Question

find g(x), where g(x) is the translation 9 units left and 4 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 \( y = f(x) \), a horizontal translation \( h \) units left is \( y = f(x + h) \), and a vertical translation \( k \) units down is \( y = f(x) - k \). The general form for a transformed absolute - value function is \( g(x)=a|x - h|+k \), where:

• If we have a horizontal translation: moving the graph of \( y = |x| \) \( h \) units left means we replace \( x \) with \( x+h \) in the function. In the form \( a|x - h|+k \), a horizontal shift \( h \) units left corresponds to \( h=- 9 \) (because \( y = |x-(-9)|=|x + 9| \)).
• For the vertical translation: moving the graph 4 units down means we subtract 4 from the function. In the form \( a|x - h|+k \), a vertical shift 4 units down corresponds to \( k=-4 \). Also, since there is no vertical stretch or compression (only translation), \( a = 1 \).

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

Given \( f(x)=|x| \), after translating 9 units left, we get \( f(x + 9)=|x + 9| \). Then, translating 4 units down, we subtract 4 from the function. So \( g(x)=|x+9|-4 \). In the form \( a|x - h|+k \), we can rewrite \( |x + 9| \) as \( 1\times|x-(-9)| \) and \( - 4\) as \(+(-4) \). So \( a = 1 \), \( h=-9 \), and \( k = - 4 \), and \( g(x)=1|x-(-9)|+(-4)=|x + 9|-4 \).

Answer:

\( g(x)=|x + 9|-4 \)