f(x)=|x| \) is V-shaped with vertex at (0,0). The graph of \( g(x) \) is a translation. Let's check the vertex of \( g(x) \). From the graph, the vertex is at (0, -1)? Wait, no, the y-axis: the line crosses the y-axis at (0, -1)? Wait, maybe the translation is vertical shift. The general form of a vertical translation of \( f(x)=|x| \) is \( g(x) = |x| + k \), where k is the vertical shift (k>0 up, k<0 down). Let's find the vertex of \( g(x) \). The vertex of \( g(x) \) is at (0, -1)? Wait, no, looking at the graph, the vertex is at (0, -1)? Wait, the graph: when x=0, y=-1. So the vertex is (0, -1). So the vertical shift is down by 1 unit. So \( g(x) = |x| - 1 \)? Wait, no, wait, let's check another point. For x=1, \( f(1)=1 \), \( g(1) \): from the graph, when x=1, y=0. So \( |1| - 1 = 0 \), which matches. For x=2, \( |2| - 1 = 1 \), and from the graph, when x=2, y=1? Wait, the graph: at x=2, y=1? Let's see the grid. The x-axis is from -10 to 10, y-axis from -10 to 10. So when x=2, the y-coordinate is 1? Yes, that matches \( |2| - 1 = 1 \). So the function \( g(x) \) is \( f(x) \) shifted down by 1 unit. So the rule is \( g(x) = |x| - 1 \). Wait, but let's confirm the vertex. The vertex of \( g(x) \) is at (0, -1), which is a vertical shift down by 1 from (0,0). So the translation is vertical shift down by 1. So the function rule is \( g(x) = |x| - 1 \).

Step2: Confirm the translation

The parent function \( f(x) = |x| \) has vertex at (0,0). The graph of \( g(x) \) has vertex at (0, -1), so it's a vertical translation down by 1 unit. The formula for vertical translation of a function \( f(x) \) is \( g(x) = f(x) + k \), where k is the vertical shift (k < 0 for down). Here, k = -1, so \( g(x) = |x| + (-1) = |x| - 1 \).

Answer:

\( g(x) = |x| - 1 \)