Question

the graph shows g(x), which is a translation 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. g(x) =

Explanation:

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

The parent function \( f(x) = |x| \) has its vertex at \( (0, 0) \). For the translated function \( g(x) \), looking at the graph, the vertex is at \( (0, -10) \). In the form \( a|x - h| + k \), \( h \) is the horizontal shift and \( k \) is the vertical shift. Here, \( h = 0 \) (no horizontal shift) and \( k = -10 \).

Step2: Determine the value of \( a \)

The slope of the parent function \( f(x) = |x| \) is \( 1 \) for \( x \geq 0 \) and \( -1 \) for \( x < 0 \). Looking at the graph of \( g(x) \), the slope of the right - hand side (for \( x\geq0 \)): we can take two points, say \( (10, 0) \) and \( (0, - 10) \). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-10)}{10 - 0}=\frac{10}{10} = 1 \). So \( a = 1 \) (since the graph has the same "steepness" as \( f(x)=|x| \), no vertical stretch or compression).

Step3: Write the function rule

Using the form \( g(x)=a|x - h|+k \), with \( a = 1 \), \( h = 0 \), and \( k=-10 \), we substitute these values in. So \( g(x)=1\times|x - 0|+(-10)=|x|-10 \).

Answer:

\( g(x)=|x|-10 \)