Question

find g(x), where g(x) is the translation 9 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) = submit

Explanation:

Step1: Recall transformation rules

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

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

Here, we have a vertical translation 9 units down. So \( a = 1 \) (no stretch/compression or reflection), \( h = 0 \) (no horizontal shift), and \( k=- 9 \) (since it's 9 units down). Substituting into \( a|x - h|+k \), we get \( 1|x - 0|+(-9) \), which simplifies to \( |x|-9 \) or \( 1|x - 0| - 9 \).

Answer:

\( g(x)=1|x - 0| - 9 \) (or simplified as \( |x|-9 \))