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 a vertex at \( (0, 0) \). For \( g(x) \), the vertex (the point where the graph changes direction) is at \( (-3, 0) \). In the form \( a|x - h| + k \), the vertex is \( (h, k) \), so \( h = -3 \) and \( k = 0 \).

Step2: Determine the value of \( a \)

The slope of the right - hand side of the absolute - value graph (for \( x\geq h \)) can be used to find \( a \). For the parent function \( f(x)=|x| \), the slope of the right - hand side (\( x\geq0 \)) is \( 1 \). Let's take two points on \( g(x) \). We know the vertex is \( (-3, 0) \), and when \( x = 0 \), from the graph, \( y = 3 \).

Using the formula for the slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \), with \( (x_1,y_1)=(-3,0) \) and \( (x_2,y_2)=(0,3) \), the slope \( m=\frac{3 - 0}{0-(-3)}=\frac{3}{3}=1 \). Since the slope of the right - hand side of the absolute - value function \( a|x - h|+k \) is \( a \) (for \( x\geq h \)), \( a = 1 \).

Step3: Write the function rule for \( g(x) \)

Substitute \( a = 1 \), \( h=-3 \), and \( k = 0 \) into the form \( a|x - h|+k \). We get \( g(x)=1\times|x-(-3)|+0=|x + 3| \)

Answer:

\( g(x)=|x + 3| \) (or in the form \( 1|x-(-3)|+0 \), but the simplified form is \( |x + 3| \))