Oblique Asymptotes

In cases where you have a rational function where the numerator has a higher degree than the denominator, you may have an oblique or parabolic asymptote. To get these asymptotes, you use long division of the numerator with the denominator and the asymptote is your quotient

Example

$f(x)=\frac{x^2}{x-1}$ long division. you asymptote is your quotient.

The process

Say we are going to want to graph from the equation: Firstly, find the V.A and x ints y = $\frac{(x^2-3x+2)(x-3)}{(x-4)(x+1)}$ y = $\frac{(x-2)(x-1)(x-3)}{(x-4)(x+1)}$ V.A = x = 4, x = -1 X int = 2, 1, 3 Ok, then expand the numerator and denominator, perform long division to find slant asymptote, the S.A is in the form: y = $x-3+\frac{6x-18}{x^2-3x-4}$ Then, we make the behavior chart of the x intercepts.

x	y
-1^-	-inf
-1^+	inf
4^-	-inf
4^+	inf
-inf	BELOW
inf	ABOVE
2	0
1	0
3	0
0	0

Quotient form is your slant asymptote. The one thing that is different with this behavior chart is that it is modified. $x+1+\frac{1}{x-1}$ is the asymptote

Other Example

Lets assign these values: $x+1$ = slant asymptote $\frac{1}{x}$ = behavior

Behavior

Lets make the behavior chart $\frac{(x-1{)}^2}{x}$

x	y
0^-	-inf
0^+	inf
-inf	from behavior. sub in inf into behavior. 1/inf = positibe. Therefore ABOVE Slant asymptote
inf	BELOW Slant Asymptote
So, you graph it out.

There is no need to find the test point