A LevelMathsEdexcelPureRevision NotesAlgebra & FunctionsGraphs of FunctionsSketching Reciprocal Graphs

Sketching Reciprocal Graphs (Edexcel A Level Maths: Pure): Revision Note

Q: How do you find the vertical and horizontal asymptotes of a reciprocal graph?

For y = a/x and y = a/x² the asymptotes are always the axes: x = 0 and y = 0. Vertical asymptote - Find the value of x that makes the denominator zero — you cannot divide by zero, so the curve never reaches it - As x gets close to that value, y shoots off towards ±∞ Horizontal asymptote - Ask what y approaches as x gets very large, positive and negative - a divided by a huge number gets closer and closer to 0, but never equals 0 Page example: y = −3/x² - x² = 0 when x = 0, so the vertical asymptote is x = 0 - As x becomes very large, −3/x² → 0, so the horizontal asymptote is y = 0 - Draw both as dotted lines (Step 4) Transformed graphs: ask the same two questions. For y = 3/(x − 2), x − 2 = 0 gives x = 2; for y = 4/x + 5, y approaches 5, so the horizontal asymptote is y = 5.

Q: How does changing the value of a affect the graph of y = a/x?

The sign of a decides where the curves sit; the size of a decides how steep or L-shaped they are. Sign of a - a > 0: curves in the top-right and bottom-left areas - a 0 puts both curves above the x-axis, a 0, so curves above the x-axis; 6 is large compared with 1, so less L-shaped. Label (1, 6) and (−1, 6) - y = −3/x²: a negative, so curves below the x-axis, through (1, −3) and (−1, −3)

Exam code: 9MA0

Written by: Paul

Reviewed by: Dan Finlay

Updated on 2 July 2026

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Sketching reciprocal curves

What are reciprocal graphs?

Reciprocal graphs involve equations with an $x$ term on the denominator e.g. $\frac{1}{x}$
There are two basic reciprocal graphs to know for A level

$y = \frac{1}{x}$ and $y = \frac{1}{x^{2}}$

The second one of these is always positive

More reciprocal graphs

You also need to recognise graphs where the numerator is not one

Reciprocal Graphs - Sketching Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

The sign of a shows which part of the graph the curves are located

The size of a shows how steep the curves are

The closer a is to 0 the more L-shaped the curves are

Reciprocal Graphs - Sketching Notes Diagram 3, A Level & AS Level Pure Maths Revision Notes

All have two asymptotes
- horizontal, y = 0 (x-axis)
- vertical, x = 0 (y-axis)

How do I sketch a reciprocal graph?

Reciprocal Graphs - Sketching Notes Diagram 4, A Level & AS Level Pure Maths Revision Notes

STEP 1
Use the sign of “a” to locate the curves and use the size of “a” to gauge the steepness of the curve
STEP 2
Sketch the graph
STEP 3
Label the points x = 1 and x = -1 as a guide to the scale of your graph
STEP 4
Draw asymptotes with dotted lines
These graphs do not intercept either axis
Graph transformations of them could cross the axes (see Translations)

Worked Example

Reciprocal Graphs - Sketching Example Diagram, A Level & AS Level Pure Maths Revision Notes

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Questions students ask about Sketching Reciprocal Graphs

Real questions from students revising this topic on Save My Exams

What does the graph of y = 1/x³ look like?

y = 1/x³ has the same basic shape and sits in the same areas as y = 1/x: one curve top-right, one curve bottom-left.

Why: x³ has the same sign as x, so y is positive when x > 0 and negative when x < 0 (odd power)
Asymptotes: x = 0 (the y-axis) and y = 0 (the x-axis), exactly as for y = 1/x
No intercepts: it never touches either axis
Guide points: (1, 1) and (−1, −1)

How it differs from y = 1/x

It approaches the asymptotes faster, so it looks more sharply L-shaped
At x = 0.5: 1/x = 2 but 1/x³ = 8, so it climbs the y-axis more steeply
At x = 2: 1/x = 0.5 but 1/x³ = 0.125, so it hugs the x-axis more closely

Pattern: odd powers (1/x, 1/x³) look like y = 1/x; even powers (1/x², 1/x⁴) look like y = 1/x², with both curves on the same side of the x-axis.

How do you find the vertical and horizontal asymptotes of a reciprocal graph?

For y = a/x and y = a/x² the asymptotes are always the axes: x = 0 and y = 0.

Vertical asymptote

Find the value of x that makes the denominator zero — you cannot divide by zero, so the curve never reaches it
As x gets close to that value, y shoots off towards ±∞

Horizontal asymptote

Ask what y approaches as x gets very large, positive and negative
a divided by a huge number gets closer and closer to 0, but never equals 0

Page example: y = −3/x²

x² = 0 when x = 0, so the vertical asymptote is x = 0
As x becomes very large, −3/x² → 0, so the horizontal asymptote is y = 0
Draw both as dotted lines (Step 4)

Transformed graphs: ask the same two questions. For y = 3/(x − 2), x − 2 = 0 gives x = 2; for y = 4/x + 5, y approaches 5, so the horizontal asymptote is y = 5.

How does changing the value of a affect the graph of y = a/x?

The sign of a decides where the curves sit; the size of a decides how steep or L-shaped they are.

Sign of a

a > 0: curves in the top-right and bottom-left areas
a < 0: the graph is reflected, so the curves are top-left and bottom-right
For y = a/x², a > 0 puts both curves above the x-axis, a < 0 puts both below it

Size of a

The closer a is to 0, the more L-shaped the curves — they hug the axes
The further a is from 0, the less L-shaped, and the curves sit further from the origin

What never changes

The asymptotes are still x = 0 and y = 0, and the graph crosses neither axis

Page examples

y = 6/x²: a = 6 > 0, so curves above the x-axis; 6 is large compared with 1, so less L-shaped. Label (1, 6) and (−1, 6)
y = −3/x²: a negative, so curves below the x-axis, through (1, −3) and (−1, −3)