Question

find g(x), where g(x) is the translation 2 units down of f(x) = |x|. write your answer in the form a|x - h| + k, where a, h, and k are integers.

Explanation:

Step1: Recall translation rules

For a function \( y = f(x) \), translating it \( k \) units down means subtracting \( k \) from the function. The general form for vertical translation is \( g(x)=f(x)+k \), where \( k \) is negative for downward translation.

Step2: Apply translation to \( f(x) = |x| \)

We need to translate \( f(x)=|x| \) 2 units down. Using the vertical translation rule, if we translate a function \( y = f(x) \) \( k \) units down, the new function is \( y=f(x)-k \). Here, \( k = 2 \) and \( f(x)=|x| \), so \( g(x)=|x|-2 \). Now, write it in the form \( a|x - h|+k \). Comparing \( |x|-2 \) with \( a|x - h|+k \), we have \( a = 1 \), \( h = 0 \), and \( k=-2 \). So \( g(x)=1|x - 0|+(-2) \) or simply \( g(x)=|x|-2 \) which is in the form \( a|x - h|+k \) with \( a = 1 \), \( h = 0 \), \( k=-2 \).

Answer:

\( g(x)=|x|-2 \) (or in the form \( 1|x - 0| - 2 \))