Rational Functions and Their Graphs

A rational function is a function of the form $y = \frac{p(x)}{q(x)}$y=q(x)p(x)​ where $p$p and $q$q are polynomials. This lesson covers how to sketch rational functions by identifying key features: asymptotes, intercepts, turning points, and behaviour at extremes.

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Vertical Asymptotes

A vertical asymptote occurs at $x = a$x=a if $q(a) = 0$q(a)=0 and $p(a) \neq 0$p(a)=0 (the denominator is zero but the numerator is not).

Near a vertical asymptote, $y \to \pm\infty$y→±∞. Check the sign on each side to determine the direction.

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Horizontal Asymptotes

The horizontal asymptote depends on the degrees of $p$p and $q$q:

Condition	Horizontal asymptote
deg(p) < deg(q)	$y = 0$y=0
deg(p) = deg(q)	$y = \frac{\text{leading coeff of } p}{\text{leading coeff of } q}$y=leading coeff of qleading coeff of p​
deg(p) = deg(q) + 1	Oblique (slant) asymptote — found by polynomial division
deg(p) > deg(q) + 1	No horizontal/oblique asymptote

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Oblique (Slant) Asymptotes

When deg(p) = deg(q) + 1, perform long division:

$y = \frac{p(x)}{q(x)} = mx + c + \frac{\text{remainder}}{q(x)}$y=q(x)p(x)​=mx+c+q(x)remainder​

The oblique asymptote is $y = mx + c$y=mx+c.

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