Question

find g(x), where g(x) is the translation 7 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 means subtracting \( k \) from the function, i.e., \( y = f(x)-k \). Also, the general form \( a|x - h|+k \): \( a \) is the vertical stretch/compression (1 here as no stretch), \( h \) is horizontal shift (0 here as no horizontal shift), \( k \) is vertical shift.

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

We need to translate \( f(x) = |x| \) 7 units down. So \( g(x)=|x| - 7 \). In the form \( a|x - h|+k \), \( a = 1 \), \( h = 0 \), \( k=-7 \). So \( g(x)=1|x - 0|+(-7) \) or simply \( |x|-7 \).

Answer:

\( |x| - 7 \) (or in the form \( 1|x - 0| - 7 \))