GCSE · Maths · AQA · Spec 8300 · Foundation

Cubic and reciprocal graphs

Join (0, 0) to (1, 1) on y = x³ with a ruler and halfway sits (½, ½). But ½ cubed is ⅛. So what is the curve really doing?

y = x³ − 4x: does the curve follow the ruler?

x: 0. y: 0

slide me along the curve

The ruler line: A table of values for x = −2 to 2 gives (−2, 0), (−1, 3), (0, 0), (1, −3) and (2, 0): slide to each one and check. This straight line joins two of them, (1, −3) and (2, 0). Halfway along, at x = 1.5, it says y = −1.5. Slide the point to x = 1.5 and read the curve instead: about −2.6. The equation settles it: 1.5³ − 4 × 1.5 = 3.375 − 6 = −2.625. The curve is right and the ruler is wrong.

Watch out: The table's lowest point is (1, −3), but that isn't the bottom of the dip. Slide from x = 1 towards x = 1.5 and the curve goes a little lower first, to about −3.08 near x = 1.15, before it climbs. The bottom of a dip doesn't have to land on a point you plotted, so you read it from the curve.

Maths · Your turn

Build the table yourself

Make a table of values for y = x³ − 3x + 2, for x from −2 to 2.

x = −2: y = (−2)³ − 3 × (−2) + 2Put a negative x in brackets before you cube it.
missing step

Which line is step 2?

= 8 + 6 + 2 = 16= −8 + 6 + 2 = 0= −8 − 6 + 2 = −12

Predict, then check

Work out 1 ÷ x for each value before you choose.

Now a different family: y = 1/x. Keep making x smaller, x = ½, then ⅓, then ¼, then 1/100. What happens to y?

Maths · Spot the slip

Where does this graph go wrong?

Draw the graph of y = 6/x for x from −6 to 6.

A student's method — which line goes wrong?

Choose x-values, leaving out x = 0: −6, −3, −2, −1, 1, 2, 3, 6
Work out y = 6 ÷ x for each: −1, −2, −3, −6, 6, 3, 2, 1
Plot (−6, −1), (−3, −2), (−2, −3), (−1, −6), (1, 6), (2, 3), (3, 2), (6, 1)
Draw a smooth curve through the four left-hand points, from (−6, −1) down to (−1, −6)
Carry the curve on from (−1, −6) through (0, 0) and up to (1, 6), then smoothly on to (6, 1)
Finished: one unbroken curve that passes through the origin

Maths · Know it on sight

Which family does each graph belong to?

Sort each equation or description into its family.

Still to sort

Linear (0)

Highest power of x is 1: a straight line.

Quadratic (0)

Highest power of x is 2.

Cubic (0)

Highest power of x is 3.

Where the line is: The number of roots doesn't decide it. A cubic can cross the x-axis once, twice or three times, and it's still cubic if the highest power of x is 3.

Reciprocal (0)

x on the bottom of a fraction, as in y = k/x.

Where the line is: Two separate branches that never touch the axes, not one curve through the middle.

9 of 9 still to sort.

Look for the highest power of x, or for x on the bottom of a fraction.

Watch out: Don't count roots to decide the family. A graph with only one root can still be a cubic.

Maths · Graphs in real life

Read a real answer off a curve

Table of values. v (m/s): 2, 3, 4, 6, 8, 12. t (s): 12, 8, 6, 4, 3, 2.

A toy car travels 24 m at a steady speed. The time it takes, t seconds, is t = 24/v, where v is its speed in m/s. Draw the graph of t against v for 2 ≤ v ≤ 12 on paper. Use your graph to estimate the speed the car needs to cover the 24 m in 2.5 seconds. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

All six points plotted correctly: (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2) (1 mark)
One smooth curve through the points, not straight line segments between them (1 mark)
A line across from t = 2.5 to the curve, then down to the v-axis (or the reading clearly described) (1 mark)
A speed between 9.3 and 9.9 m/s (the exact value is 9.6 m/s) (1 mark)

Model answer

I plotted (2, 12), (3, 8), (4, 6), (6, 4), (8, 3) and (12, 2) and drew one smooth curve through them, bending between the points rather than using a ruler. From 2.5 on the time axis I went across to the curve and down to the speed axis, and read v ≈ 9.6 m/s.

Check your reading in the equation: 24 ÷ 9.6 = 2.5. A ruler line from (8, 3) to (12, 2) would cross t = 2.5 at v = 10, outside the accepted range. That's why the curve matters.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Not just the points in your table. So a cubic curves between them, and y = 1/x never reaches its axes.

What you need to know

A table of values gives coordinate pairs: substitute each x into the equation to calculate its y.
Join plotted points with one smooth curve, never a ruler. The points in between must fit the equation too.
A cubic (highest power of x is 3) has one, two or three roots, where it crosses the x-axis, and can have turning points: a local maximum and a local minimum.
The reciprocal graph y = k/x has two separate branches, the axes as its two asymptotes, and no point at x = 0.
To answer a problem from a graph, read across to the curve and down to the other axis. The answer is an estimate.

The big picture

You draw cubic and reciprocal graphs like any graph: substitute each x to build a table of values, plot the pairs, then draw the curve. The shapes are what's new. A cubic (highest power of x is 3) is one smooth curve with up to three roots and can have a local maximum and minimum. A reciprocal graph, y = k/x, is two separate branches that hug the axes without touching them. Between plotted points the curve still obeys its equation, so you draw it freehand and can read approximate answers from it.

Key points

1Cube a negative in brackets: (−2)³ = −8, because a cube keeps the sign.

2Turning points are read from the curve, and they need not land on a plotted point.

3For y = 1/x, as a positive x shrinks towards 0 the value of y grows without limit; as x grows large, y shrinks towards 0 but never reaches it.

4The solutions of an equation such as x³ − 4x = 2 are the x-values where the curve meets the line y = 2.

5Graph readings are approximate. Give a sensible estimate and, where you can, check it in the equation.

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Worked example

Problem

Use the graph of y = x³ − 4x (the curve at the top of this lesson) to estimate the solutions of x³ − 4x = 2.

⚠ Watch out

Joining the dots with a ruler. Between plotted points the graph follows the equation, not a straight line: a cubic curves, and a reciprocal graph has a gap at x = 0 you must never join across. Unsure? Test an in-between x.

🧠

Memory hook

Ruler for lines, freehand for curves, and lift your pencil at the y-axis for y = k/x.

✓

Check yourself

For y = x³ + x, find y at x = −2. Is y = 2x + x³ − 5 linear, quadratic or cubic? (Answers: −10; cubic, as the highest power is 3.)

Flashcards

(13)

What does a table of values give you?

Coordinate pairs that satisfy the equation. Substitute each x into the equation to calculate its y, then plot each (x, y).

What is (−3)³?

−27. A cube keeps the sign: −3 × −3 = 9, then 9 × −3 = −27.

Why shouldn't you join the plotted points of a cubic with a ruler?

The points between them must fit the equation too, and they don't lie on those straight lines. Draw one smooth curve through every plotted point.

What are the roots of a cubic graph?

The places where the graph intersects the x-axis, where y = 0.

What are the turning points of a cubic graph?

Its local maximum and local minimum: the top of a hump and the bottom of a dip.

Why is a turning point called a 'local' maximum?

It's the highest point nearby, not on the whole graph. Further along, the curve climbs higher still.

Do turning points always fall on a point from your table?

No. For y = x³ − 4x the table gives (1, −3), but the curve dips a little lower, to about (1.15, −3.08). Read turning points from the curve.

How many roots can a cubic graph have?

One, two or three. They're all cubics, because the highest power of x is 3.

What is an asymptote?

A line that a curve gets closer and closer to but never touches.

What are the asymptotes of y = k/x?

The two axes: the y-axis (the line x = 0) and the x-axis (the line y = 0).

Why is there no point on y = 1/x at x = 0?

1 ÷ 0 has no value, so x ≠ 0. The graph never meets the y-axis.

How do you draw a reciprocal graph such as y = 6/x?

Leave x = 0 out of the table, then draw two separate smooth branches. Never join them across the y-axis.

How do you estimate an answer from a graph?

Find the value on one axis, go across to the curve, then down (or up) to the other axis and read it off. The answer is an approximation.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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y = x³ − 4x: does the curve follow the ruler?

Build the table yourself

Where does this graph go wrong?

Which family does each graph belong to?

Read a real answer off a curve

More AQA GCSE Maths topics