Cubic and Reciprocal Graphs: Edexcel GCSE Foundation maths revision notes

Cubic and reciprocal graphs

A cubic graph comes from an equation where the highest power of x is x^3, such as y = x^3 or y = x^3 + 4

A reciprocal graph comes from an equation where a number is divided by x, such as y = \frac{1}{x} or y = \frac{20}{x}

Both graphs are curves, not straight lines. You need to know their shapes, work out values from their equations, and draw them from a table of values.

In the exam you are usually asked to match equations such as y = x^3 and y = \frac{1}{x} to sketch graphs, for 1 or 2 marks. Near the end of a calculator paper you may be asked to complete a table of values for a cubic or reciprocal graph and then draw the graph, for about 4 marks.

Cubing numbers, including negatives

x^3 means x \times x \times x. So 2^3 = 2 \times 2 \times 2 = 8, not 6

A negative number cubed is negative: (-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8. This is different from squaring, where (-2)^2 = 4 is positive.

When you substitute a negative value of x, put it in a bracket, and use the brackets on your calculator too: (-3)^3 = -27

Worked example 1: Working out values from a cubic equation

A graph has equation y = x^3 - 5. Work out the value of y when (a) x = 2, (b) x = -3

(a) Put 2 in place of x: y = 2^3 - 5
2^3 = 8, so y = 8 - 5 = 3
Your turn. (b) Put -3 in a bracket: y = (-3)^3 - 5. First work out (-3)^3

Your answer

(-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27
So y = -27 - 5 = -32
Answer: (a) y = 3 (b) y = -32

The shape of a cubic graph

Here is a table of values for y = x^3: when x = -2, -1, 0, 1 and 2, y = -8, -1, 0, 1 and 8

So the graph of y = x^3 goes from the bottom left to the top right, with a bend in the middle where it flattens out as it passes through the origin, O.

Adding a number moves the whole curve up or down. Every value of y = x^3 + 2 is 2 more than for y = x^3, so the curve is the same shape, moved up by 2. It crosses the y-axis at (0, 2). In the same way, y = x^3 - 5 crosses the y-axis at (0, -5)

A minus sign in front of x^3 turns the curve the other way. y = -x^3 goes from the top left down to the bottom right. So does y = 6 - x^3, which crosses the y-axis at (0, 6)

To draw a cubic graph, plot the points from the table with small crosses. Then join them with one smooth curve through every point, drawn freehand. Never use a ruler, and don't join the points with straight lines.

Worked example 2: Completing a table and drawing a cubic graph

Make a table of values for y = x^3 + 3 for values of x from -2 to 2. Then draw the graph of y = x^3 + 3

Use x = -2, -1, 0, 1 and 2. For x = -2: y = (-2)^3 + 3 = -8 + 3 = -5
Your turn. Work out y when x = -1.

Your answer

y = (-1)^3 + 3 = -1 + 3 = 2
In the same way, x = 0, 1 and 2 give y = 3, 4 and 11
The y-values go up by 7, 1, 1 and 7. The steps are not equal, so the points do not lie on a straight line.
Plot the five points (-2, -5), (-1, 2), (0, 3), (1, 4) and (2, 11) with small crosses.
Answer: join the points with one smooth curve from (-2, -5) to (2, 11)

The shape of a reciprocal graph

The reciprocal of a number is 1 divided by that number. The reciprocal of 4 is \frac{1}{4}. So the graph of y = \frac{1}{x} shows the reciprocal of every value of x.

You cannot divide by zero, so x = 0 cannot be used. The graph has no point on the y-axis. And 1 \div x is never 0, so the graph never meets the x-axis either.

When x is positive, y is positive, so one part is in the top right. When x is negative, y is negative, so the other part is in the bottom left.

As x gets bigger, y gets smaller: x = 10 gives y = 0.1. The curve gets closer and closer to the x-axis but never touches it. Near x = 0, y is very big, so the curve gets closer to the y-axis without touching it.

A graph such as y = \frac{20}{x} has the same shape. For every point on it, x \times y = 20: for example (4, 5) and (-2, -10)

Worked example 3: Completing a table and drawing a reciprocal graph

Make a table of values for y = \frac{8}{x} for x = 0.5, 1, 2, 4 and 8. Then draw the graph of y = \frac{8}{x} for values of x from 0.5 to 8

\frac{8}{x} means 8 \div x. For x = 1: y = 8 \div 1 = 8. For x = 2: y = 8 \div 2 = 4
Your turn. Work out y when x = 0.5

Your answer

y = 8 \div 0.5 = 16, because there are 16 halves in 8
In the same way, x = 4 gives y = 2, and x = 8 gives y = 1
Plot (0.5, 16), (1, 8), (2, 4), (4, 2) and (8, 1) with small crosses. Check the scales: the two axes are numbered differently.
Answer: join the points with one smooth curve from (0.5, 16) to (8, 1). It gets closer to the x-axis as x gets bigger, but never touches it.

Recognising graphs from their equations

Many questions show some sketch graphs and ask which one matches an equation. Learn the shape that goes with each kind of equation.

y = x, or an equation such as y = 3x - 1: a straight line.
y = x^2: a U shape, the same on both sides of its lowest point.
y = x^3: from bottom left to top right, flattening out in the middle.
y = -x^3: from top left to bottom right, flattening out in the middle.
y = \frac{1}{x}: two separate parts, in opposite corners, that never touch the axes.

If two graphs have the same shape, find where each crosses the y-axis by putting x = 0 into its equation, or substitute a value such as x = 2 to see which way the curve goes.

Worked example 4: Matching equations to sketch graphs

Here are four sketch graphs, A, B, C and D. Match each of these equations to its graph: y = x^3 + 2, y = 2 - x^3, y = x^2 + 2 and y = \frac{1}{x}

Graph A is in two separate parts that never touch the axes, so it is y = \frac{1}{x}
Graph C is a U shape, so it is y = x^2 + 2
Graphs B and D are both cubic curves through (0, 2). Try x = 2 in y = x^3 + 2: 2^3 + 2 = 10. This curve goes up on the right, so it is graph D.
Your turn. For y = 2 - x^3, work out y when x = 2

Your answer

y = 2 - 2^3 = 2 - 8 = -6. This curve goes down on the right, so it is graph B.
Answer: y = \frac{1}{x} is A, y = 2 - x^3 is B, y = x^2 + 2 is C and y = x^3 + 2 is D

Reading a graph and checking points

To find y for a given value of x, go straight up or down from that value on the x-axis to the curve, then straight across to the y-axis. To find x for a given y, do it the other way round.

Check the scale on each axis before you count squares. A value read from a curve is an estimate, so an answer close to the true value gets the marks.

To check whether a point is on a graph, substitute its x-value into the equation. Is (-3, 27) on y = x^3? (-3)^3 = -27, not 27, so it is not.

On a reciprocal graph such as y = \frac{18}{x}, the two coordinates of every point multiply to give 18. This helps you find a missing coordinate.

Worked example 5: Reading values from a cubic graph

Here is the graph of y = x^3 - 3 for values of x from -2 to 2. (a) Use the graph to find the value of y when x = -1 (b) Use the graph to find an estimate for the value of x when y = 2

Check the scales: on the x-axis each square is 0.5, and on the y-axis each square is 1.
Your turn. (a) Go down from x = -1 to the curve, then across to the y-axis. What is the value of y?

Your answer

The curve is level with -4 on the y-axis, so y = -4. Check: (-1)^3 - 3 = -1 - 3 = -4
(b) Go across from y = 2 to the curve, then down to the x-axis. You arrive between 1.5 and 2, a little less than halfway, at about 1.7
Check with the equation: 1.7^3 - 3 = 4.913 - 3 = 1.913, which is close to 2. So 1.7 is a good estimate.
Answer: (a) y = -4 (b) x is about 1.7

Worked example 6: Finding missing coordinates on a reciprocal graph

The points (6, p) and (q, -2) are on the graph of y = \frac{18}{x}. Work out the value of p and the value of q.

For (6, p), x = 6. So p = 18 \div 6 = 3
For (q, -2), the coordinates multiply to give 18: q \times (-2) = 18
Your turn. Work out the value of q.

Your answer

q = 18 \div (-2) = -9
Check: 18 \div (-9) = -2. A negative x gives a negative y, on the bottom-left part of the graph.
Answer: p = 3 and q = -9

Common mistakes

Treating the table as if it were for a straight line. Work out every y-value from the equation; the steps in a cubic or reciprocal table are not equal.
Getting the sign wrong when cubing a negative number. (-2)^3 is -8, not 8, so put negative values in brackets.
Joining the points with straight lines. Draw one smooth curve through every point, freehand, not with a ruler.
Mixing up y = x^2 and y = x^3. x^2 gives a U shape; x^3 goes from one corner to the opposite corner with a bend in the middle.
Mixing up y = x^3 and y = -x^3. Substitute x = 2: a positive y means the curve goes up on the right.
Letting a reciprocal graph touch the axes. y = \frac{1}{x} never meets either axis, and its two parts are top right and bottom left.