Question

How do you graph `f(x)=1/(x^2-2x)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

Vertical asymptotes at `x = 0, x = 2`
Horizontal asymptote at `y = 0`
No holes, no x-intercepts

Explanation:

Rational equation `f(x) = (N(x))/(D(x)) = (a_nx^n+...)/(b_nx^m+...)`

Factor both the numerator and denominator:
`f(x) = 1/ (x(x-2))`

Find x-intercepts `N(x) = 0`:
There are no factors in the numerator.

Find holes:
Holes occur when factors can be cancelled because they are found both in the numerator and denominator. This does not occur in this problem.

Find the vertical asymptotes `D(x) = 0`:
Vertical asymptotes at `x = 0, x = 2`

Find horizontal asymptotes `m>n, y = 0`:
`m = 2, n = 0` so there is a horizontal asymptote at `y = 0`

graph{1/(x^2-2x) [-10, 10, -5, 5]}