Rational functions and asymptotes

IB Mathematics: Applications and Interpretation HL· 2.6 Rational functions and graphs· 15 min read

1. What is a Rational Function?★★☆☆☆⏱ 3 min

📘 Definition

Rational function

f(x) = \frac{p(x)}{q(x)}

A function that can be written as the ratio of two polynomials $p (x)$ (numerator) and $q (x)$ (denominator), where $q (x)$ is a non-zero polynomial.

Example:

$f (x) = \frac{x ^{2} - 4}{x + 1}$ is rational; $f (x) = \frac{x}{x + 2}$ is not.

The domain of a rational function excludes any $x$ that makes the denominator zero. These excluded values are closely linked to the location of vertical asymptotes and holes on the graph.

📐 Worked Example

Determine which of the following are rational functions: (a) $f (x) = \frac{3 x ^{3} - 2 x + 1}{x ^{2} - 5}$ , (b) $g (x) = 2 x + 5$

1
Check for (a): Both the numerator $3 x^{3} - 2 x + 1$ and denominator $x^{2} - 5$ are polynomials. So $f (x)$ is a rational function.
2
Check for (b): Any linear polynomial can be written as $\frac{2 x + 5}{1}$ , where $1$ is a constant polynomial. So $g (x)$ is also a rational function.

Exam tip:

Always simplify a rational function first before finding asymptotes, as common factors create holes instead of asymptotes.

2. Vertical Asymptotes and Holes★★☆☆☆⏱ 4 min

📘 Definition

Vertical asymptote

A vertical line $x = a$ that the graph approaches but never crosses, where the function tends to $+ \infty$ or $- \infty$ as $x$ approaches $a$ .

The process to find vertical asymptotes and holes is:

Fully factor the numerator and denominator
Cancel any common shared factors
Canceled factors correspond to holes at that $x$ value
Remaining roots of the denominator are vertical asymptotes

📐 Worked Example

Find all vertical asymptotes and holes of $f (x) = \frac{x ^{2} - 3 x + 2}{x ^{2} - 1}$

1
Factor numerator and denominator:
2
$x^2 - 3x + 2 = (x-1)(x-2), \quad x^2 - 1 = (x-1)(x+1)$
3
Cancel the common factor $(x - 1)$ , giving the simplified function $f (x) = \frac{x - 2}{x + 1}, x \neq = 1$
4
A hole occurs at $x = 1$ . Calculate the $y$ -coordinate by substituting into the simplified function:
5
$y = \frac{1 - 2}{1 + 1} = -\frac{1}{2} \implies \text{hole at } (1, -\frac{1}{2})$
6
Vertical asymptotes occur where the simplified denominator is zero: $x + 1 = 0 ⟹ x = - 1$

3. Horizontal and Oblique Asymptotes★★★☆☆⏱ 5 min

Horizontal and oblique (slant) asymptotes describe the end behaviour of a rational function as $x \to + \infty$ and $x \to - \infty$ . They are found by comparing the degree of the numerator ( $n$ ) and the degree of the denominator ( $d$ ).

If $n < d$ : Horizontal asymptote at $y = 0$
If $n = d$ : Horizontal asymptote at $y = \frac{leading coefficient of numerator}{leading coefficient of denominator}$
If $n = d + 1$ : Oblique asymptote equal to the quotient of polynomial division of numerator by denominator
If $n \geq d + 2$ : No horizontal or oblique asymptote

📐 Worked Example

Find the end behaviour asymptote of $f (x) = \frac{2 x ^{2} + 3 x - 2}{x - 1}$

1
Compare degrees: numerator degree $n = 2$ , denominator degree $d = 1$ . Since $n = d + 1$ , an oblique asymptote exists.
2
Divide the numerator by the denominator:
3
$2x^2 + 3x - 2 = (x - 1)(2x + 5) + 3$
4
The quotient is $2 x + 5$ , and the remainder $3$ approaches $0$ as $x \to \pm \infty$ .
5
So the oblique asymptote is:
6
$y = 2x + 5$

4. Sketching Rational Functions★★★★☆⏱ 6 min

To sketch a complete graph of a rational function, follow this structured process to find all key features before drawing:

Find all $x$ and $y$ intercepts
Find all vertical asymptotes and holes
Find the horizontal or oblique end behaviour asymptote
Test the sign of the function in each interval separated by vertical asymptotes
Draw asymptotes as dashed lines, plot key points, then draw the graph

📐 Worked Example

Find all key features and sketch $f (x) = \frac{x + 2}{x - 3}$

1
Find intercepts: $x$ -intercept when numerator = 0: $x = - 2 ⟹ (- 2, 0)$ . $y$ -intercept when $x = 0$ : $y = \frac{2}{- 3} = - \frac{2}{3} ⟹ (0, - \frac{2}{3})$ .
2
Find vertical features: no common factors, so vertical asymptote at $x = 3$ , no holes.
3
Find end behaviour: degrees are equal ( $n = d = 1$ ), so horizontal asymptote at $y = \frac{1}{1} = 1$ .
4
Test sign: for $x < 3$ , $f (x) < 0$ ; for $x > 3$ , $f (x) > 0$ .
5
Draw dashed lines for asymptotes, plot intercepts, and draw two separate branches approaching the asymptotes.

✓ Quick check

Test your understanding of asymptote rules:

What type of asymptote does $f (x) = \frac{3 x ^{3} + 2 x}{x ^{2} + 1}$ have?
- A: Horizontal asymptote $y = 3$
- B: Oblique asymptote
- C: Vertical asymptote $x = 1$
- D: No asymptotes
Reveal answer
B —
Degree of numerator is 3, degree of denominator is 2, so $n = d + 1$ , which means we have an oblique asymptote.

5. Common Pitfalls

Wrong move:

Forgetting to cancel common factors before finding vertical asymptotes

Why:

Canceled common factors create holes, not vertical asymptotes

Correct move:

Always fully factor and cancel common factors first, then find asymptotes from the simplified function

Wrong move:

Claiming an oblique asymptote exists when the numerator degree is 2 more than the denominator

Why:

Oblique asymptotes only exist when the numerator degree is exactly 1 greater than the denominator

Correct move:

Check the difference of degrees: $n - d = 1$ for oblique asymptotes, any other difference gives no oblique asymptote

Wrong move:

Claiming a graph can never cross a horizontal or oblique asymptote

Why:

Only vertical asymptotes cannot be crossed; graphs can cross horizontal/oblique asymptotes at finite $x$

Correct move:

Remember the 'no crossing' rule only applies to vertical asymptotes

Wrong move:

Using the ratio of constant terms for horizontal asymptotes when degrees are equal

Why:

Leading coefficients (coefficients of the highest power term) are used, not constant terms

Correct move:

When $n = d$ , horizontal asymptote is the ratio of the leading coefficients of numerator and denominator

6. Quick Reference Cheatsheet

Case (n = deg numerator, d = deg denominator)	Vertical asymptotes	End behaviour asymptote
Any n, after canceling common factors	x = a for all roots of simplified denominator	No end behaviour asymptote here
n < d	As above	Horizontal: $y = 0$
n = d	As above	Horizontal: $y = \frac{a _{n}}{b _{d}}$ (leading coefficient ratio)
n = d + 1	As above	Oblique: $y =$ quotient of $p (x) \div q (x)$
n ≥ d + 2	As above	No horizontal or oblique asymptote

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find all asymptotes of a rational function
2022 · 2
Sketch rational function graph using asymptotes

Going deeper

videoPolynomial division review for oblique asymptotesFree review of core prerequisite skill

What's Next

Now that you can identify asymptotes and sketch rational functions, you are ready to solve problems involving rational equations and inequalities, which are frequently tested in IB AI HL exams. Rational functions also appear in many applied contexts, including optimization, rate problems, and inverse variation, so understanding their graphical behaviour is critical for applied question success. You will also reuse asymptote concepts later when studying reciprocal trigonometric functions and other non-polynomial functions in the course.