Question

graph the given functions, f and g, in the same rectangular coordinate system. describe how the graph of g is related to the graph of f.
$f(x) = |x|$
$g(x) = |x| - 6$
use the graphing tool to graph the functions
click to enlarge graph

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) + k \), if \( k>0 \), the graph shifts up; if \( k<0 \), the graph shifts down. Here, \( f(x)=|x| \) and \( g(x)=|x|-6 \).

Step2: Analyze the transformation

Comparing \( g(x) \) with \( f(x) \), we see that \( g(x) = f(x) - 6 \). By the vertical shift rule, subtracting 6 from the function \( f(x) \) shifts the graph of \( f(x) \) down by 6 units.

Answer:

The graph of \( g(x)=|x|-6 \) is the graph of \( f(x)=|x| \) shifted vertically down by 6 units.