Graphing Rational Functions

Graphing Rational Functions — Moosa Academy

A rational function is one polynomial divided by another. Sketching it is not guesswork: the denominator tells you where the curve breaks, comparing the two degrees tells you what happens far out to the sides, and a shared factor tells you where a single point has been quietly punched out.

Concept What counts as a rational function

A rational function has the form

f(x) = \dfrac{a(x)}{b(x)}

, where

a(x)

and

b(x)

are polynomials with no common factor. That last condition matters: if the two share a factor, the graph gains a hole rather than a break, as the last example shows.

Theorem The three asymptote rules

Vertical asymptotes come from the denominator; the horizontal asymptote is decided by comparing the degree of the numerator with the degree of the denominator.

Vertical: find every value of $x$ that makes the denominator zero.

Horizontal: three cases, decided by degree alone.

Condition	Horizontal asymptote
degree of numerator < degree of denominator	$y = 0$
degree of numerator = degree of denominator	$y =$ leading coefficient of numerator $\div$ leading coefficient of denominator
degree of numerator > degree of denominator	None

Only the leading terms matter for the horizontal rule. Everything lower down becomes negligible once $x$ is large enough, which is precisely the situation a horizontal asymptote describes.

Concept Four steps to a sketch

Find the domain — every real number except those that zero the denominator.
Find the vertical and horizontal asymptotes using the rules above.
Find the zeros — the values that make the numerator zero.
Build a table of values on both sides of each vertical asymptote, then draw.

Step 4 is the one students skip. Asymptotes tell you where the curve cannot go, but only sample points tell you which way each branch turns.

Example A function with no horizontal asymptote

Sketch $f(x) = \dfrac{x^{2}}{x - 1}$ .

Denominator zero:

x - 1 = 0

, so

x = 1

Domain: all real numbers except

x = 1

Vertical asymptote:

x = 1

Degrees: numerator

2

, denominator

1

, so

2 > 1

Numerator zero:

x^{2} = 0

, so the curve crosses at

(0,\,0)

Vertical asymptote $x = 1$ ; no horizontal asymptote

$x$	−3	−2	−1	0	1.5	2	3
$f(x)$	−2.25	−1.33	−0.5	0	4.5	4	4.5

Because the numerator outranks the denominator, the outputs keep growing instead of levelling off. There is no horizontal line for the curve to settle against.

Example A point of discontinuity

A point of discontinuity — a hole — appears when

(x - c)

is a factor of both the numerator and the denominator. It cancels, so the curve behaves normally everywhere except at that one abandoned

x

-value, drawn as an open circle.

Sketch $f(x) = \dfrac{x^{2} - 16}{x - 4}$ .

Factor the numerator:

x^{2} - 16 = (x - 4)(x + 4)

Cancel the shared factor:

f(x) = x + 4

, provided

x \neq 4

x = 4

4 + 4 = 8

The graph is the line $f(x) = x + 4$ with a hole at $(4,\,8)$

The shared factor is why the definition insisted on "no common factor". Cancel first, and what looked like an asymptote turns out to be a single missing point.

Note Break or hole?

Both come from a zero denominator, so always factor before deciding.

Factor cancels — the graph has a hole at that

x

-value

Factor does not cancel — the graph has a vertical asymptote there

Summary

A rational function is one polynomial over another, with no factor in common.
Vertical asymptotes sit where the denominator is zero and the factor does not cancel.
If the numerator's degree is lower, the horizontal asymptote is $y = 0$ .
If the degrees are equal, divide the leading coefficients to get the horizontal asymptote.
If the numerator's degree is higher, there is no horizontal asymptote.
A factor shared by numerator and denominator produces a hole, not a break.
Sketch in four steps: domain, asymptotes, zeros, then a table of values.