Rational Functions and Asymptotes

Find the asymptotes and axis crossings of curves like y = (ax + b)/(cx + d), sketch them, find unknown constants, and solve equations by drawing a line.

In the exam for Edexcel Further Pure Maths 4PM1 · 17 real past-paper questions below

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Lines a curve never reaches

A class hires a bus for a trip. The more students who join, the less each one pays. The share keeps falling towards 0, but it never reaches 0. Graphs with a fraction like this have lines they get close to but never touch.

Look at $y = \frac{1}{x}$ . You cannot divide by 0, so the curve has no point at $x = 0$ . Near $x = 0$ , y becomes huge. Far out to the left or right, y gets very close to 0.

Rule

An asymptote is a straight line that a curve gets closer and closer to without ever reaching it. For $y = \frac{1}{x}$ , the asymptotes are the lines $x = 0$ and $y = 0$ .

In this lesson the curves are rational functions: one expression divided by another, where the bottom is linear, like $x + 2$ or $2 x - 3$ . Their asymptotes are parallel to the axes.

Rule

An asymptote is a line, so always write it as an equation: $x = - 2$ or $y = 3$ . Writing only $- 2$ or $3$ loses the mark.

Key idea

The word asymptote comes from Greek and means “not falling together”: the curve and the line never meet.

Finding the asymptotes

Most questions use a curve of this shape:

Formula

y = \frac{a x + b}{c x + d}

$a$ — the number in front of x on top.
$c$ — the number in front of x on the bottom.
$d$ — the number on its own on the bottom. With c, it decides where the bottom is 0.
$b$ — the number on its own on top.

Vertical asymptote. The curve breaks where the bottom equals 0, because you cannot divide by 0. Solve $c x + d = 0$ :

Formula

x = - \frac{d}{c}

$d$ — the number on its own on the bottom.
$c$ — the number in front of x on the bottom. The answer is a vertical line.

Horizontal asymptote. When x is huge, the numbers b and d hardly matter. The curve behaves like $\frac{a x}{c x}$ , which is $\frac{a}{c}$ :

Formula

y = \frac{a}{c}

$a$ — the x-number on top.
$c$ — the x-number on the bottom. The answer is a horizontal line.

Worked example

1
Find the asymptotes of $y = \frac{3 x - 6}{x + 2}$ .
2
$x + 2 = 0 \Rightarrow x = - 2$ — The bottom is 0 here. Vertical asymptote: $x = - 2$ .
3
$y \approx \frac{3 x}{x} = 3$ — For huge x, only the x terms matter. Horizontal asymptote: $y = 3$ .

Worked example

1
Find the asymptotes of $y = \frac{4}{x - 1} + 2$ .
2
$x - 1 = 0 \Rightarrow x = 1$ — Vertical asymptote: $x = 1$ .
3
$\frac{4}{x - 1} \to 0 \Rightarrow y \to 2$ — For huge x the fraction is tiny. Horizontal asymptote: $y = 2$ .

Rule

If the top has an $x^{2}$ term and the bottom is linear, there is still a vertical asymptote where the bottom is 0, but no horizontal one.

Common mistake

Sign slips. The bottom $x + 2$ is 0 when $x = - 2$ , not $x = 2$ .

Key idea

Vertical: make the bottom 0. Horizontal: divide the x-numbers, top over bottom.

Crossing the axes

Next find where the curve crosses each axis. These points help you place the curve in the sketch.

Crosses the x-axis: put $y = 0$ . A fraction is 0 only when its top is 0.
Crosses the y-axis: put $x = 0$ .

Worked example

1
Find where $y = \frac{3 x - 6}{x + 2}$ crosses the axes.
2
$3 x - 6 = 0 \Rightarrow x = 2$ — Set the top equal to 0. The point is (2, 0).
3
$y = \frac{3 ( 0 ) - 6}{0 + 2} = - 3$ — Put x = 0. The point is (0, −3).

Worked example

1
Find where $y = \frac{4}{x - 1} + 2$ crosses the axes.
2
$\frac{4}{x - 1} = - 2$ — Put y = 0 and subtract 2.
3
$x - 1 = - 2 \Rightarrow x = - 1$ — Rearrange: $4 = - 2 (x - 1)$ . The point is (−1, 0).
4
$y = \frac{4}{- 1} + 2 = - 2$ — Put x = 0. The point is (0, −2).

Common mistake

Setting the bottom equal to 0 to find the x-axis crossing. The bottom gives the asymptote. The top gives the crossing.

Key idea

Give each crossing as coordinates, like (2, 0) and (0, −3). Examiners look for the pair.

Sketching the curve

A sketch is not an accurate drawing. It shows the shape and the key facts. The two asymptotes cross and cut the page into four corners. The curve has two branches, in two opposite corners.

Steps

1
1. Draw both asymptotes as dashed lines and write their equations on them.
2. Mark the axis crossings and label their coordinates.
3. Find which corner each point is in. One branch passes through them.
4. The other branch sits in the opposite corner. Test one x-value if you are not sure.
5. Draw two smooth branches that bend towards the asymptotes. They never cross or touch them.

Worked example

1
Sketch $y = \frac{3 x - 6}{x + 2}$ .
2
$x = - 2, y = 3$ — The asymptotes cross at (−2, 3).
3
$(2, 0), (0, - 3)$ — Both points are right of $x = - 2$ and below $y = 3$ : the bottom-right corner.
4
$x = - 3 : y = \frac{- 15}{- 1} = 15$ — A test point on the left. It is above $y = 3$ : the top-left corner.
5
$bottom-right and top-left$ — Draw one branch through (0, −3) and (2, 0), and one branch in the top-left corner.

Rule

The mark scheme looks for three things: two branches of the right shape, both asymptotes drawn with their equations, and the axis crossings labelled.

Common mistake

Letting a branch cross an asymptote, or bend back away from it. Each end must get closer and closer to its line.

Key idea

Asymptotes first, crossings second, then let the points tell you which two corners the curve lives in.

Find the constants

Some questions give you the asymptote or a crossing point and ask for the unknown constants. Use the same facts backwards.

Worked example

1
The curve $y = \frac{a x - 4}{2 x + b}$ has the vertical asymptote $x = \frac{3}{2}$ and crosses the x-axis at (2, 0). Find a and b.
2
$2 (\frac{3}{2}) + b = 0$ — The bottom is 0 on the vertical asymptote.
3
$b = - 3$ — Since $3 + b = 0$ .
4
$2 a - 4 = 0$ — At (2, 0) the top is 0.
5
$a = 2$ — So the curve is $y = \frac{2 x - 4}{2 x - 3}$ .
6
$y = \frac{2}{2} = 1$ — Now the horizontal asymptote is $y = 1$ .

Example

A point on the curve works too. Here the y-axis crossing is $y = \frac{- 4}{- 3} = \frac{4}{3}$ , the point $(0, \frac{4}{3})$ . If a question gives you this point, put it into the equation to find a constant.

Common mistake

Mixing up which fact uses which part. An asymptote parallel to the y-axis uses the bottom. An x-axis crossing uses the top.

Key idea

Each fact you are given becomes one equation. Two unknowns need two facts.

Solving with a drawn line

4PM1 also asks you to plot a curve from a table of values, then use it to solve a harder equation. The curve may be rational, a polynomial, or an exponential or log curve. Plot each point and join them with one smooth curve.

The idea: rearrange the equation so one side is the curve you drew. The other side is a straight line. Draw it. Where the line crosses the curve, the x-values are the solutions.

Steps

1
1. Write: curve $= m x + c$ .
2. Clear the fraction and collect everything on one side.
3. Compare with the given equation to find m and c.
4. Draw $y = m x + c$ on the grid and read the x-values where it crosses the curve.

Worked example

1
The graph of $y = \frac{2 x + 1}{x - 1}$ is drawn. Find the line to draw to solve $x^{2} - 2 x - 2 = 0$ .
2
$\frac{2 x + 1}{x - 1} = m x + c$ — Set the curve equal to a line.
3
$2 x + 1 = m x^{2} + (c - m) x - c$ — Multiply both sides by $x - 1$ and expand.
4
$m x^{2} + (c - m - 2) x - c - 1 = 0$ — Collect everything on one side.
5
$m = 1, c - 1 - 2 = - 2, - c - 1 = - 2$ — Match with $x^{2} - 2 x - 2 = 0$ , term by term.
6
$y = x + 1$ — So m = 1 and c = 1. Check: −c − 1 = −2. ✓
7
$x \approx - 0.7 or x \approx 2.7$ — Read the two crossings from the graph.

Example

Check with algebra: $x^{2} - 2 x - 2 = 0$ gives $x = 1 \pm 3$ , which is −0.73 or 2.73. The readings agree.

Common mistake

Reading the y-values of the crossings. The solutions of an equation in x are the x-values.

Key idea

Curve on one side, straight line on the other. The line is the only thing you have to draw.

Which step? Read the question

Read what each part asks, then use the matching tool.

You see	You do
“Asymptote parallel to the y-axis”	Bottom = 0. Write $x = \dots$
“Asymptote parallel to the x-axis”	Divide the x-numbers, top over bottom. Write $y = \dots$
“Crosses the x-axis”	Top = 0, then give the point $(\dots, 0)$
“Crosses the y-axis”	Put $x = 0$ , then give the point $(0, \dots)$
“Sketch the curve”	Dashed asymptotes with equations, labelled crossings, two branches
Unknown constants in the equation	Turn each given fact into an equation
“By drawing a suitable straight line”	Curve $= m x + c$ , compare, draw, read the x-values

Common mistake

Forgetting to label. A correct sketch without the asymptote equations and crossing coordinates loses marks.

Key idea

Bottom zero: vertical asymptote. Top zero: x-axis crossing. Huge x: horizontal asymptote.

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