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 translated function \( g(x) \) has its vertex at \((2,0)\) (from the graph). The general form of a horizontal translation of \( |x| \) is \( g(x)=|x - h|+k \), where \((h,k)\) is the vertex. Here, \( h = 2 \), \( k = 0 \), so initially \( g(x)=|x - 2| \), but we also check the y - intercept.

Step2: Check y - intercept and adjust

When \( x = 0 \), from the graph, \( g(0)=2 \). Let's substitute \( x = 0 \) into \( g(x)=|x - 2|+k \). We know vertex is \((2,0)\), so \( k = 0 \) is wrong? Wait, no, wait. Wait, the vertex is \((2,0)\), so the general form is \( g(x)=|x - 2|+0 \)? But when \( x = 0 \), \( |0 - 2|=2 \), which matches \( g(0)=2 \). Oh right! So the function is \( g(x)=|x - 2| \)? Wait, no, wait. Wait, the parent function \( f(x)=|x| \) has vertex at \((0,0)\). The graph of \( g(x) \) has vertex at \((2,0)\) and passes through \((0,2)\). The transformation for horizontal shift: if we shift \( f(x)=|x| \) 2 units to the right, the function becomes \( f(x - 2)=|x - 2| \). Let's check \( x = 0 \): \( |0 - 2|=2 \), which matches the y - intercept (when \( x = 0 \), \( y = 2 \)). And the vertex at \( x = 2 \), \( y = 0 \), which matches the graph. So the function rule for \( g(x) \) is \( g(x)=|x - 2| \)? Wait, no, wait, wait. Wait, the standard form for translation of \( |x| \) is \( g(x)=|x - h|+k \), where \((h,k)\) is the vertex. Here, vertex is \((2,0)\), so \( h = 2 \), \( k = 0 \), so \( g(x)=|x - 2|+0=|x - 2| \). Let's verify with another point. Take \( x = 4 \), \( g(4)=|4 - 2| = 2 \)? Wait, no, from the graph, when \( x = 4 \), what's \( y \)? The graph at \( x = 4 \), let's see the grid. The line from \((2,0)\) to \((10,8)\): slope is \( (8 - 0)/(10 - 2)=1 \), so equation for \( x\geq2 \) is \( y=x - 2 \). For \( x\leq2 \), slope is \( (0 - 2)/(2 - 0)=- 1 \), so equation is \( y=-x + 2 \). Which is equivalent to \( y=|x - 2| \), because \( |x - 2|=

$$\begin{cases}x - 2, & x\geq2\\-(x - 2)=-x + 2, & x<2\end{cases}$$

\), which matches the two linear pieces. So yes, the function is \( g(x)=|x - 2| \)? Wait, no, wait, when \( x = 0 \), \( |0 - 2|=2 \), which is correct. When \( x = 2 \), \( |2 - 2|=0 \), correct. When \( x = 10 \), \( |10 - 2|=8 \), correct. So the function rule is \( g(x)=|x - 2| \). Wait, but let's check again. The parent function \( f(x)=|x| \): vertex at \((0,0)\), opens up. The graph of \( g(x) \) has vertex at \((2,0)\), so it's a horizontal shift 2 units to the right. So the transformation is \( f(x - 2)=|x - 2| \), which is \( g(x)=|x - 2| \).

Answer:

\( g(x)=|x - 2| \)