Question

draw a graph of $f(x) = \frac{-3x}{x-2}$ by first placing the horizontal and vertical asymptotes, then plotting an additional point on the graph.

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $x-2=0 \implies x=2$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator:
Since degrees are equal (both 1), horizontal asymptote is ratio of leading coefficients: $y=\frac{-3}{1}=-3$

Step3: Plot an additional point

Choose $x=0$: $f(0)=\frac{-3(0)}{0-2}=\frac{0}{-2}=0$
Point: $(0, 0)$

Answer:

1. Vertical asymptote: $x=2$ (dashed vertical line at $x=2$)
2. Horizontal asymptote: $y=-3$ (dashed horizontal line at $y=-3$)
3. Plotted point: $(0, 0)$

The graph is a hyperbola approaching these asymptotes, passing through $(0,0)$.