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. g(x) =

Explanation:

Step1: Recall translation rules

For a function \( y = f(x) \), translating it \( k \) units down means subtracting \( k \) from the function, so \( y = f(x)-k \). Here, \( f(x)=\vert x\vert \) and we translate 2 units down, so \( k = 2 \). Also, the form is \( a\vert x - h\vert + k \). For the parent function \( f(x)=\vert x\vert \), \( a = 1 \), \( h = 0 \) (since there's no horizontal shift), and after vertical shift down 2, \( k=- 2 \).

Step2: Construct \( g(x) \)

Using the form \( a\vert x - h\vert + k \), with \( a = 1 \), \( h = 0 \), and \( k=-2 \) (because shifting down 2 units), we get \( g(x)=1\vert x - 0\vert+(-2)=\vert x\vert - 2 \).

Answer:

\( \vert x\vert - 2 \) (or in the form \( 1\vert x - 0\vert - 2 \))