Chapter 3 of 8

All 8 chapters
  1. 1Function notation, readunderstood
  2. 2Composite and inverse functions, readunderstood
  3. 3Cubic, reciprocal and circle graphs, readunderstood
  4. 4Exponential graphs, readunderstood
  5. 5Graphs of sin, cos and tan, readunderstood
  6. 6Moving graphs, readunderstood
  7. 7Gradients and areas under graphs, readunderstood
  8. 8Solving with graphs and trials, readunderstood

M4.10.3 Chapter 3

Cubic, reciprocal and circle graphs

What does the graph of y = 1/x look like? Does it ever touch the axes?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Sketch y = x³ and y = 1/x
  • Sketch the circle x² + y² = 25
  • Match an equation to the shape of its graph

You know that y = 2x + 1 gives a straight line, and y = x² gives a U shaped curve. In this chapter you meet three more shapes. Each one comes from a table of values, and each has features you can learn to spot.

First, the cubic y = x³. It means x × x × x. Make a table, with negative values of x as well. Take care: three negative numbers multiplied give a negative answer.

For x from −3 to 3, the table gives y = −27, −8, −1, 0, 1, 8 and 27. The y values are negative when x is negative.

Worked example

Find y when x = −2 for y = x³.

  1. x³ means x × x × x. So y = (−2) × (−2) × (−2).
  2. Multiply the first two: (−2) × (−2) = 4.
  3. Then multiply by the last: 4 × (−2) = −8.

Answer: y = −8.

A graph with the x axis x and the y axis y. Seven dots, (−3, −27), (−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8) and (3, 27), lie on one smooth curve shaped like a stretched S. Two points of the curve are named, A at (2, 8) and B at (−2, −8), and a dashed line joins them through the origin.AB−3123−30−20−101020300xy
The graph of y = x³. Plot the points of the table and join them with one smooth curve.

Look at A and B. They are the same distance from the origin, on opposite sides of it. The graph looks the same when it is turned half way round about the origin.

The graph of y = x³ passes through the origin. For large x it climbs steeply. For very negative x it drops steeply. It is flat for an instant at the origin, and it never turns round.

Next, the reciprocal graph y = 1/x. The reciprocal of a number is 1 divided by it. For example, the reciprocal of 4 is 0.25, and the reciprocal of −5 is −0.2.

Worked example

Find y for y = 1/x when x = 4, and when x = −5.

  1. y = 1/x means y = 1 ÷ x. When x = 4: y = 1 ÷ 4 = 0.25.
  2. When x = −5: y = 1 ÷ (−5). A positive divided by a negative is negative.
  3. So y = −0.2.

Answer: When x = 4, y = 0.25. When x = −5, y = −0.2.

You cannot work out y when x = 0, because nothing can be divided by 0. So there is a gap in the graph at x = 0. The graph comes in two parts, called branches.

Draw it first

The graph of y = 1/x

  1. A graph with the x axis x and the y axis y. Step 1 plots the points (1, 1), (2, 0.5), (4, 0.25) and the points (−1, −1), (−2, −0.5), (−4, −0.25). Step 2 draws two separate branches through them, one in the top right of the graph and one in the bottom left.−5−412345−412340xy

    1 Plot points for positive and negative x. Use small values like 1, 2 and 4.

  2. A graph with the x axis x and the y axis y. Step 1 plots the points (1, 1), (2, 0.5), (4, 0.25) and the points (−1, −1), (−2, −0.5), (−4, −0.25). Step 2 draws two separate branches through them, one in the top right of the graph and one in the bottom left.−5−412345−412340xy

    2 Join the points with x above 0 into one smooth curve, and the points with x below 0 into a second curve. The two branches never join.

Look at how the branches behave. When x is big, y = 1/x is tiny but still positive. So the curve gets close to the x axis. When x is very small and positive, y is huge. So the curve climbs close to the y axis.

A graph with the x axis x and the y axis y. A curve for y = 1/x runs from the point (1, 1) down to the right, sinking towards the x axis. The points (5, 0.2) and (10, 0.1) are marked. The curve stays above the x axis.5100.20.40.60.810xy
For big x, y = 1/x is close to the x axis. At x = 10 it is still 0.1.
A graph with the x axis x and the y axis y. A curve for y = 1/x climbs steeply on the left, close to the y axis. The points (0.25, 4) and (0.5, 2) are marked. The curve stays to the right of the y axis.0.250.50.75124680xy
For small x, y = 1/x is huge. The curve climbs close to the y axis but never reaches it.

The axes here are called asymptotes of the graph. An asymptote is a line that a curve gets nearer and nearer to, without ever touching it.

See it three ways

With things

Leila has a rectangle of area 1 m². Its width is x m and its length is y m. With width 2 m the length is 0.5 m. With width 4 m the length is 0.25 m.

As a picture

A graph with the x axis width (m) and the y axis length (m). A curve falls from the point (1, 1) through (2, 0.5) and (4, 0.25) and keeps falling towards the x axis without touching it.123450.250.50.7510width (m)length (m)

In words

A wider rectangle must be shorter. The length gets closer to 0 m, but it never reaches 0.

In symbols

y = 1/x, so y = 0.5 when x = 2, and y = 0.25 when x = 4.

The third shape is a circle. The points on a circle with its centre at the origin are all the same distance from the origin. Take the point P with coordinates (3, 4). Pythagoras gives its distance from the origin: 3² + 4² = 9 + 16 = 25, and √25 = 5.

Any point (x, y) that is 5 units from the origin has x² + y² = 25. That is the equation of the circle with radius 5 and centre at the origin. In general, x² + y² = r² is the circle with radius r.

Draw it first

The circle x² + y² = 25

  1. A graph with the x axis x and the y axis y, with equal units. Step 1 plots the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (−3, 4), (−4, 3), (−5, 0), (−4, −3), (−3, −4), (0, −5), (3, −4) and (4, −3). Step 2 joins them into a circle round the origin, marks the point P at (3, 4) and draws a dashed line from the origin to P.−6−4−2246−6−4−22460xy

    1 Find points 5 units from the origin, such as (5, 0), (3, 4) and (4, 3). Copy them into the other quarters.

  2. A graph with the x axis x and the y axis y, with equal units. Step 1 plots the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (−3, 4), (−4, 3), (−5, 0), (−4, −3), (−3, −4), (0, −5), (3, −4) and (4, −3). Step 2 joins them into a circle round the origin, marks the point P at (3, 4) and draws a dashed line from the origin to P.P−6−4−2246−6−4−22460xy

    2 Join them into one smooth round curve, a circle with centre at the origin. The dashed line from the origin to P is 5 units long, because 3² + 4² = 25.

A circle is not the graph of a function. For x = 3 it has two y values, 4 and −4. A function must give one output for each input.

Worked example

Find the y values on the circle x² + y² = 25 when x = 3.

  1. Put x = 3 into x² + y² = 25. First 3² = 9.
  2. So 9 + y² = 25. Take away 9: y² = 16.
  3. The square roots of 16 are 4 and −4. So y = 4 or y = −4.
  4. The circle has two points above and below x = 3: (3, 4) and (3, −4).

Answer: y = 4 or y = −4.

Try it

Get close to zero

You need: a calculator, or paper and a pencil.

  1. Work out 1 ÷ x for x = 1, 10, 100 and 1000.
  2. Write the answers in a table.
  3. Look at the answers. Do they ever reach 0?

What you will see: The answers fall from 1 to 0.001 and keep getting smaller, but each one is still above 0. The curve gets close to the x axis and never touches it.

If you cannot try it, this is what happens.
A table with the heads x and 1/x. Rows: 1 and 1; 10 and 0.1; 100 and 0.01; 1000 and 0.001.x1/x11100.11000.0110000.001

A wrong idea many people have

The two parts of y = 1/x join up across the y axis.

Why it feels right: Smooth curves usually run on without a gap, and the two parts face each other.

The test that breaks it: To cross the y axis the curve would need a value at x = 0. But 1 ÷ 0 has no answer. The curve is above the axis for x above 0 and below it for x below 0.

What is true: y = 1/x has two separate branches. The graph never touches the y axis or the x axis.

A wrong idea many people have

The circle x² + y² = 25 has radius 25.

Why it feels right: The number 25 is the number in the equation, and a radius seems to be that number.

The test that breaks it: The point (5, 0) is on the circle, because 25 + 0 = 25. It is 5 units from the origin. The point (25, 0) gives 625 + 0, which is not 25.

What is true: In x² + y² = r², the number on the right is r², the radius squared. The radius is its square root, so here it is 5.

Key words

cubic
a graph or rule with x³ as its highest power, such as y = x³.
reciprocal
the reciprocal of a number other than 0 is 1 divided by it. y = 1/x is the reciprocal graph.
asymptote
a line that a curve gets nearer and nearer to, without touching it.

Back to your guess

The answer is A: Two separate curves that come close to the axes but never touch them.

The graph of y = 1/x has two parts, one in the top right and one in the bottom left. Each part comes closer and closer to both axes. But 1/x is never 0, and x can never be 0, so the curve never touches an axis.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Find y when x = −5 for y = x³.

    Tap your answer.

    1. Not this one. You multiplied only two of the three factors. x³ means x × x × x: (−5) × (−5) × (−5). Go to M4.10.3. Try another.

    2. Not this one. You worked out 3 × x. But x³ means x × x × x, with three factors, not 3 times x. Go to M4.10.3. Try another.

    3. Not this one. The size is right, but the sign is wrong. Three negative factors give a negative answer: (−5) × (−5) = 25, and 25 × (−5) = −125. Go to M4.10.3. Try another.

    4. Yes, that is right. (−5) × (−5) = 25, and 25 × (−5) = −125. A negative number cubed is negative.

    See the answer

    −125

    (−5) × (−5) = 25, and 25 × (−5) = −125. A negative number cubed is negative.

    • If you chose A: You multiplied only two of the three factors. x³ means x × x × x: (−5) × (−5) × (−5). Go to M4.10.3.
    • If you chose B: You worked out 3 × x. But x³ means x × x × x, with three factors, not 3 times x. Go to M4.10.3.
    • If you chose C: The size is right, but the sign is wrong. Three negative factors give a negative answer: (−5) × (−5) = 25, and 25 × (−5) = −125. Go to M4.10.3.

  2. Which equation has this graph?

    A graph with the x axis x and the y axis y. Two separate curves: one in the top right that falls towards the x axis, and one in the bottom left that falls away from the x axis. Neither curve touches an axis.−412341230xy
    Tap your answer.

    1. Not this one. The graph of y = x³ is one unbroken S shape through the origin. This graph has two separate parts. Go to M4.10.3. Try another.

    2. Yes, that is right. Two separate branches that come close to the axes without touching them is the shape of y = 1/x. Check one point: when x = 1, y = 1.

    3. Not this one. The graph of y = x² is a U that touches the origin and never goes below the x axis. This graph has a part below it. Go to M4.9.6. Try another.

    See the answer

    y = 1/x

    Two separate branches that come close to the axes without touching them is the shape of y = 1/x. Check one point: when x = 1, y = 1.

    • If you chose A: The graph of y = x³ is one unbroken S shape through the origin. This graph has two separate parts. Go to M4.10.3.
    • If you chose C: The graph of y = x² is a U that touches the origin and never goes below the x axis. This graph has a part below it. Go to M4.9.6.

  3. The circle x² + y² = 36 has its centre at the origin. What is its radius?

    Tap your answer.

    1. Not this one. The 36 is the radius squared. The radius is the square root of 36. Go to M4.10.3. Try another.

    2. Not this one. You halved 36. But the equation gives r², so you need the square root, not half. Go to M4.10.3. Try another.

    3. Yes, that is right. In x² + y² = r², the right side is r². So r² = 36, and the radius is the square root of 36, which is 6. Check: the point (6, 0) has 36 + 0 = 36.

    See the answer

    6

    In x² + y² = r², the right side is r². So r² = 36, and the radius is the square root of 36, which is 6. Check: the point (6, 0) has 36 + 0 = 36.

    • If you chose A: The 36 is the radius squared. The radius is the square root of 36. Go to M4.10.3.
    • If you chose B: You halved 36. But the equation gives r², so you need the square root, not half. Go to M4.10.3.

  4. Which of these points is on the circle x² + y² = 100?

    Tap your answer.

    1. Yes, that is right. Square each coordinate and add: 6² + 8² = 36 + 64 = 100. That is the number in the equation, so (6, 8) is on the circle.

    2. Not this one. You may have thought both coordinates equal the radius, 10. But 10² + 10² = 100 + 100 = 200, which is more than 100. Go to M4.10.3. Try another.

    3. Not this one. You may have kept x = 6 and used the radius 10 for y. But 6² + 10² = 36 + 100 = 136, which is more than 100. Go to M4.10.3. Try another.

    See the answer

    (6, 8)

    Square each coordinate and add: 6² + 8² = 36 + 64 = 100. That is the number in the equation, so (6, 8) is on the circle.

    • If you chose B: You may have thought both coordinates equal the radius, 10. But 10² + 10² = 100 + 100 = 200, which is more than 100. Go to M4.10.3.
    • If you chose C: You may have kept x = 6 and used the radius 10 for y. But 6² + 10² = 36 + 100 = 136, which is more than 100. Go to M4.10.3.

  5. y = 1/x. What happens to y as x gets bigger and bigger?

    Tap your answer.

    1. Not this one. 1 divided by a positive number is never zero. For example 1 ÷ 1000 = 0.001, which is still above zero. Go to M4.10.3. Try another.

    2. Yes, that is right. For big x, 1/x is a tiny positive number: 1 ÷ 10 = 0.1, 1 ÷ 100 = 0.01, and so on. The x axis is an asymptote.

    3. Not this one. You are thinking of y = x. For y = 1/x, a bigger x divides 1 into smaller pieces, so y shrinks. Go to M4.10.3. Try another.

    See the answer

    It gets closer to zero and never reaches it.

    For big x, 1/x is a tiny positive number: 1 ÷ 10 = 0.1, 1 ÷ 100 = 0.01, and so on. The x axis is an asymptote.

    • If you chose A: 1 divided by a positive number is never zero. For example 1 ÷ 1000 = 0.001, which is still above zero. Go to M4.10.3.
    • If you chose C: You are thinking of y = x. For y = 1/x, a bigger x divides 1 into smaller pieces, so y shrinks. Go to M4.10.3.

  6. Which equation has this graph?

    A graph with the x axis x and the y axis y. One unbroken curve shaped like a stretched S rises from the bottom left, passes through the origin and climbs to the top right.−22−8−4480xy
    Tap your answer.

    1. Not this one. The graph of y = x² never goes below the x axis. This graph goes below it on the left. Go to M4.9.6. Try another.

    2. Not this one. The graph of y = 1/x has two separate parts. This graph is one unbroken curve through the origin. Go to M4.10.3. Try another.

    3. Not this one. The graph of y = 3x is a straight line. This graph bends. Go to M3.9.7. Try another.

    4. Yes, that is right. One unbroken S shape through the origin, rising to the right and falling to the left, is a cubic. Check: at x = 2 the graph is at 8, and 2³ = 8.

    See the answer

    y = x³

    One unbroken S shape through the origin, rising to the right and falling to the left, is a cubic. Check: at x = 2 the graph is at 8, and 2³ = 8.

    • If you chose A: The graph of y = x² never goes below the x axis. This graph goes below it on the left. Go to M4.9.6.
    • If you chose B: The graph of y = 1/x has two separate parts. This graph is one unbroken curve through the origin. Go to M4.10.3.
    • If you chose C: The graph of y = 3x is a straight line. This graph bends. Go to M3.9.7.

  7. From an earlier chapter

    f(x) = x³ and g(x) = x + 1. Find fg(2).

    Tap your answer.

    1. Yes, that is right. fg(2) means f(g(2)). First g(2) = 2 + 1 = 3. Then f(3) = 3³ = 27.

    2. Not this one. You put 2 into f first, to get 8, then added 1. But in fg(2) the function g goes first: g(2) = 3. Go to M4.10.2. Try another.

    3. Not this one. You added f(2) = 8 and g(2) = 3. A composite puts the answer of g into f. It does not add answers. Go to M4.10.2. Try another.

    See the answer

    27

    fg(2) means f(g(2)). First g(2) = 2 + 1 = 3. Then f(3) = 3³ = 27.

    • If you chose B: You put 2 into f first, to get 8, then added 1. But in fg(2) the function g goes first: g(2) = 3. Go to M4.10.2.
    • If you chose C: You added f(2) = 8 and g(2) = 3. A composite puts the answer of g into f. It does not add answers. Go to M4.10.2.

  8. y = x³. Find y when x = −4.

    See the answer

    −64

    (−4) × (−4) = 16, and 16 × (−4) = −64.

  9. A point on the circle x² + y² = 169 has x = 5. Find its positive y value.

    See the answer

    12

    Put x = 5: 25 + y² = 169, so y² = 144. The positive square root of 144 is 12.

Keep it

Copy this into your exercise book.

Cubic: an S through the origin. Reciprocal: two branches that never touch the axes. Circle: x² + y² = r², radius r.

Draw: Three small graphs side by side: an S curve, two branches in opposite corners, and a circle round the origin.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (956 KB). It opens on any phone.