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.

Explanation:

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

The parent function \( f(x) = |x| \) has its vertex at \( (0, 0) \). For the translated function \( g(x) = a|x - h| + k \), the vertex is at \( (h, k) \). From the graph, the vertex of \( g(x) \) is at \( (0, -8) \), so \( h = 0 \) and \( k = -8 \).

Step2: Determine the value of \( a \)

The parent function \( f(x) = |x| \) has \( a = 1 \) (since it opens upwards with a slope of 1 for \( x \geq 0 \) and -1 for \( x < 0 \)). The graph of \( g(x) \) has the same shape as \( f(x) \) (no vertical stretch or compression), so \( a = 1 \).

Step3: Write the function rule

Substitute \( a = 1 \), \( h = 0 \), and \( k = -8 \) into the form \( a|x - h| + k \). We get \( g(x) = 1|x - 0| + (-8) \), which simplifies to \( g(x) = |x| - 8 \).

Answer:

\( g(x) = |x| - 8 \) (or in the form \( 1|x - 0| - 8 \))