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 \( g(x) \) has its vertex at \((8,0)\), so it is a horizontal translation of \( f(x) \) 8 units to the right. The general form for a horizontal translation of \( |x| \) is \( |x - h| \) where \( h \) is the horizontal shift. Here, \( h = 8 \), so initially we consider \( |x - 8| \), but wait, let's check the slope. Wait, no, wait the left side of the graph (for \( x < 8 \)) has a slope. Wait, the parent function \( f(x)=|x| \) has two pieces: \( y = x \) for \( x \geq 0 \) and \( y=-x \) for \( x < 0 \). Let's find the equation of \( g(x) \).

Looking at the graph, when \( x = 0 \), \( y = 8 \)? Wait no, when \( x = 0 \), the point is on the line. Wait, let's take two points on the left part (before the vertex at \( (8,0) \)). Let's take \( (0,8) \) and \( (8,0) \). The slope between these two points is \( \frac{0 - 8}{8 - 0} = \frac{-8}{8} = -1 \). So for \( x \leq 8 \), the equation is \( y = -1(x - 8) \)? Wait, no. Wait, the vertex is at \( (8,0) \), so the equation of the left branch (for \( x \leq 8 \)): using point-slope form. The slope \( m \) between \( (0,8) \) and \( (8,0) \) is \( \frac{0 - 8}{8 - 0} = -1 \). So the equation is \( y - 0 = -1(x - 8) \) when \( x \leq 8 \), which simplifies to \( y = -x + 8 \). For \( x \geq 8 \), the slope: take \( (8,0) \) and \( (10,2) \), slope is \( \frac{2 - 0}{10 - 8} = 1 \), so equation is \( y - 0 = 1(x - 8) \), so \( y = x - 8 \). So combining these two, we get \( y = |x - 8| \)? Wait no, wait \( y = -x + 8 \) for \( x \leq 8 \) and \( y = x - 8 \) for \( x \geq 8 \), which is exactly \( y = |x - 8| \)? Wait no, \( |x - 8| \) is \( x - 8 \) when \( x \geq 8 \) and \( 8 - x \) when \( x < 8 \), which is the same as \( -x + 8 \) when \( x < 8 \) and \( x - 8 \) when \( x \geq 8 \). Wait, but when \( x = 0 \), \( |0 - 8| = 8 \), which matches the point \( (0,8) \). When \( x = 8 \), \( |8 - 8| = 0 \), which matches the vertex. When \( x = 10 \), \( |10 - 8| = 2 \), which matches the point \( (10,2) \). So actually, the function \( g(x) = |x - 8| \)? Wait, no, wait the parent function \( f(x)=|x| \) is translated 8 units to the right, so \( g(x) = |x - 8| \). Wait, but let's confirm. Wait, the original function \( f(x)=|x| \) has vertex at (0,0). The new vertex is at (8,0), so horizontal shift right by 8, so \( g(x) = |x - 8| \). Wait, but when we plug in \( x = 0 \), we get \( |0 - 8| = 8 \), which matches the y-intercept at (0,8). When \( x = 8 \), we get 0, which is the vertex. When \( x = 10 \), we get 2, which matches the point (10,2). So that works. Wait, but let's check the slope again. For \( f(x)=|x| \), the left slope is -1, right slope is 1. For \( g(x) \), left slope (x < 8) is -1, right slope (x > 8) is 1, which matches \( |x - 8| \), because \( |x - 8| = -(x - 8) = -x + 8 \) for \( x < 8 \) (slope -1) and \( |x - 8| = x - 8 \) for \( x > 8 \) (slope 1). So yes, the function rule is \( g(x) = |x - 8| \)? Wait, no, wait when x=0, y=8, which is |0 -8|=8, correct. When x=8, y=0, correct. When x=10, y=2, which is |10-8|=2, correct. So that's the function. Wait, but wait, is there a vertical shift? No, because the vertex is at y=0, same as the parent function's vertex y=0, but the y-intercept is 8, which is due to the horizontal shift. Wait, no, the horizontal shift of |x| to |x -8| moves the vertex to (8,0), and when x=0, it's |-8|=8, so that's correct. So the function rule is \( g(x) = |x - 8| \)? Wait, no, wait, wait I think I made a mistake. Wait, the parent function is f(x)=|x|, which is V-shaped with vertex at (0,0). The graph of g(x) is a V-shaped graph with vertex at (8,0), and the left side goes through (0,8). So the equation is g(x) = |x - 8|? Wait, let's check: |x -8| at x=0 is 8, at x=8 is 0, at x=10 is 2. Yes, that's exactly the points. So the function is a horizontal translation of f(x)=|x| 8 units to the right, so g(x) = |x - 8|. Wait, but wait, is there a reflection? No, because the left side has negative slope (like -x) and right side positive slope (like x), same as |x|. So the transformation is just a horizontal shift 8 units to the right. So the function rule is \( g(x) = |x - 8| \). Wait, but let's confirm with the slope. For f(x)=|x|, the left branch is y=-x (slope -1), right branch y=x (slope 1). For g(x), left branch (x <8) is y= -x +8 (slope -1), right branch (x >=8) is y= x -8 (slope 1), which is exactly |x -8|, because |x -8| = 8 -x when x <8 and x -8 when x >=8. So yes, that's correct.

Wait, but initially I thought maybe a vertical shift, but no, the vertex is at y=0, same as f(x). The y-intercept is 8 because when x=0, |0 -8|=8. So the function is \( g(x) = |x - 8| \).

Step2: Confirm the function

Let's verify with the graph. At x=8, y=0 (vertex), correct. At x=0, y=8, which is on the graph, correct. At x=10, y=2, which is on the graph, correct. The slope of the left side (x <8) is -1, which is the same as the left side of |x| (slope -1), and the slope of the right side (x >8) is 1, same as the right side of |x| (slope 1). So the function is a horizontal translation 8 units to the right of f(x)=|x|, so the function rule is \( g(x) = |x - 8| \). Wait, but wait, is there a vertical stretch or compression? No, because the slope is -1 and 1, same as |x|. So no stretch, just translation. So the function is \( g(x) = |x - 8| \).

Wait, but wait, I think I messed up earlier. Wait, the parent function is f(x)=|x|. The graph of g(x) has vertex at (8,0), so it's f(x - 8), which is |x - 8|. Yes, that's the horizontal translation formula: f(x - h) shifts the graph h units to the right. Here, h=8, so g(x) = f(x - 8) = |x - 8|.

Answer:

\( g(x) = |x - 8| \)