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 vertex of \( f(x) = |x| \) is \( (0,0) \). For \( g(x) \), the vertex (from the graph) is also \( (0,0) \), so \( h = 0 \) and \( k = 0 \).

Step2: Determine the value of \( a \)

The general form is \( g(x) = a|x - h| + k \). Substitute \( h = 0 \), \( k = 0 \): \( g(x) = a|x| \).
Take a point on \( g(x) \), e.g., \( (2, 5) \)? Wait, no, looking at the graph, when \( x = 2 \), \( y = 5 \)? Wait, no, let's check the grid. Wait, when \( x = 2 \), the \( y \)-value: from the graph, at \( x = 2 \), the line passes through \( (2, 5) \)? Wait, no, wait the grid lines: each square is 1 unit. Wait, when \( x = 2 \), the \( y \)-coordinate: let's see, the right side of the V: from \( (0,0) \) to \( (4, 10) \)? Wait, no, the top points are at \( x = -4 \) and \( x = 4 \), \( y = 10 \). So when \( x = 4 \), \( y = 10 \).
Substitute \( x = 4 \), \( y = 10 \) into \( g(x) = a|x - 0| + 0 \): \( 10 = a|4| \) → \( 10 = 4a \) → \( a = \frac{10}{4} = \frac{5}{2} \)? Wait, no, wait the graph: when \( x = 2 \), what's \( y \)? Let's check the slope. From \( (0,0) \) to \( (4, 10) \), the slope is \( \frac{10 - 0}{4 - 0} = \frac{10}{4} = \frac{5}{2} \). So the equation is \( g(x) = \frac{5}{2}|x| \)? Wait, no, wait the standard form is \( a|x - h| + k \). Wait, maybe I made a mistake. Wait, the vertex is \( (0,0) \), so \( h = 0 \), \( k = 0 \). Then, take a point like \( (2, 5) \)? Wait, no, when \( x = 2 \), the \( y \)-value: looking at the graph, at \( x = 2 \), the line is at \( y = 5 \)? Wait, no, the graph: the right arm goes from \( (0,0) \) to \( (4, 10) \), so the slope is \( \frac{10}{4} = 2.5 = \frac{5}{2} \). So \( a = \frac{5}{2} \)? Wait, but let's check \( x = 1 \): \( y = \frac{5}{2}(1) = 2.5 \), which matches the graph? Wait, no, maybe I misread the graph. Wait, the top points are at \( x = -4 \) and \( x = 4 \), \( y = 10 \). So when \( x = 4 \), \( y = 10 \). So substituting into \( g(x) = a|x - 0| + 0 \): \( 10 = a*4 \) → \( a = \frac{10}{4} = \frac{5}{2} \). So the function is \( g(x) = \frac{5}{2}|x - 0| + 0 \), which simplifies to \( g(x) = \frac{5}{2}|x| \). Wait, but let's check another point: \( x = -2 \), \( y = \frac{5}{2}*2 = 5 \), which matches the graph (at \( x = -2 \), \( y = 5 \))? Wait, no, the graph at \( x = -2 \): looking at the left arm, from \( (0,0) \) to \( (-4, 10) \), so at \( x = -2 \), \( y = 5 \), yes. So that works.

Wait, but maybe I made a mistake. Wait, the original function is \( f(x) = |x| \), which has a slope of 1 for \( x > 0 \). The transformed function has a steeper slope. So the vertical stretch factor \( a \) is \( \frac{5}{2} \)? Wait, no, when \( x = 4 \), \( y = 10 \), so \( a = \frac{10}{4} = 2.5 = \frac{5}{2} \). So the function is \( g(x) = \frac{5}{2}|x - 0| + 0 \), so \( g(x) = \frac{5}{2}|x| \).

Wait, but let's confirm the form: \( a|x - h| + k \). Here, \( h = 0 \), \( k = 0 \), \( a = \frac{5}{2} \). So the function rule is \( g(x) = \frac{5}{2}|x - 0| + 0 \), which is \( g(x) = \frac{5}{2}|x| \).

Wait, maybe I misread the graph. Let me check again. The graph: at \( x = 2 \), the \( y \)-value is 5? Wait, no, when \( x = 2 \), the line is at \( y = 5 \)? Wait, the grid: each square is 1 unit. So from \( (0,0) \) to \( (4, 10) \), so the slope is \( 10/4 = 2.5 \), so \( a = 2.5 = 5/2 \). So the equation is \( g(x) = \frac{5}{2}|x| \).

Answer:

\( g(x) = \frac{5}{2}|x - 0| + 0 \) or simplified \( g(x) = \frac{5}{2}|x| \)