Question

draw a graph of $f(x) = \frac{-4}{2x - 4}$ 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:
$$2x - 4 = 0$$
Solve for $x$:
$$2x = 4 \implies x = 2$$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator.
Numerator degree = 0, denominator degree = 1. For $\frac{\text{lower degree}}{\text{higher degree}}$, horizontal asymptote is $y=0$.

Step3: Plot an additional point

Choose $x=0$:
$$f(0) = \frac{-4}{2(0)-4} = \frac{-4}{-4} = 1$$
Point: $(0, 1)$

Answer:

1. Vertical asymptote: $x=2$ (dashed vertical line at $x=2$)
2. Horizontal asymptote: $y=0$ (dashed horizontal line along the x-axis)
3. Plot the point $(0, 1)$, then draw the hyperbola branches: one branch in the second quadrant (approaching $x=2$ from the left and $y=0$) and one branch in the fourth quadrant (approaching $x=2$ from the right and $y=0$).