Question

find g(x), where g(x) is the translation 5 units down of f(x) = |x|. write your answer in the form a|x - h| + k, where a, h, and k are integers. g(x) =

Explanation:

Step1: Recall translation rules

For a function \( y = f(x) \), translating it \( k \) units down changes the function to \( y = f(x)-k \). Also, the general form \( a|x - h|+k \): \( a \) is vertical stretch/compression, \( h \) is horizontal shift, \( k \) is vertical shift.

Step2: Identify parameters for \( f(x)=|x| \)

For \( f(x)=|x| \), \( a = 1 \), \( h = 0 \) (no horizontal shift). Translating 5 units down, so \( k=- 5 \) (since down is negative vertical shift).

Step3: Form \( g(x) \)

Substitute into \( a|x - h|+k \): \( a = 1 \), \( h = 0 \), \( k=-5 \). So \( g(x)=1|x - 0|+(-5)=|x|-5 \).

Answer:

\( |x| - 5 \)