# Graphs of polynomials and rational functions

> Edexcel International GCSE Further Pure Mathematics · 4PM1 (2016 spec, Higher)
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s4-graphs-of-polynomials-and-rational/

This guide teaches you to sketch polynomial graphs and rational functions with linear denominators for your Edexcel IGCSE Further Pure Mathematics exam. You will practise identifying intercepts, end behaviour and horizontal/vertical asymptotes, plus follow official exam labelling rules.

**Prerequisites:** [Solving linear and quadratic equations to find intercepts](https://www.owlsprep.com/study/edexcel-igcse-fpm-algebra-solving-equations/); Understanding standard coordinate system labelling conventions

## Learning objectives

- Sketch polynomial graphs by identifying shape, intercepts and end behaviour
- Identify vertical and horizontal asymptotes for rational functions with linear denominators
- Sketch rational functions with linear denominators following exam labelling conventions
- Avoid common exam errors for graph sketch questions

## Sketching Polynomial Graphs

**Polynomial end behaviour** — The direction the graph extends as $x \to +\infty$ and $x \to -\infty$, determined by the polynomial degree (even/odd) and the sign of the leading coefficient.

To sketch a polynomial graph, follow 3 core steps: 1. Find all x-intercepts by solving $f(x)=0$, and the y-intercept by calculating $f(0)$. 2. Identify end behaviour: even degree polynomials have both ends pointing the same direction, odd degree have ends pointing opposite directions. Positive leading coefficients have the right end pointing upwards, negative have it pointing downwards. 3. Draw a smooth curve through all intercepts, matching the end behaviour, and label all required features.

**Worked example:** Sketch the graph of $f(x) = x^3 - 4x$, labelling all intercepts and showing correct shape.

1. Find y-intercept: substitute $x=0$, $f(0)=0$, so y-intercept at $(0, 0)$
2. Find x-intercepts: solve $x^3 - 4x = 0 \to x(x^2 -4) = 0 \to x(x-2)(x+2)=0$, so intercepts at $(-2, 0)$, $(0, 0)$, $(2, 0)$
3. Identify end behaviour: leading term is $x^3$, degree 3 (odd), leading coefficient positive. So as $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to -\infty$
4. Draw a smooth curve passing through all intercepts, matching the end behaviour. Label the graph $f(x)=x^3 -4x$, axes, and all intercept coordinate pairs.

> **Exam tip:** Always label every intercept with its full coordinate pair, not just the numerical value on the axis, to avoid losing easy marks.

*Calculator:* allowed

## Identifying Asymptotes for Linear Denominator Rational Functions

**Coordinate-parallel asymptotes** — Vertical asymptotes occur at x-values that make the linear denominator zero (with non-zero numerator at that x). Horizontal asymptotes are the fixed y-value the graph approaches as $x \to \pm\infty$.

For rational functions of the form $f(x) = \frac{P(x)}{cx + d}$ where $P(x)$ is a polynomial and denominator is linear: 1. Vertical asymptote: solve $cx + d = 0$ to get $x = -\frac{d}{c}$. 2. Horizontal asymptote: if degree of $P(x) < 1$, $y=0$; if degree of $P(x) = 1$, $y = \frac{\text{leading coefficient of } P(x)}{c}$.

**Worked example:** Find the vertical and horizontal asymptotes of $f(x) = \frac{2x + 3}{x - 1}$

1. Find vertical asymptote: set denominator equal to zero: $x - 1 = 0 \to x = 1$
2. Find horizontal asymptote: degree of numerator (1) equals degree of denominator (1). Leading coefficient of numerator is 2, denominator is 1, so $y = \frac{2}{1} = 2$
3. Verify: as $x \to \pm\infty$, constant terms become negligible, so $f(x) \approx \frac{2x}{x} = 2$, confirming the horizontal asymptote.

> **tip**
>
> Asymptotes must always be drawn as dashed lines on sketches, with their equation written clearly next to the line to earn full marks.

*Calculator:* allowed

## Sketching Linear Denominator Rational Functions

To sketch a linear denominator rational function, follow these steps after finding asymptotes: 1. Find x-intercept by solving numerator = 0, y-intercept by substituting $x=0$. 2. Test values of x on either side of the vertical asymptote to find whether the graph approaches $+\infty$ or $-\infty$ on each side. 3. Draw the two separate branches of the graph, approaching the asymptotes, passing through intercepts, and label all required features.

**Worked example:** Sketch $f(x) = \frac{2x + 3}{x - 1}$, labelling all intercepts and asymptotes.

1. Use previously found asymptotes: draw dashed lines for $x=1$ (vertical) and $y=2$ (horizontal), label both equations.
2. Find y-intercept: substitute $x=0$, $f(0) = \frac{3}{-1} = -3$, so intercept at $(0, -3)$
3. Find x-intercept: set numerator equal to zero: $2x + 3 = 0 \to x = -1.5$, so intercept at $(-1.5, 0)$
4. Test values left of $x=1$: $x=0.5$ gives $f(x) = \frac{4}{-0.5} = -8$, so left branch approaches $-\infty$ near $x=1$ and $y=2$ as $x \to -\infty$, passing through $(-1.5, 0)$ and $(0, -3)$
5. Test values right of $x=1$: $x=2$ gives $f(x)=7$, so right branch approaches $+\infty$ near $x=1$ and $y=2$ as $x \to +\infty$
6. Label all intercept coordinates, axes, and the function equation next to the curve.

> **Exam tip:** Never let your graph cross a vertical asymptote, even if you are drawing quickly. You will lose marks if your curve intersects a vertical asymptote line.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Drawing asymptotes as solid lines instead of dashed lines
  - Why it fails: The exam rubric explicitly requires dashed lines for asymptotes to distinguish them from the curve, costing 1 mark per incorrectly drawn asymptote.
  - Correct: Use a dashed line for all asymptotes, and write their full equation next to the line.
- **Wrong:** Forgetting to label the function equation on the sketched curve
  - Why it fails: The specification requires you to write the function equation on each curve, so unlabelled curves lose marks even if the shape is correct.
  - Correct: Write the full function equation next to the curve immediately after drawing it.
- **Wrong:** Assuming all rational functions have horizontal asymptote $y=0$
  - Why it fails: If the numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients, not zero.
  - Correct: Always compare the degree of the numerator and denominator to calculate the horizontal asymptote value.
- **Wrong:** Calculating stationary points to draw polynomial graphs for this sub-topic
  - Why it fails: Stationary point analysis is not required here, and wasting time on these calculations uses up time needed for other questions.
  - Correct: Only use intercepts and end behaviour to sketch polynomial graphs for this sub-topic, unless explicitly told to find stationary points.

## Cheatsheet

| Graph Type | Core Steps | Required Labels |
| --- | --- | --- |
| Polynomial | 1. Find x/y intercepts 2. Identify end behaviour from leading term 3. Draw smooth curve | Intercept coordinates, axes, function equation, correct shape |
| Rational (linear denominator) | 1. Find vertical asymptote (denominator=0) 2. Find horizontal asymptote 3. Find intercepts 4. Test values around vertical asymptote | Intercept coordinates, dashed asymptote lines + equations, axes, function equation |

## What's next

Now that you can sketch polynomial and rational function graphs, you are ready to move on to related topics in the Edexcel IGCSE Further Pure Mathematics S4 unit. Next, you will learn how to use these graph sketches to solve equations, including intersections between different graph types. Later, you will combine this knowledge with calculus from S9 to analyse stationary points, maxima and minima, and refine your graph sketches with more precise detail. Make sure you practise sketching multiple examples of both graph types before moving on, as these foundational graphing skills are tested regularly across multiple sections of the 4PM1 exam, including algebra and calculus questions.

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