Question

How do you graph `f(x)=(x^2-2x)/(x^2-4)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

We can factor `f(x)`.

`f(x) = (x(x - 2))/((x + 2)(x - 2))`

`f(x) = x/(x+ 2)`

This means that there will be a hole at `x = 2`. There will be a vertical asymptote at `x = -2`.

The horizontal asymptote will occur at the ratio between the highest power in the numerator and in the denominator (only if the powers are equal).

Hence, there will be a horizontal asymptote at `y = 1/1 = 1`.

As for intercepts, the graph will pass through the origin, and the origin will serve as the x and y intercept.

Here is the graph:

graph{(x^2 - 2x)/(x^2 - 4) [-10, 10, -5, 5]}

Hopefully this helps!