Question

find an equation for the graph sketched below

Explanation:

Step1: Identify function type

The graph has a vertical asymptote at $x=0$ (y-axis) and curves down from left, then flatens—characteristic of a reciprocal function shifted vertically. General form: $f(x)=\frac{a}{x}+b$.

Step2: Find vertical shift $b$

As $x\to\pm\infty$, the graph approaches $y=-1$, so $b=-1$. Now $f(x)=\frac{a}{x}-1$.

Step3: Determine $a$ using a point

Pick the point $(1,0)$ on the graph. Substitute $x=1,y=0$: $0=\frac{a}{1}-1\Rightarrow a=1$.

Answer:

$f(x)=\frac{1}{x}-1$