# Rational functions and graphs

> CIE A-Level Further Mathematics · Further Pure 1
> Source: https://www.owlsprep.com/study/cie-9231-u1-rational-functions-and-graphs/

This sub-topic covers how to analyze and sketch graphs of rational functions, including finding all types of asymptotes, intercepts, and key features to produce accurate, exam-ready sketches.

**Prerequisites:** [Polynomial division and factoring](https://www.owlsprep.com/study/cie-9231-u1-polynomials/); [Basic coordinate geometry and curve sketching](https://www.owlsprep.com/study/cie-a-level-mathematics-coordinate-geometry/)

## Learning objectives

- Define rational functions and identify their key features
- Find vertical, horizontal and oblique asymptotes of rational functions
- Calculate intercepts and intersection points with asymptotes
- Produce accurate, exam-ready sketches of rational functions
- Use graphs of rational functions to solve simple problems

## Core Definitions and Intercepts

**Rational Function** — A function written as the ratio of two polynomials $P(x)$ (numerator) and $Q(x)$ (denominator), where $Q(x)$ is not the zero polynomial. The domain excludes all $x$ that make $Q(x) = 0$.

*Notation:* f(x) = \frac{P(x)}{Q(x)}

*Example:* $f(x) = \frac{x^2 - 1}{x - 2}$ is a rational function; $f(x) = \frac{\sqrt{x}}{x+1}$ is not.

To start analyzing any rational function, first calculate the intercepts, the key points where the graph crosses the axes.

**Worked example:** Find all intercepts of $f(x) = \frac{x^2 - 2x - 3}{2x - 4}$

1. Find the y-intercept by substituting $x = 0$:
2. $$f(0) = \frac{0 - 0 - 3}{0 - 4} = \frac{3}{4}$$
3. The y-intercept is at $(0, \frac{3}{4})$.
4. Find x-intercepts by setting the numerator equal to zero (check denominator is non-zero):
5. $$x^2 - 2x - 3 = 0 \implies (x-3)(x+1) = 0 \implies x = 3, x = -1$$
6. Denominator at $x=3$ is $2 \neq 0$, at $x=-1$ is $-6 \neq 0$, so x-intercepts are at $(3, 0)$ and $(-1, 0)$.

> **Exam tip:** Always check for common factors between numerator and denominator: a shared root creates a hole, not an x-intercept.

## Vertical and Horizontal Asymptotes

**Asymptote** — A line that the graph of a rational function approaches arbitrarily closely as $x$ or $f(x)$ approaches $\pm\infty$.

Vertical asymptotes occur at $x$-values that make the denominator zero (after removing removable discontinuities). For horizontal asymptotes, we compare the degrees of the numerator ($\deg P$) and denominator ($\deg Q$):

- If $\deg P < \deg Q$: horizontal asymptote at $y = 0$
- If $\deg P = \deg Q$: horizontal asymptote at $y = \frac{a_n}{b_m}$, where $a_n$ = leading coefficient of $P$, $b_m$ = leading coefficient of $Q$
- If $\deg P > \deg Q$: no horizontal asymptote

**Worked example:** Find all vertical and horizontal asymptotes of $f(x) = \frac{3x^2 - 3}{2x^2 - 8}$

1. Factor numerator and denominator to check for common factors:
2. $$f(x) = \frac{3(x-1)(x+1)}{2(x-2)(x+2)}$$
3. No common factors, so denominator is zero at $x=2$ and $x=-2$, so vertical asymptotes are $x=2$ and $x=-2$.
4. Compare degrees: both numerator and denominator are degree 2, so calculate horizontal asymptote:
5. $$y = \frac{\text{Leading coefficient of } P}{\text{Leading coefficient of } Q} = \frac{3}{2}$$
6. Horizontal asymptote is $y = \frac{3}{2}$.

## Oblique (Slant) Asymptotes

Oblique asymptotes only occur when the degree of the numerator is exactly one greater than the degree of the denominator. We find the equation using polynomial division.

**Oblique Asymptote** — A non-vertical, non-horizontal straight line that the graph approaches as $x \to \pm\infty$. The equation is given by the quotient of the polynomial division (the remainder can be ignored as it tends to zero).

**Worked example:** Find the oblique asymptote of $f(x) = \frac{2x^2 - 3x + 1}{x + 2}$

1. Confirm degrees: numerator degree 2, denominator degree 1, difference is 1, so oblique asymptote exists.
2. Divide numerator by denominator:
3. $$2x^2 - 3x + 1 = (x+2)(2x - 7) + 15$$
4. Rewrite $f(x)$:
5. $$f(x) = 2x - 7 + \frac{15}{x+2}$$
6. As $x \to \pm\infty$, $\frac{15}{x+2} \to 0$, so $f(x)$ approaches $y = 2x - 7$, which is the oblique asymptote.

## Full Sketching Process

Follow this consistent step-by-step process to produce an exam-quality sketch that includes all required key features:

1. Factor numerator and denominator, identify any holes (removable discontinuities)
2. Find all x-intercepts and the y-intercept
3. Find all vertical, horizontal or oblique asymptotes
4. Check for any intersection points between the curve and non-vertical asymptotes
5. Test the sign of $f(x)$ in regions separated by vertical asymptotes, then sketch

**Worked example:** List all key features for a sketch of $f(x) = \frac{x^2 + x}{x - 2}$

1. Factor: $f(x) = \frac{x(x+1)}{x-2}$, no common factors, no holes.
2. Intercepts: roots at $x=0$ and $x=-1$, so intercepts at $(0,0)$ and $(-1, 0)$.
3. Asymptotes: vertical asymptote at $x=2$. Degree of numerator is 1 greater than denominator, so find oblique asymptote by division:
4. $$x^2 + x = (x-2)(x+3) + 6 \implies \text{Oblique asymptote } y = x+3$$
5. Check for intersection between $f(x)$ and $y=x+3$:
6. $$\frac{x^2+x}{x-2} = x+3 \implies x^2 + x = x^2 +x -6 \implies 0 = -6$$
7. No solution, so the curve never crosses the oblique asymptote.

> **Exam tip:** When asked to find the number of solutions to an equation, rearrange to get the rational function equal to a straight line, then count intersections on your sketch.

## Common pitfalls

- **Wrong:** Marking a hole (removable discontinuity) as a vertical asymptote
  - Why it fails: Common factors create holes, not asymptotes, and you will lose marks for incorrect asymptote labels
  - Correct: Always factor both numerator and denominator first, cancel common factors, and only mark asymptotes at remaining roots of the denominator
- **Wrong:** Claiming a horizontal asymptote exists when the degree of the numerator is greater than the denominator
  - Why it fails: Horizontal asymptotes only exist when the degree of the numerator is less than or equal to the degree of the denominator
  - Correct: Compare degrees first: if the difference is 1, you have an oblique asymptote instead of a horizontal one
- **Wrong:** Using the remainder from polynomial division as the oblique asymptote
  - Why it fails: The remainder tends to zero as $x$ approaches infinity, so it does not form the asymptote equation
  - Correct: The oblique asymptote is always the linear quotient term from the division, ignore the remainder
- **Wrong:** Assuming curves never cross horizontal asymptotes
  - Why it fails: Unlike vertical asymptotes, curves can cross horizontal asymptotes at finite values of $x$, only approaching them as $x \to \pm\infty$
  - Correct: Always check for intersections by setting $f(x)$ equal to the asymptote equation to find crossing points if they exist
- **Wrong:** Forgetting to mark intercepts on your sketch
  - Why it fails: Exam markers require all key features (intercepts, asymptotes, holes) to be clearly labelled to get full marks
  - Correct: Always calculate and label both x-intercepts and the y-intercept before drawing the curve

## Cheatsheet

| Degree Comparison | Type of Non-Vertical Asymptote | How to Find Equation |
| --- | --- | --- |
| $\deg(P) < \deg(Q)$ | Horizontal | $y = 0$ |
| $\deg(P) = \deg(Q)$ | Horizontal | $y = \frac{\text{leading } P}{\text{leading } Q}$ |
| $\deg(P) = \deg(Q) + 1$ | Oblique | Quotient from $P \div Q$, ignore remainder |
| $\deg(P) > \deg(Q) + 1$ | None (for 9231) | Not assessed in CIE 9231 |

## What's next

Mastery of rational function graphing is a foundational skill for many topics in CIE A-Level Further Mathematics, including solving rational inequalities, sketching polar curves, and finding areas under curves. The polynomial division and graphical analysis skills you learned here will also support you when you work with parametric curves and reciprocal graphs. A strong understanding of this topic also helps you quickly answer common exam questions asking for the number of solutions to an equation, relying on graphical interpretation rather than lengthy algebraic calculation.

- [Summation of Series](https://www.owlsprep.com/study/cie-9231-u1-summation-of-series/)
- [Matrices](https://www.owlsprep.com/study/cie-9231-u1-matrices/)
- [Polar coordinates](https://www.owlsprep.com/study/cie-9231-u1-polar-coordinates/)

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