Question

find g(x), where g(x) is the reflection across the x-axis 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 reflection over x-axis

To reflect a function \( y = f(x) \) across the \( x \)-axis, we use the transformation \( g(x)=-f(x) \).

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

Given \( f(x) = |x| \), reflecting across the \( x \)-axis gives \( g(x)=-|x| \). Now, we write this in the form \( a|x - h|+k \). Here, \( a=-1 \), \( h = 0 \) (since there is no horizontal shift), and \( k = 0 \) (since there is no vertical shift other than the reflection). So \( g(x)=-1|x - 0|+0 \), which simplifies to \( g(x)=-|x| \).

Answer:

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