Question

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

Explanation:

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

The parent function \( f(x) = |x| \) has its vertex at \( (0, 0) \). For \( g(x) \), from the graph, the vertex is at \( (-5, 0) \)? Wait, no, looking at the graph, the vertex (the point of the V - shape) is at \( (-5, 0) \)? Wait, no, let's check the grid. Wait, the graph of \( g(x) \): when \( x=-5 \), \( y = 0 \)? Wait, no, looking at the graph, the vertex is at \( (-5, 0) \)? Wait, no, let's re - examine. Wait, the parent function \( f(x)=|x| \) is a V - shaped graph with vertex at \( (0,0) \). The graph of \( g(x) \) is a translation. Let's find two points. When \( x = 0 \), \( g(0)=5 \)? Wait, no, the y - intercept: looking at the graph, when \( x = 0 \), \( y = 5 \)? Wait, no, the grid lines: each square is 1 unit. The vertex of \( g(x) \) is at \( (-5, 0) \)? Wait, no, let's see the slope. For the right side of the V (where \( x\geq - 5 \)), let's take two points. When \( x=-5 \), \( y = 0 \); when \( x = 0 \), \( y = 5 \). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 0}{0-(-5)} = 1 \). So the equation of the right side (for \( x\geq - 5 \)) is \( y-0 = 1\times(x + 5) \), so \( y=x + 5 \). For the left side ( \( x\lt - 5 \)), the slope is \( - 1 \), and using the vertex \( (-5,0) \), the equation is \( y-0=-1\times(x + 5) \), so \( y=-x - 5 \). Combining these, we get \( g(x)=|x + 5| \)? Wait, no, wait, when \( x = 0 \), \( |0 + 5|=5 \), which matches the y - intercept. Wait, but let's check another point. When \( x = 1 \), \( g(1)=|1 + 5|=6 \)? Wait, no, the graph at \( x = 1 \) should be \( y=6 \)? Wait, maybe I made a mistake. Wait, let's look at the graph again. Wait, the original problem's graph: the vertex is at \( (-5, 0) \), and the slope of the right arm is 1. So the transformation of \( f(x)=|x| \) to \( g(x) \): if we translate \( f(x)=|x| \) 5 units to the left, the function becomes \( g(x)=|x+5| \). Wait, but let's check the y - intercept. When \( x = 0 \), \( g(0)=|0 + 5| = 5 \), which matches the graph (the y - intercept is at \( (0,5) \)). Alternatively, maybe it's a vertical translation? No, because the vertex has moved horizontally. The general form of a translated absolute - value function is \( g(x)=|x - h|+k \), where \( (h,k) \) is the vertex. Here, the vertex is at \( (-5,0) \), so \( h=-5 \), \( k = 0 \). So \( g(x)=|x-(-5)|+0=|x + 5| \). Wait, but let's check with \( x=-5 \), \( g(-5)=| - 5+5|=0 \), which is correct. When \( x = 0 \), \( g(0)=|0 + 5| = 5 \), which matches the graph. So the function rule for \( g(x) \) is \( g(x)=|x + 5| \).

Wait, maybe I misread the graph. Let's start over. The parent function is \( f(x)=|x| \), vertex at \( (0,0) \). The graph of \( g(x) \) has its vertex at \( (-5,0) \) (since the V - shape is at \( x=-5,y = 0 \)). The transformation from \( f(x) \) to \( g(x) \) is a horizontal translation 5 units to the left. The formula for a horizontal translation of \( f(x)=|x| \) by \( h \) units to the left is \( g(x)=|x + h| \). Here, \( h = 5 \), so \( g(x)=|x+5| \).

Step2: Verify the function

Let's check some points. When \( x=-5 \), \( g(-5)=|-5 + 5|=0 \), which matches the vertex. When \( x = 0 \), \( g(0)=|0 + 5|=5 \), which matches the y - intercept. When \( x = 5 \), \( g(5)=|5 + 5| = 10 \), and from the graph, as \( x \) increases, \( y \) increases with a slope of 1, so at \( x = 5 \), \( y = 10 \), which is correct. When \( x=-10 \), \( g(-10)=|-10 + 5|=|-5| = 5 \), and the left side of the V - shape at \( x=-10 \) should have \( y = 5 \), which is correct.

Answer:

\( g(x)=|x + 5| \)