Question

find g(x), where g(x) is the translation 8 units left and 2 units up 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 \( h \) units left replaces \( x \) with \( x + h \), and translating \( k \) units up adds \( k \) to the function. The general form for a transformed absolute - value function is \( g(x)=a|x - h|+k \), where:

• If we translate the parent function \( f(x)=|x| \) \( c \) units to the left, we replace \( x \) with \( x + c \) in the function. So, for a left - translation of 8 units, we have \( x\to x+8 \), which in the form \( |x - h| \) means \( h=- 8 \) (because \( |x-(-8)|=|x + 8| \)).
• For a vertical translation of 2 units up, we add 2 to the function. Also, since there is no vertical stretch or compression (the problem only mentions translation), \( a = 1 \).

Step2: Construct the function \( g(x) \)

Using the general form \( g(x)=a|x - h|+k \), with \( a = 1 \), \( h=-8 \) (from the 8 - unit left translation) and \( k = 2 \) (from the 2 - unit up translation), we substitute these values into the formula:
\( g(x)=1\times|x-(-8)|+2=|x + 8|+2 \)

Answer:

\( |x + 8|+2 \)