Question

use translations to graph the given function.
$g(x)=\frac{1}{x}-3$

Explanation:

Step1: Identify parent function

Parent function: $f(x)=\frac{1}{x}$

Step2: Analyze transformation type

Vertical shift down by 3 units: $g(x)=f(x)-3$

Step3: Transform key points/asymptotes

• Parent asymptotes: $x=0$, $y=0$
• New asymptotes: $x=0$, $y=0-3=-3$
• Example point: $(1,1)$ on $f(x)$ becomes $(1,1-3)=(1,-2)$; $(-1,-1)$ becomes $(-1,-1-3)=(-1,-4)$

Answer:

The graph of $g(x)=\frac{1}{x}-3$ is the graph of $y=\frac{1}{x}$ shifted vertically downward by 3 units. Its vertical asymptote is $x=0$ and horizontal asymptote is $y=-3$, with key points such as $(1,-2)$ and $(-1,-4)$ plotted to match the translated curve.