What's the difference between a horizontal and an oblique asymptote?

A horizontal asymptote ($y=c$) occurs when the degree of the numerator is less than or equal to the degree of the denominator. An oblique asymptote ($y=mx+c$) occurs when the degree of the numerator is exactly one greater than the degree of the denominator. A graph cannot have both.

Can the graph of a rational function cross its asymptote?

Yes, a graph can cross its horizontal or oblique asymptote. These asymptotes only describe the end behaviour of the function as x approaches positive or negative infinity. However, a graph can NEVER cross a vertical asymptote, as the function is undefined at that x-value.

Do I always need to find the stationary points for a sketch?

For a full, detailed sketch, yes. Stationary points are crucial for showing the correct shape of the curve, such as local maxima and minima. However, if an exam question only asks for asymptotes and intercepts, you may not need to find them. Always read the question carefully to see what features are required.

What if the numerator and denominator have a common factor?

If a factor $(x-a)$ cancels out, the graph has a 'hole' (a point of removable discontinuity) at $x=a$, not a vertical asymptote. The graph will look like the simplified function, but with an open circle at the point where $x=a$ to show it's undefined there.

Where can I learn Rational functions and graphs for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 1.2 (Rational functions and graphs) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/1-2-rational-functions-and-graphs.

Is Rational functions and graphs (1.2) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Rational functions and graphs.

Rational functions and graphs 9231 — Free…

Rational functions and graphs

Welcome to the world of rational functions. These are functions formed by dividing one polynomial by another, creating intriguing graphs with unique features. Unlike polynomials which produce smooth, continuous curves, rational functions can have breaks, jumps, and boundaries called asymptotes. Mastering the techniques to sketch these graphs is a key skill in Further Mathematics, allowing you to visualise complex mathematical relationships and solve related problems like inequalities.

BY THE END, YOU CAN…

1Identify and determine the equations of vertical, horizontal, and oblique asymptotes for a rational function.
2Find the coordinates of intercepts and stationary points to aid in sketching.
3Sketch the graph of a rational function, showing its key features and behaviour near asymptotes.
4Use graphical sketches to solve inequalities involving rational functions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Decoding Rational Graphs

Rational functions are essentially fractions made of polynomials. Their graphs often have 'breaks' called asymptotes, which are invisible boundary lines that the curve approaches but (usually) never touches.

THINK OF IT LIKE…

Imagine you're navigating a maze in a video game with invisible walls. You can run right up to these walls, getting infinitely closer, but you can never pass through them. These walls are like the asymptotes of a rational function's graph; they dictate the overall shape and guide the path of the curve, especially as it extends towards infinity.

1
Find Asymptotes: Locate vertical asymptotes by setting the denominator to zero. Determine horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator.
2
Find Intercepts: Calculate the y-intercept by setting x=0. Find the x-intercept(s) by setting the numerator to zero.
3
Find Stationary Points: Differentiate the function using the quotient rule, set the derivative to zero, and solve for x to find the coordinates of any turning points.
4
Sketch and Refine: Draw the asymptotes as dashed lines and plot the intercepts and stationary points. Sketch the curve, checking its behaviour near the asymptotes to complete the graph.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · Function: each input x maps to exactly one output f(x)

Loading simulation…

Fetching from GeoGebra…

GeoGebra Rational Function Grapher

Vertical line test — one y per x on graph.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

Function: each input x maps to exactly one output f(x)

Function: each input x maps to exactly one output f(x).

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$y = \frac{P(x)}{Q(x)}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Identifying Asymptotes: The Skeleton of the Graph

Asymptotes form the fundamental structure of a rational function's graph. They are lines that the curve approaches but does not cross (for vertical asymptotes) or describes the end behaviour of the function (for horizontal and oblique asymptotes). There are three types to look for.

Vertical Asymptotes (VA): These occur where the denominator is zero (and the numerator is non-zero). If $y = \frac{P(x)}{Q(x)}$ , the lines $x=a$ are vertical asymptotes for each root $a$ of $Q(x)=0$ . The function is undefined at these x-values.
Horizontal Asymptotes (HA): These describe the behaviour as $x \to \pm \infty$ . Compare the degrees of the numerator (deg(P)) and denominator (deg(Q)).
If deg(P) < deg(Q), the horizontal asymptote is the x-axis, $y=0$ .
If deg(P) = deg(Q), the horizontal asymptote is $y = \frac{\text{leading coefficient of P}}{\text{leading coefficient of Q}}$ .
If deg(P) > deg(Q), there is no horizontal asymptote. Look for an oblique one instead.

Oblique Asymptotes: The Slanted Boundary

When the degree of the numerator is exactly one more than the degree of the denominator, the graph will have an oblique (or slant) asymptote. This is a straight line of the form $y=mx+c$ that the curve approaches as $x \to \pm \infty$ . To find its equation, you must perform polynomial long division.

If $y = \frac{P(x)}{Q(x)}$ with deg(P) = deg(Q) + 1, perform long division to get: \ $y = (ax+b) + \frac{R(x)}{Q(x)}$ \ The equation of the oblique asymptote is $y = ax+b$ .

Completing the Picture: Stationary Points and Curve Behaviour

To refine the sketch, especially between intercepts and asymptotes, we find stationary points. This involves differentiating the function using the quotient rule, setting the derivative to zero, and solving for $x$ . The sign of the second derivative (or the shape of the graph) will tell you if they are maxima or minima. It's also useful to check if the curve crosses its horizontal or oblique asymptote by setting the function equal to the asymptote's equation.

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Sketch the graph of $y = \frac{x^2 - 2x - 3}{x-1}$ . State the equations of any asymptotes and the coordinates of any points where the graph crosses the axes.

1
Asymptotes:

EXAMPLE 2

The curve C has equation $y = \frac{2x^2 + 5}{x^2 - 4}$ . (i) Find the equations of the asymptotes of C. (ii) Find the coordinates of the stationary point of C. (iii) Sketch the curve C.

1
(i) Asymptotes: - Vertical Asymptotes: Set denominator to zero: $x^2 - 4 = 0 \implies (x-2)(x+2)=0$ . The vertical asymptotes are $x=2$ and $x=-2$ . - Horizontal Asymptote: The degree of the numerator (2) equals the degree of the denominator (2). The asymptote is $y = \frac{\text{leading coeff of num}}{\text{leading coeff of den}} = \frac{2}{1} = 2$ . The horizontal asymptote is $y=2$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEARational functions and graphs

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 10

What is a rational function?

TAP OR SPACE TO FLIP

ANSWER

A function that can be expressed as a ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$ , where $Q(x) \neq 0$ .

TAP OR SPACE TO FLIP BACK

1 / 10 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Vertical Asymptotes (VA): These occur where the denominator is zero (and the numerator is non-zero). If $y = \frac{P(x)}{Q(x)}$ , the lines $x=a$ are vertical asymptotes for each root $a$ of $Q(x)=0$ . The function is undefined at these x-values.
✓
Horizontal Asymptotes (HA): These describe the behaviour as $x \to \pm \infty$ . Compare the degrees of the numerator (deg(P)) and denominator (deg(Q)).
✓
If deg(P) < deg(Q), the horizontal asymptote is the x-axis, $y=0$ .
✓
If deg(P) = deg(Q), the horizontal asymptote is $y = \frac{\text{leading coefficient of P}}{\text{leading coefficient of Q}}$ .
✓
If deg(P) > deg(Q), there is no horizontal asymptote. Look for an oblique one instead.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

ExamSolutions: Rational Functions↗

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.1 Roots of polynomial equations NEXT →1.3 Summation of series

Rational functions and graphs

BY THE END, YOU CAN…

1Identify and determine the equations of vertical, horizontal, and oblique asymptotes for a rational function.
2Find the coordinates of intercepts and stationary points to aid in sketching.
3Sketch the graph of a rational function, showing its key features and behaviour near asymptotes.
4Use graphical sketches to solve inequalities involving rational functions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Decoding Rational Graphs

THINK OF IT LIKE…

1
Find Asymptotes: Locate vertical asymptotes by setting the denominator to zero. Determine horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator.
2
Find Intercepts: Calculate the y-intercept by setting x=0. Find the x-intercept(s) by setting the numerator to zero.
3
Find Stationary Points: Differentiate the function using the quotient rule, set the derivative to zero, and solve for x to find the coordinates of any turning points.
4
Sketch and Refine: Draw the asymptotes as dashed lines and plot the intercepts and stationary points. Sketch the curve, checking its behaviour near the asymptotes to complete the graph.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · Function: each input x maps to exactly one output f(x)

Loading simulation…

Fetching from GeoGebra…

GeoGebra Rational Function Grapher

Vertical line test — one y per x on graph.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

Function: each input x maps to exactly one output f(x)

Function: each input x maps to exactly one output f(x).

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$y = \frac{P(x)}{Q(x)}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Identifying Asymptotes: The Skeleton of the Graph

Vertical Asymptotes (VA): These occur where the denominator is zero (and the numerator is non-zero). If $y = \frac{P(x)}{Q(x)}$ , the lines $x=a$ are vertical asymptotes for each root $a$ of $Q(x)=0$ . The function is undefined at these x-values.
Horizontal Asymptotes (HA): These describe the behaviour as $x \to \pm \infty$ . Compare the degrees of the numerator (deg(P)) and denominator (deg(Q)).
If deg(P) < deg(Q), the horizontal asymptote is the x-axis, $y=0$ .
If deg(P) = deg(Q), the horizontal asymptote is $y = \frac{\text{leading coefficient of P}}{\text{leading coefficient of Q}}$ .
If deg(P) > deg(Q), there is no horizontal asymptote. Look for an oblique one instead.

Oblique Asymptotes: The Slanted Boundary

If $y = \frac{P(x)}{Q(x)}$ with deg(P) = deg(Q) + 1, perform long division to get: \ $y = (ax+b) + \frac{R(x)}{Q(x)}$ \ The equation of the oblique asymptote is $y = ax+b$ .

Completing the Picture: Stationary Points and Curve Behaviour

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Sketch the graph of $y = \frac{x^2 - 2x - 3}{x-1}$ . State the equations of any asymptotes and the coordinates of any points where the graph crosses the axes.

1
Asymptotes:

EXAMPLE 2

The curve C has equation $y = \frac{2x^2 + 5}{x^2 - 4}$ . (i) Find the equations of the asymptotes of C. (ii) Find the coordinates of the stationary point of C. (iii) Sketch the curve C.

1
(i) Asymptotes: - Vertical Asymptotes: Set denominator to zero: $x^2 - 4 = 0 \implies (x-2)(x+2)=0$ . The vertical asymptotes are $x=2$ and $x=-2$ . - Horizontal Asymptote: The degree of the numerator (2) equals the degree of the denominator (2). The asymptote is $y = \frac{\text{leading coeff of num}}{\text{leading coeff of den}} = \frac{2}{1} = 2$ . The horizontal asymptote is $y=2$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEARational functions and graphs

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 10

What is a rational function?

TAP OR SPACE TO FLIP

ANSWER

A function that can be expressed as a ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$ , where $Q(x) \neq 0$ .

TAP OR SPACE TO FLIP BACK

1 / 10 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Vertical Asymptotes (VA): These occur where the denominator is zero (and the numerator is non-zero). If $y = \frac{P(x)}{Q(x)}$ , the lines $x=a$ are vertical asymptotes for each root $a$ of $Q(x)=0$ . The function is undefined at these x-values.
✓
Horizontal Asymptotes (HA): These describe the behaviour as $x \to \pm \infty$ . Compare the degrees of the numerator (deg(P)) and denominator (deg(Q)).
✓
If deg(P) < deg(Q), the horizontal asymptote is the x-axis, $y=0$ .
✓
If deg(P) = deg(Q), the horizontal asymptote is $y = \frac{\text{leading coefficient of P}}{\text{leading coefficient of Q}}$ .
✓
If deg(P) > deg(Q), there is no horizontal asymptote. Look for an oblique one instead.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

ExamSolutions: Rational Functions↗

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.1 Roots of polynomial equations NEXT →1.3 Summation of series

Revise Rational functions and graphs for Cambridge 9231 (1.2)

In simple terms

Explore the concept

Key formulas

Full topic notes

Identifying Asymptotes: The Skeleton of the Graph

Oblique Asymptotes: The Slanted Boundary

Completing the Picture: Stationary Points and Curve Behaviour

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Identifying Asymptotes: The Skeleton of the Graph

Oblique Asymptotes: The Slanted Boundary

Completing the Picture: Stationary Points and Curve Behaviour

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Rational functions and graphs