Question

use translations to graph the given function.
$d(x)=\frac{1}{x + 2}-3$

Explanation:

Step1: Identify parent function

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

Step2: Horizontal translation left 2 units

Transform to: $f(x+2)=\frac{1}{x+2}$

Step3: Vertical translation down 3 units

Transform to: $d(x)=\frac{1}{x+2}-3$

Step4: Find asymptotes

Vertical asymptote: $x=-2$ (from $x+2=0$)
Horizontal asymptote: $y=-3$ (vertical shift)

Step5: Plot key points

For parent $f(x)$: $(1,1), (-1,-1)$
Transformed points: $(1-2,1-3)=(-1,-2)$; $(-1-2,-1-3)=(-3,-4)$

Answer:

The graph of $d(x)=\frac{1}{x+2}-3$ is the parent function $f(x)=\frac{1}{x}$ shifted left 2 units and down 3 units, with vertical asymptote $x=-2$ and horizontal asymptote $y=-3$, passing through points like $(-1,-2)$ and $(-3,-4)$.