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Maths · Graphs

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Cubic, Reciprocal and Other Graphs

Recognising and plotting cubic, reciprocal, exponential and circle graphs.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Recognise the graphs of linear, quadratic, cubic and reciprocal functions from their equations.
  2. 2Complete tables of values for cubic and reciprocal functions, and plot them.
  3. 3Recognise the graph of an exponential function \(y = a^x\).
  4. 4Recognise and use the equation of a circle, \(x^2 + y^2 = r^2\) (Higher tier).

Every equation has a shape

Once you know the standard shapes you can often tell which equation a graph belongs to without plotting a single point. The question usually shows two or three curves and asks you to match them to equations, or asks you to plot one from a table. On Paper 1 the tables are small and the numbers are simple, so the marks come from remembering the shapes and from taking care with negatives, especially in cubes and reciprocals.

Cubic graphs

A cubic has an \(x^3\) term as its highest power, and its graph is an S shape.

  • Positive \(x^3\)

    The curve starts at the bottom left and finishes at the top right.

  • Negative \(x^3\)

    It starts at the top left and finishes at the bottom right, like \(y = -x^3\).

  • Turning points

    A cubic can have a hill and a valley, like \(y = x^3 - 3x\), or none at all, like \(y = x^3\).

  • Tables of values

    For \(y = x^3\) the values for \(x = -2, -1, 0, 1, 2\) are \(-8, -1, 0, 1, 8\). Cubing a negative gives a negative.

Completing a cubic table

Complete the table for \(y = x^3 - 3x\) for \(x = -2, -1, 0, 1, 2\).

Show the solutionHide the solution
  1. 1 \(x = -2\) \((-2)^3 - 3 \times (-2) = -8 + 6 = -2\).
  2. 2 \(x = -1\) \(-1 + 3 = 2\).
  3. 3 \(x = 0\) \(0 - 0 = 0\).
  4. 4 \(x = 1\) and \(x = 2\) \(1 - 3 = -2\) and \(8 - 6 = 2\).

Answer\(-2,\ 2,\ 0,\ -2,\ 2\)

Reciprocal graphs

The graph of \(y = \dfrac{1}{x}\) comes in two pieces, because you cannot divide by zero.

  • No value at x = 0

    The curve never touches the \(y\)-axis. It gets closer and closer without reaching it.

  • Never reaches y = 0

    The curve gets very close to the \(x\)-axis for large \(x\) but never touches it.

  • Two branches

    One is in the top right (positive \(x\), positive \(y\)) and one in the bottom left (negative \(x\), negative \(y\)).

  • Tables of values

    For \(y = \dfrac{4}{x}\), \(x = 1, 2, 4\) give \(4, 2, 1\), and \(x = -1, -2, -4\) give \(-4, -2, -1\).

Exponential graphs

In \(y = a^x\) the unknown is in the power. The graph climbs very quickly.

  • Passes through (0, 1)

    Any number to the power 0 is 1, so \(y = 2^x\) meets the \(y\)-axis at \((0, 1)\).

  • Grows fast

    \(y = 2^x\) gives \(1, 2, 4, 8, 16\) for \(x = 0, 1, 2, 3, 4\), doubling every time.

  • Never negative

    The curve stays above the \(x\)-axis. For negative \(x\) it gets close to the axis: \(2^{-1} = \dfrac{1}{2}\) and \(2^{-2} = \dfrac{1}{4}\).

  • Decay

    If the base is a fraction, such as \(y = \left(\dfrac{1}{2}\right)^x\), the curve falls instead.

The circle (Higher tier)

A circle with its centre at the origin has a very simple equation.

  • Equation

    \(x^2 + y^2 = r^2\), where \(r\) is the radius. \(x^2 + y^2 = 25\) is a circle of radius 5.

  • Points on it

    \((3, 4)\) is on this circle because \(9 + 16 = 25\). So are \((4, 3)\), \((5, 0)\) and \((0, -5)\).

  • Not a function

    A circle has two \(y\)-values for most \(x\)-values, so it is not a single curve of the form \(y = \ldots\).

  • Use the radius

    The radius is the square root of the number on the right, so \(x^2 + y^2 = 9\) has radius 3.

Matching equations to graphs

Which of these equations gives a cubic graph: \(y = 3x + 2\), \(y = x^2 - 4\), \(y = \dfrac{2}{x}\), \(y = x^3 + 1\)?

Show the solutionHide the solution
  1. 1 Look at the highest power A cubic has \(x^3\) as its highest power.
  2. 2 Rule out the others \(y = 3x + 2\) is linear, \(y = x^2 - 4\) is quadratic, and \(y = \dfrac{2}{x}\) is reciprocal.
  3. 3 Choose \(y = x^3 + 1\) is the cubic.

Answer\(y = x^3 + 1\)

Test yourself

  1. 1

    What shape is the graph of \(y = x^3\)?

    Show answerHide answer

    An S shape through the origin.

  2. 2

    What is special about the graph of \(y = \dfrac{1}{x}\)?

    Show answerHide answer

    It has two branches and never touches either axis.

  3. 3

    Where does \(y = 3^x\) cross the y-axis?

    Show answerHide answer

    At \((0, 1)\).

  4. 4

    What is the value of \(y = x^3\) when \(x = -3\)?

    Show answerHide answer

    \(-27\).

  5. 5

    What is the radius of \(x^2 + y^2 = 36\) (Higher tier)?

    Show answerHide answer

    6.

Exam technique: recognising graphs

Matching questions can be solved quickly by testing a point.

  • Test a value

    Put \(x = 0\) or \(x = 1\) into each equation and see which graph passes through the matching point.

  • Look for symmetry

    A U shape is symmetrical about the \(y\)-axis for \(y = x^2 + k\), and a cubic turns round the origin.

  • Negative numbers

    Put brackets round negative values in tables, like \((-2)^3\), to avoid sign mistakes.

  • Smooth curves

    Draw curves by hand, and leave the gap at \(x = 0\) on a reciprocal graph.

Summary and exam focus

  • Linear graphs are straight, quadratics are U-shaped, cubics are S-shaped and reciprocals have two branches.
  • Exponential graphs pass through \((0, 1)\) and grow very quickly.
  • A cubed negative number is negative, so check each entry in a table.
  • \(x^2 + y^2 = r^2\) is a circle of radius \(r\) centred on the origin (Higher tier).

Exam focus

Complete the table of values for \(y = x^3\) for \(x = -2, -1, 0, 1, 2\). (2 marks) (2 marks)

The answers are \(-8, -1, 0, 1, 8\). Do not forget that cubing a negative gives a negative. Writing \((-2)^3 = -8\) in the margin shows the method.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Cubic
An equation whose highest power of \(x\) is \(x^3\), with an S-shaped graph.
Reciprocal
One divided by a number or expression, such as \(\dfrac{1}{x}\).
Exponential
A relationship where the unknown is in the power, such as \(y = 2^x\).
Asymptote
A line that a curve gets closer and closer to without ever touching.
Branch
One of the separate pieces of a graph such as \(y = \dfrac{1}{x}\).
Function
A rule that gives one output for each input.
Circle equation
\(x^2 + y^2 = r^2\) for a circle with centre the origin and radius \(r\).
Power
The small number showing how many times to multiply, such as 3 in \(x^3\).
Smooth curve
A curve drawn freehand with no corners or breaks.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Match 3 marks Core

The diagram shows three graphs, \(A\), \(B\) and \(C\). The three equations are \(y = 2^x\), \(y = 4 - x^2\) and \(y = 3 - x\). Match each equation to the correct graph. [3 marks]

Three graphs: an upside-down U-shaped curve, a rapidly rising curve above the x-axis and a falling straight line.

Mark scheme — 3 marks available

  • \(A\) is \(y = 4 - x^2\) — B1
  • \(B\) is \(y = 2^x\) — B1
  • \(C\) is \(y = 3 - x\) — B1

Model answer

Graph \(A\) is an upside-down U, so \(y = 4 - x^2\). Graph \(B\) rises quickly and stays above the \(x\)-axis, so \(y = 2^x\). Graph \(C\) is a falling straight line, so \(y = 3 - x\).

2. Exam question Complete 2 marks Easier

Complete the table of values for \(y = x^3 + 1\). \(x = -2, -1, 0, 1, 2\)

Mark scheme — 2 marks available

  • At least three correct values — M1
  • \(-7, 0, 1, 2, 9\) — A1

Model answer

The values are \(-7, 0, 1, 2, 9\).

3. Exam question Complete 3 marks Core

(a) Complete the table of values for \(y = \dfrac{6}{x}\). \(x = 1, 2, 3, 6, -2, -3\) [2 marks] (b) How many separate parts, or branches, does the graph of \(y = \dfrac{6}{x}\) have? [1 mark]

Mark scheme — 3 marks available

  • (a) At least four correct values — M1
  • (a) \(6, 3, 2, 1, -3, -2\) — A1
  • (b) 2 — B1

Model answer

(a) The values are \(6, 3, 2, 1, -3, -2\). (b) The graph has 2 branches.

4. Exam question Complete 3 marks Core

(a) Complete the table of values for \(y = 3^x\). \(x = 0, 1, 2, 3\) [2 marks] (b) Write down the \(y\)-intercept of the graph of \(y = 3^x\). [1 mark]

Mark scheme — 3 marks available

  • (a) At least two correct values — M1
  • (a) \(1, 3, 9, 27\) — A1
  • (b) 1 — B1

Model answer

(a) The values are \(1, 3, 9, 27\). (b) The \(y\)-intercept is 1, because \(3^0 = 1\).

5. Exam question Work out 3 marks Core

The graph of \(y = x^2 - 4\) is symmetrical. (a) Write down the equation of its line of symmetry. [1 mark] (b) Work out the coordinates of the points where the graph crosses the \(x\)-axis. [2 marks]

Mark scheme — 3 marks available

  • (a) \(x = 0\) — B1
  • (b) \(x^2 = 4\) — M1
  • (b) \((2, 0)\) and \((-2, 0)\) — A1

Model answer

(a) The line of symmetry is the \(y\)-axis, \(x = 0\). (b) When \(y = 0\), \(x^2 = 4\), so \(x = 2\) or \(x = -2\). The points are \((2, 0)\) and \((-2, 0)\).

6. Exam question Show that 3 marks Stretch

A circle has equation \(x^2 + y^2 = 100\). (a) Write down the radius of the circle. [1 mark] (b) Show that the point \((6, 8)\) lies on the circle. [2 marks]

Mark scheme — 3 marks available

  • (a) 10 — B1
  • (b) \(6^2 + 8^2\) or \(36 + 64\) — M1
  • (b) 100 with a conclusion — Q1

Model answer

(a) The radius is \(\sqrt{100} = 10\). (b) \(6^2 + 8^2 = 36 + 64 = 100\), so the point lies on the circle.

7. Multiple choice 1 mark Easier

What shape is the graph of \(y = x^3\)?

  1. A An S-shaped curve through the origin Correct
  2. B A U-shaped curve
  3. C Two separate branches
  4. D A straight line

Why: Cubic graphs have an S shape.

8. Multiple choice 1 mark Core

What is special about the graph of \(y = \dfrac{1}{x}\)?

  1. A It passes through the origin
  2. B It is a straight line
  3. C It is a closed curve
  4. D It has two branches and never touches either axis Correct

Why: You cannot divide by 0, and \(\dfrac{1}{x}\) is never 0.

9. Multiple choice 1 mark Core

Where does the graph of \(y = 3^x\) cross the \(y\)-axis?

  1. A \((0, 3)\)
  2. B \((0, 0)\)
  3. C \((0, 1)\) Correct
  4. D \((1, 0)\)

Why: \(3^0 = 1\).

10. Multiple choice 1 mark Easier

What is the value of \(x^3\) when \(x = -3\)?

  1. A \(27\)
  2. B \(-27\) Correct
  3. C \(-9\)
  4. D \(9\)

Why: \((-3) \times (-3) \times (-3) = -27\).

11. Multiple choice 1 mark Easier

Which of these equations gives a cubic graph?

  1. A \(y = x^3 + 1\) Correct
  2. B \(y = 3x + 2\)
  3. C \(y = x^2 - 4\)
  4. D \(y = \dfrac{2}{x}\)

Why: A cubic has \(x^3\) as its highest power.

12. Multiple choice 1 mark Core

Work out \(y\) when \(x = 2\) on \(y = x^3 - 3x\).

  1. A \(14\)
  2. B \(-2\)
  3. C \(6\)
  4. D \(2\) Correct

Why: \(8 - 6 = 2\).

13. Multiple choice 1 mark Easier

What is \(2^3\)?

  1. A \(6\)
  2. B \(9\)
  3. C \(8\) Correct
  4. D \(5\)

Why: \(2 \times 2 \times 2 = 8\).

14. Multiple choice 1 mark Core

What shape is the graph of \(y = -x^2\)?

  1. A A U shape
  2. B An upside-down U Correct
  3. C An S shape
  4. D A straight line

Why: A negative \(x^2\) term turns the parabola upside down.

15. Multiple choice 1 mark Stretch

What is the radius of the circle \(x^2 + y^2 = 36\)?

  1. A \(6\) Correct
  2. B \(36\)
  3. C \(18\)
  4. D \(72\)

Why: The radius is \(\sqrt{36} = 6\).