Question

the graph shows g(x), which is a transformation of f(x) = |x|. write the function rule for g(x).
write your answer in the form a|x - h| + k, where a, h, and k are integers or simplified fractions.

Explanation:

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

The parent function \( f(x) = |x| \) has vertex at \( (0,0) \). For \( g(x) \), from the graph, the vertex (the point of the V - shape) is at \( (0, - 7) \)? Wait, no, looking at the graph, the vertex is at \( (0, - 7) \)? Wait, no, let's check the grid. The graph of \( g(x) \) has its vertex at \( (0, - 7) \)? Wait, no, the orange graph: let's see the y - coordinate of the vertex. The vertex is at \( (0, - 7) \)? Wait, no, the grid lines: each grid is 1 unit. The vertex is at \( (0, - 7) \)? Wait, no, looking at the graph, the vertex is at \( (0, - 7) \)? Wait, no, let's re - examine. The parent function \( f(x)=|x| \) has vertex \( (0,0) \). The transformed function \( g(x) \) has vertex at \( (h,k)=(0, - 7) \)? Wait, no, the graph of \( g(x) \): the vertex is at \( (0, - 7) \)? Wait, no, the y - axis: from 0 down, the vertex is at \( (0, - 7) \)? Wait, no, the graph shows that the vertex is at \( (0, - 7) \)? Wait, no, let's count the grid lines. The vertex is at \( (0, - 7) \)? Wait, no, the correct vertex: let's see, the parent function \( f(x)=|x| \) is a V - shape with vertex at (0,0). The transformed function \( g(x) \): let's find two points. For example, when \( x = 0 \), \( g(0)=-7 \)? Wait, no, the graph: the vertex is at \( (0, - 7) \)? Wait, no, looking at the graph, the vertex is at \( (0, - 7) \)? Wait, no, the y - coordinate of the vertex: from the origin (0,0) down 7 units? Wait, no, the graph's vertex is at \( (0, - 7) \)? Wait, no, let's check the slope. The parent function \( f(x)=|x| \) has a slope of 1 for \( x\geq0 \) and - 1 for \( x < 0 \). For \( g(x) \), let's take two points. Let's take \( x = 1 \), what's \( g(1) \)? From the graph, when \( x = 1 \), \( g(1)=-8 \)? Wait, no, the vertex is at \( (0, - 7) \)? Wait, no, maybe I made a mistake. Wait, the general form of the absolute value function transformation is \( g(x)=a|x - h|+k \), where \( (h,k) \) is the vertex.

Looking at the graph, the vertex of \( g(x) \) is at \( (h,k)=(0, - 7) \)? Wait, no, the graph: the vertex is at \( (0, - 7) \)? Wait, no, let's look again. The graph of \( g(x) \) has vertex at \( (0, - 7) \)? Wait, no, the correct vertex: let's see, when \( x = 0 \), \( g(0)=-7 \). When \( x = 1 \), \( g(1)=-8 \), when \( x=-1 \), \( g(-1)=-8 \). So the slope \( a \): for \( x\geq0 \), the slope is \( \frac{g(1)-g(0)}{1 - 0}=\frac{-8-(-7)}{1}=-1 \). For \( x < 0 \), the slope is \( \frac{g(-1)-g(0)}{-1 - 0}=\frac{-8-(-7)}{-1}=1 \). Wait, no, the slope for \( x\geq0 \): from \( (0, - 7) \) to \( (1, - 8) \), the slope is \( \frac{-8 - (-7)}{1-0}=-1 \). From \( (0, - 7) \) to \( (-1, - 8) \), the slope is \( \frac{-8 - (-7)}{-1-0}=1 \). So the value of \( a \) is - 1? Wait, no, the parent function \( f(x)=|x| \) has \( a = 1 \). If the slope is - 1 for \( x\geq0 \) and 1 for \( x < 0 \), that would be a reflection? Wait, no, the slope for \( x\geq0 \) in \( g(x) \) is - 1, which is the same as the slope of \( f(x) \) for \( x\leq0 \). Wait, no, let's re - express.

The general form is \( g(x)=a|x - h|+k \). The vertex \( (h,k) \): from the graph, the vertex is at \( (0, - 7) \), so \( h = 0 \), \( k=-7 \). Now, let's find \( a \). Take a point on \( g(x) \), say \( x = 1 \). What's \( g(1) \)? From the graph, when \( x = 1 \), \( g(1)=-8 \). Plug into the formula: \( g(1)=a|1 - 0|+(-7)=a - 7 \). We know \( g(1)=-8 \), so \( a-7=-8 \), which gives \( a=-1 \). Let's check another point, \( x = 2 \). \( g(2)=a|2 - 0|+k=-2 - 7=-9 \). From the graph, when \( x = 2 \), \( g(2)=-9 \), which matches. Also, for \( x=-1 \), \( g(-1)=a|-1 - 0|+k=-1 - 7=-8 \), which also matches the graph.

Wait, but maybe I made a mistake in the vertex. Wait, the graph: the vertex is at \( (0, - 7) \)? Wait, no, the correct vertex: let's see, the parent function \( f(x)=|x| \) has vertex (0,0). The transformed function \( g(x) \) is a vertical shift down and maybe a reflection? Wait, no, the slope of \( g(x) \) for \( x\geq0 \) is - 1, which is the same as the slope of \( f(x) \) for \( x\leq0 \), and the slope for \( x < 0 \) is 1, which is the same as the slope of \( f(x) \) for \( x\geq0 \). So \( a=-1 \), \( h = 0 \), \( k=-7 \). So the function is \( g(x)=-|x - 0|+(-7)=-|x|-7 \). Wait, but let's check the vertex again. The vertex of \( f(x)=|x| \) is (0,0). The vertex of \( g(x) \) is (0, - 7), so the vertical shift is down 7 units, and the reflection over the x - axis (since \( a=-1 \)).

Step2: Write the function rule

The general form is \( g(x)=a|x - h|+k \). We found that \( h = 0 \), \( k=-7 \), and \( a=-1 \). So substituting these values into the formula, we get \( g(x)=-|x - 0|+(-7)=-|x|-7 \).

Answer:

\( g(x)=-|x|-7 \) (or in the form \( a|x - h|+k \), it is \( -|x - 0|+(-7) \))