Reciprocal Functions

Reciprocal graphs y = k/x: completing the table, drawing the smooth curve, and where the curves sit for negative k.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 22 real past-paper questions below

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The Shape of y = k/x

A reciprocal graph has two separate curves. It never touches either axis, because x cannot be 0 and k/x is never 0.

Rule

As x gets bigger, y gets smaller. Double x and y halves: $x \times y = k$ .

Key idea

The axes are asymptotes: lines the curve gets closer and closer to but never touches. For $y = \frac{k}{x}$ they are $x = 0$ and $y = 0$ .

Key idea

Plan a 240 km trip. At 60 km/h it takes 4 hours, at 80 km/h 3 hours, at 120 km/h just 2 hours. Speed × time is always 240, so the graph of time against speed is a reciprocal curve: $t = \frac{240}{v}$ .

Completing the Table

The 2–3 mark question: complete the table for $y = \frac{10}{x}$ . Divide 10 by each x. Marks are lost on minus signs and decimals, so take one value at a time.

Worked example

1
$x = 2 : y = 10 \div 2 = 5$ — Put x into the rule. $\frac{10}{x}$ means 10 divided by x.
2
$x = - 2 : y = 10 \div (- 2) = - 5$ — Positive ÷ negative is negative. So each negative x gives the same number as before, with a minus.
3
$x = 4 : y = 10 \div 4 = 2.5$ — 4 does not go into 10 exactly, so the answer is a decimal. Write it as 2.5, not 2 r 2.
4
$x = 3 : y = 10 \div 3 = 3.333 \dots$ — This decimal never ends. Work it out first, then round on the next line.
5
$y \approx 3.3$ — Round to 1 decimal place, because that is how the other values in the table are given. Copy the table's accuracy.
6
$x = 0.5 : y = 10 \div 0.5 = 20$ — There are 20 halves in 10, so dividing by a half doubles. The answer gets bigger, not smaller.
7
$2 \times 5 = 10, (- 2) \times (- 5) = 10, 4 \times 2.5 = 10$ — Check: x × y must give 10 every time. If a pair does not, that value is wrong.

Common mistake

Writing 5 for x = 0.5. That is 10 × 0.5. Dividing by 0.5 doubles: 10 ÷ 0.5 = 20.

Common mistake

Dropping the minus sign. 10 ÷ (−2) is −5, not 5. Every negative x in this table gives a negative y.

Drawing the Curve

Plot every point from your table, then join them with smooth curves. $y = \frac{10}{x}$ has two pieces: one for positive x, one for negative x.

Worked example

1
$(1, 10), (2, 5), (3, 3.3), (4, 2.5), (5, 2)$ — Plot the positive-x points first. They all sit in the top-right corner of the grid.
2
$curve 1$ — Join them with one smooth curve, freehand. A curve has no corners, so never use a ruler.
3
$(- 1, - 10), (- 2, - 5), (- 4, - 2.5), (- 5, - 2)$ — Now plot the negative-x points. They sit in the bottom-left corner.
4
$curve 2$ — Join these with a second smooth curve. Do not join it to the first one: x = 0 is not allowed, so there is a gap at the y-axis.
5
$1 \leq x \leq 5 and - 5 \leq x \leq - 1$ — Stop each curve at the end points of the table. Do not stretch it to touch the axes.

Key idea

Most of the drawing marks are for points plotted in the right place, and they are checked against your own table. So plot every point carefully, even if one table value was wrong.

Common mistake

Joining the two pieces across the y-axis, using straight lines, or letting the curve touch an axis.

Negative k

If k is negative, every y changes sign. The curves move to the other two corners of the grid.

Worked example

1
$y = \frac{- 3}{x}$ — Here k = −3. The minus belongs to the top, so y = −3 ÷ x.
2
$x = 1 : y = (- 3) \div 1 = - 3$ — Negative ÷ positive is negative. So when x is positive, the point is below the x-axis.
3
$x = 3 : y = (- 3) \div 3 = - 1$ — A bigger x gives a y closer to 0, just as for positive k.
4
$x = - 1 : y = (- 3) \div (- 1) = 3$ — Negative ÷ negative is positive. So when x is negative, the point is above the x-axis.
5
$x = - 3 : y = (- 3) \div (- 3) = 1$ — Check: x × y = −3 for every pair, so the values are right.
6
$top-left and bottom-right$ — So the two curves sit in the corners that $y = \frac{10}{x}$ leaves empty.

Common mistake

Losing a minus sign: (−3) ÷ (−1) is +3, not −3.

Symmetry of the Graph

Exams ask for the symmetry of a reciprocal graph for 1–3 marks. The answers never change, so learn them once.

Rule

Order of rotational symmetry: 2, about the point (0, 0). Lines of symmetry: $y = x$ and $y = - x$ .

Worked example

1
$(2, 5)$ — Start with a point on $y = \frac{10}{x}$ . It is on the curve because 2 × 5 = 10.
2
$(- 2, - 5)$ — Turn it half a turn about (0, 0): both signs change. (−2) × (−5) = 10, so it is on the curve too. That is rotational symmetry of order 2.
3
$(5, 2)$ — Reflect (2, 5) in the line y = x: x and y swap. 5 × 2 = 10, so y = x is a line of symmetry.
4
$(- 5, - 2)$ — Reflect (2, 5) in the line y = −x: swap them and change both signs. (−5) × (−2) = 10, so y = −x is a line of symmetry.

Key idea

One line of symmetry never meets the curve. For a positive k it is $y = - x$ ; for a negative k it is $y = x$ .

Common mistake

Giving the x-axis or the y-axis as a line of symmetry. Fold along the x-axis and the top-right curve lands in the bottom-right corner, where there is no curve.

Solving k/x = c from the Graph

“Use your graph to solve $\frac{10}{x} = 4$ ” is a 1-mark part. The answer is the x where the curve meets the line y = 4.

Worked example

1
$\frac{10}{x} = 4$ — The left side is your curve $y = \frac{10}{x}$ . The right side is the line y = 4.
2
$draw y = 4$ — Rule a horizontal line across the grid at y = 4, if it is not drawn already.
3
$x \approx 2.5$ — From the point where the line crosses the curve, read straight down to the x-axis.
4
$10 = 4 x$ — Check without the graph: multiply both sides by x.
5
$x = 10 \div 4 = 2.5 ✓$ — Divide both sides by 4. It matches the reading, so the answer is right.

If the number is negative, the line is below the x-axis and meets the other curve.

Worked example

1
$\frac{10}{x} = - 4$ — Draw the line y = −4. It meets the bottom-left curve, so the answer is negative.
2
$x \approx - 2.5$ — Read down from the crossing point. Check: 10 ÷ (−2.5) = −4.

Key idea

Answers read from a graph are accepted within a small range, so read to the nearest small square. $\frac{10}{x} = 0$ has no answer: the curve never meets the x-axis.

Asymptotes of y = k/x + c (Extended)

Extended papers add a number: $y = \frac{4}{x} + 2$ . The curve keeps its shape but moves up 2, and one asymptote moves with it.

Worked example

1
$y = \frac{4}{x} + 2$ — The question. The $\frac{4}{x}$ part is the reciprocal curve; the +2 lifts every point up 2.
2
$x \neq = 0 \Rightarrow asymptote x = 0$ — You cannot divide by 0, so the curve never touches the y-axis. $x = 0$ is one asymptote.
3
$\frac{4}{x} \neq = 0 \Rightarrow y \neq = 0 + 2$ — 4 ÷ x is never 0, so y can never be exactly 2.
4
$y = 2$ — So the line $y = 2$ is the other asymptote: the curve gets closer and closer to it but never reaches it.

Rule

$y = \frac{k}{x} + c$ has asymptotes $x = 0$ and $y = c$ .

The same papers ask where the curve crosses the x-axis.

Worked example

1
$\frac{4}{x} + 2 = 0$ — On the x-axis y = 0, so set the rule equal to 0.
2
$\frac{4}{x} = - 2$ — Take 2 from both sides.
3
$4 = - 2 x$ — Multiply both sides by x to clear the fraction.
4
$x = 4 \div (- 2) = - 2$ — Divide both sides by −2. Check: 4 ÷ (−2) + 2 = 0, so it crosses at (−2, 0).

Common mistake

Writing y = 0 as the second asymptote. The +2 lifts it to y = 2.

Which Method? Read the Question

Look at where x is in the equation.

You see	You do
y = k/x	two curves, never touching the axes
Complete the table	k ÷ x for each value
Draw the graph	smooth curve, no ruler
Symmetry of the graph	order 2; lines y = x and y = −x
Use your graph to solve k/x = c	draw y = c, read x where it crosses
Asymptotes of y = k/x + c	x = 0 and y = c
y = x² or x³	Recognising and Sketching Graph Shapes
y inversely proportional to x	Inverse Proportion: y = k/x

Key idea

Check the table: x × y should be the same number every time.

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