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Cubic and reciprocal graphs How can you sketch a cubic or y = k/x before you plot a single point?

Foundation and Higher · AQA · 45 minutes

Students move a cubic and find it can meet the x-axis three times, twice or once, but never miss it. Then they read its roots from brackets and sketch y = k/x with its asymptotes. Slides to present, a vote that shows who's stuck, practice and homework.

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A quadratic meets the x-axis twice, once or never. Move c: how many times can the cubic y = x³ − 3x + c meet it? Can it ever miss?

−2−112

3 times: at x = −1.73, 0, 1.73

The idea in one minute

Look at the ends first. In y = x³ a large positive x gives a huge positive y, and a large negative x a huge negative y, because a negative number cubed is negative. So a cubic starts low and ends high (or the other way round when x³ has a minus in front): it must cross the x-axis at least once, and it can bend back to cross it three times. Written as brackets, y = (x + 1)(x − 1)(x − 2), it shows where it crosses, because a product is zero only when one of its brackets is. y = k/x has no value at x = 0 and is never 0: its two branches get closer and closer to the axes, its asymptotes, without ever touching them.

The lesson goes picture first. Students move a cubic and count where it meets the x-axis before anyone explains, then reason about the ends and the middle before plotting anything: what happens for large x, for negative x, close to 0. Then they read roots from brackets and from tables, and sketch y = k/x for k positive and negative. The core, recognising and plotting y = x³ and y = 1/x and reading roots from a graph, a table or brackets, is for Foundation and Higher; a repeated root, a minus in front of x³ and a negative k are Higher.

By the end, students can

  • recognise the graphs of y = x³ and y = k/x from their equations, and sketch them
  • explain why a cubic meets the x-axis at least once and at most three times
  • read the roots of a cubic from its graph, a table or its brackets
  • sketch y = k/x with its asymptotes, for k positive or negative

Key words

  • cubic
  • reciprocal
  • root
  • asymptote
  • turning point
  • y-intercept

Before this lesson

Students should plot a curve from a table of values, cube a negative number ((−2)³ = −8) and find a quadratic's roots from its brackets.

Taught in Quadratic equations.

The lesson, step by step

60 minutes in total. Each step says what changes for this length.

  1. 1 · Think5 min

    Can a cubic miss the axis?

    A quadratic: twice, once or never.

    y = x³ − 3x + c is a cubic.

    How many times can it meet

    the x-axis? Can it miss it?

    Vote first. Then move c.

    5 min 1 · Hook · students think · the slides say “Think”

    Can a cubic miss the axis?

    Students predict how often a cubic meets the x-axis before anyone explains.

    Say

    “A quadratic meets the x-axis twice, once or never. This is a cubic, y = x³ − 3x + c. Before I move c: how many times can it meet the x-axis? Can it miss it altogether?”

    Ask
    • What is x³ − 3x when x is 10? When x is −10?
    • So where is the curve at the far left, and at the far right?
    Watch for

    Saying it can miss, as a quadratic can: keep it. The next step shows why a cubic can't.

    For the teacher

    Give a minute of silent thinking, then vote. Many will say yes, because a quadratic can sit above the axis.

    Move c slowly. Three crossings, then two (one is a touch), then one, but never none. Don't explain yet: the question is why it can never miss.

    Take a vote with Ask the class: can a cubic miss the x-axis, yes or no? Then move c from −4 to 4 and count the crossings at each value.

    Shorter: Vote only; move c in the next step (−2 min).

  2. 2 · Watch12 min

    y = x³: look at the ends

    x = 2: y = 8

    x = −2: y = (−2)³ = −8

    Far right: up. Far left: down.

    So a cubic must cross the x-axis.

    x³ − 3x bends back: 3 crossings.

    Low on the left, high on the right.

    2 · Watch

    Roots from brackets

    y = (x + 1)(x − 1)(x − 2)

    Each bracket is 0 at one x:

    x = −1, x = 1, x = 2

    At x = 0: 1 × (−1) × (−2) = 2

    It crosses the y-axis at 2.

    A bracket is 0, so y is 0.

    2 · Watch

    A touch, and upside down

    y = (x + 2)(x − 1)²

    It crosses at x = −2.

    (x − 1)² is never negative:

    at x = 1 it touches, turns back.

    A minus before x³ flips it:

    high on the left, low on the right.

    A bracket twice: a touch.

    2 · Watch

    y = 1/x: never 0, nothing at 0

    x = 1, 2, 4: y = 1, 0.5, 0.25

    Small, but never 0.

    x = 0.5, 0.25: y = 2, 4

    1 ÷ 0 has no value: a gap at 0.

    The axes are its asymptotes.

    Two branches. Never join them.

    2 · Watch

    y = k/x: where do they go?

    x × y = k, every time.

    k = 4: top right, bottom left

    k = −4: x = 2 gives y = −2,

    so top left and bottom right.

    Predict, then move k.

    12 min 2 · Explain · you explain · the slides say “Watch”

    Ends, brackets, then y = k/x

    The ends of a cubic, its roots from brackets, a repeated root; then y = 1/x and y = k/x.

    Say

    “Look at the ends before anything else: big x, negative x, and x close to 0.”

    Ask
    • (−2)³: is it 8, −8 or −6?
    • y = (x + 1)(x − 1)(x − 2): which x makes the first bracket 0?
    • What is 1 ÷ 0.01? And 1 ÷ 1000? Is 1 ÷ x ever 0?
    Watch for

    Reading the root of (x − 1) as −1: the bracket is 0 when x = 1.

    Roots from brackets

    y = (x + 1)(x − 1)(x − 2)

    Each bracket is 0 at one x: x = −1, x = 1, x = 2

    At x = 0: 1 × (−1) × (−2) = 2, so it crosses the y-axis at 2

    A bracket twice: a touch

    y = (x + 2)(x − 1)²

    It crosses the x-axis at x = −2

    (x − 1)² is never negative, so at x = 1 it touches and turns back

    y = 1/x is never 0

    1/x = 0 would need 1 = 0 × x

    No x does that, so the curve never meets the x-axis

    At x = 0, 1 ÷ 0 has no value: no point on the y-axis either

    For the teacher

    First slide: y = x³. At x = 2, y = 8; at x = −2, y = −8. Large positive x: huge positive y; large negative x: huge negative y. So it starts low on the left and ends high on the right, and must cross the x-axis somewhere. y = x³ − 3x has the same ends but bends back in the middle, so it crosses three times. That answers the hook.

    Second slide: in brackets, each bracket gives a root, with the opposite sign: (x + 1) is 0 at x = −1. Put x = 0 into every bracket for the y-intercept. Third slide (Higher): a bracket squared gives a touch, not a crossing; a minus in front of x³ turns the curve upside down, so it starts high and ends low.

    Fourth slide: y = 1/x. Make a short table: x = 4, 2, 1, 0.5, 0.25 gives y = 0.25, 0.5, 1, 2, 4. y gets small but never 0; near x = 0 it shoots up. Negative x gives negative y. Two branches, never joined; the axes are its asymptotes. Fifth slide: y = k/x keeps x × y = k; a negative k puts the branches top left and bottom right.

    Vote before each move (Ask the class): where will y = (x + 1)(x − 1)(x − 2) cross the x-axis? Which corners will y = −4/x be in?

    Shorter: Skip the repeated root and the negative k (the Higher slide and move k as a demonstration only) (−4 min).

  3. 3 · Your turn8 min

    Worked, then yours

    I do: y = (x − 1)(x + 3)(x − 4)

    Roots: x = 1, x = −3, x = 4

    y-intercept: (−1) × 3 × (−4) = 12

    You do: y = (x + 2)(x − 3)(x − 5)

    Which x makes each bracket 0?

    Write here

    3 · Your turn

    Two more pairs

    I do: y = x³ − 4x, x = −2 to 2: y = 0, 3, 0, −3, 0

    You do: y = x³ − x for x = −2, −1, 0, 1, 2

    I do: y = 6/x: x = 1, 2, 3, 6 gives y = 6, 3, 2, 1

    x × y = 6: top right and bottom left

    You do: y = −8/x for x = 1, 2, 4, 8. Which corners?

    A negative number cubed is negative.

    Write here

    Its answers' slide comes with your copy.

    8 min 3 · Your turn · students write

    Worked, then yours

    Each worked example beside a twin that students solve themselves.

    Say

    “Watch me do one. Then you do its twin, which is almost the same.”

    Ask
    • Which x makes this bracket 0?
    • Is a negative number cubed positive or negative?
    Watch for

    Working out (−2)³ as 8 or −6: it is (−2) × (−2) × (−2) = −8.

    Roots and the y-intercept

    y = (x − 1)(x + 3)(x − 4)

    Roots: x = 1, x = −3, x = 4

    y-intercept: (−1) × 3 × (−4) = 12

    Then theirs

    y = (x + 2)(x − 3)(x − 5): its roots and its y-intercept

    A table, then the roots

    y = x³ − 4x for x = −2, −1, 0, 1, 2

    y = 0, 3, 0, −3, 0

    It crosses at x = −2, 0 and 2: one smooth S, not straight lines

    Then theirs

    y = x³ − x for x = −2, −1, 0, 1, 2: the table and the roots

    y = k/x from a table

    y = 6/x for x = 1, 2, 3, 6: y = 6, 3, 2, 1

    For x = −1, −2, −3, −6: y = −6, −3, −2, −1

    x × y = 6 every time: top right and bottom left

    Then theirs

    y = −8/x for x = 1, 2, 4, 8: the table, and which corners?

    For the teacher

    Work the first example aloud, writing each line. Then students solve its twin while you watch who changes the sign from the bracket to the root.

    Do the same for each pair. The second pair catches students who cube negative numbers wrongly; the third, students who put a negative k in the same corners as a positive one.

    Open the answers slide once most have finished their twins.

    Shorter: Do the first and third pairs only (−3 min).

  4. 4 · Vote and practise25 min

    Vote first

    Where does y = (x − 2)(x + 1)(x + 3) cross the axes?

    A: x-axis 2, −1, −3; y-axis −6

    B: x-axis −2, 1, 3; y-axis 6

    C: x-axis 2, −1, −3; y-axis 6

    D: x-axis −2, 1, 3; y-axis −6

    Choose on your own, then say why.

    4 · Vote and practise

    Practise: show your working

    1. On y = x³, y when x = −3?

    2. On y = 12/x, y when x = 4? When x = −2?

    3. y = (x + 4)(x − 1)(x − 5): where on the x-axis?

    4. And where does it cross the y-axis?

    Stuck on 3? Which x makes (x + 4) zero?

    Write here

    4 · Vote and practise

    Reason it out

    5. Can a cubic miss the x-axis? Why?

    6. What is true of y = 1/x and the axes?

    7. y = −6/x: where are its two branches?

    8. y = x³ − 2x: y when x = −1? When x = 2?

    Stretch: y = (x − 2)²(x + 1). Where does it touch, and cross?

    25 min 4 · Who's stuck? · students vote · the slides say “Vote and practise”

    Vote, then practise

    One question shows who's stuck; then practice in levels, and a stretch.

    For this length: The stretch, then: sketch y = (x − 2)²(x + 1) and y = −(x − 2)²(x + 1) on one pair of axes

    Say

    “Choose an answer on your own. Nobody talks until everyone has chosen.”

    Ask
    • Which x makes (x − 2) zero?
    • Put x = 0 into every bracket: what do you get?
    Watch for

    Answers not checked: each root should make its own bracket 0, and the y-intercept is the value at x = 0.

    Show your working: for each bracket, write the x that makes it zero. The stretch is optional.

    The question that shows who's stuck

    1. Where does y = (x − 2)(x + 1)(x + 3) cross the axes?
      1. The x-axis at 2, −1 and −3; the y-axis at −6
      2. The x-axis at −2, 1 and 3; the y-axis at 6
      3. The x-axis at 2, −1 and −3; the y-axis at 6
      4. The x-axis at −2, 1 and 3; the y-axis at −6
    In your Classroom, as they answer (an example)
    • A9
    • B7
    • C6
    • D3

    Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.

    1. On y = x³, when x = −3, y = ___
    2. On y = 12/x, when x = 4, y = ___, and when x = −2, y = ___
    3. y = (x + 4)(x − 1)(x − 5) crosses the x-axis at the smallest x = ___, then x = ___, then the largest x = ___
    4. Where does y = (x + 4)(x − 1)(x − 5) cross the y-axis? Give the y-value.
    5. Can the graph of a cubic miss the x-axis altogether?
      1. No: one end is far below the axis and the other far above it
      2. Yes, like y = x² + 1 does
      3. Yes, if it has no brackets
    6. Which is true of the graph of y = 1/x?
      1. It never meets either axis
      2. It goes through (0, 0)
      3. It crosses the x-axis at x = 1
    7. y = −6/x. Where are the two branches of its graph?
      1. Top left and bottom right
      2. Top right and bottom left
      3. Top left and top right
      4. One branch, through the origin
    8. Complete the table for y = x³ − 2x: when x = −1, y = ___, and when x = 2, y = ___
    9. Stretch: y = (x − 2)²(x + 1) touches the x-axis at x = ___ and crosses it at x = ___

    Support · For question 3: a bracket such as (x + 7) is 0 when x = −7, so each root has the opposite sign to the number in its bracket. For question 4, put x = 0 into every bracket and multiply, signs and all.

    For the teacher

    Everyone chooses before anyone explains. For the roots, ask which x makes each bracket zero. For the y-intercept, ask them to put x = 0 into each bracket, signs and all.

    Questions 1–2 build fluency; 3–4 change the look; 5–7 ask for reasons; 8 is a table as exams set it. The stretch is a repeated root.

    Students show their working on paper or the board; the quiz keeps their final answers.

    Whatever the vote, before they practise: ask everyone to put each root they chose back into its bracket, and to work out (0 − 2) × (0 + 1) × (0 + 3) out loud.

    Shorter: After the vote, questions 1–4 only (−6 min).

    Longer: The stretch, then: sketch y = (x − 2)²(x + 1) and y = −(x − 2)²(x + 1) on one pair of axes (+10 min).

  5. 5 · Show what you know10 min

    Sketch one from its brackets

    y = (x + 1)(x − 2)(x − 3)

    Mark where it crosses both axes.

    How do you know which way its ends go?

    A sketch, then one sentence.

    Write here

    10 min 5 · Exit · students show · the slides say “Show what you know”

    Sketch one from its brackets

    One sketch shows who reads roots, the y-intercept and the ends correctly.

    For this length: Sketch y = 4/x and y = −4/x on one pair of axes

    Say

    “Sketch y = (x + 1)(x − 2)(x − 3). Mark where it crosses both axes, and say how you know which way its ends go.”

    Watch for

    Roots with the wrong sign, or a U shape in place of an S.

    A sketch with the crossings marked, and one sentence.

    1. Sketch y = (x + 1)(x − 2)(x − 3). Mark where it crosses the x-axis and the y-axis, and say in one sentence how you know which way its ends go.
    For the teacher

    Sketch y = (x + 1)(x − 2)(x − 3) and mark where it crosses both axes.

    Then say in one sentence how you know which way its ends go.

    The sketches show who still reads roots with the wrong sign or draws a cubic as a U: start the next lesson with the hook again. Set the homework.

    Longer: Sketch y = 4/x and y = −4/x on one pair of axes (+5 min).

One to one: before each sketch, ask the student what y does when x is 10, when x is −10 and when x is 0, and only then let them draw; check the sketch with one point from a calculator.

Homework: tables, roots and sketches

Show your working. Draw each curve smoothly, never with straight lines between the points, and label where it crosses the axes.

  1. Complete a table for y = x³ − 3x for x = −2, −1, 0, 1, 2, then plot the graph.
  2. Where does y = (x + 3)(x − 1)(x − 2) cross the x-axis? Where does it cross the y-axis?
  3. Complete a table for y = 8/x for x = −8, −4, −2, −1, 1, 2, 4, 8, then plot the graph.
  4. Explain why the graph of y = 1/x never crosses the x-axis, and why it has no point on the y-axis.
  5. Sketch y = x³ and y = −x³ on one pair of axes. Where do they meet?
  6. A cubic crosses the x-axis at −2, 1 and 3, and starts low on the left. Write its equation in brackets, and find where it crosses the y-axis.

Print the worksheet with working space and homework. Teacher answers come with your Library copy.

Teaching cubic and reciprocal graphs: common questions

How do you tell a cubic graph from a quadratic one?

Look at the ends. A quadratic's ends go the same way, both up or both down, so it is a U or an upside-down U. A cubic's ends go opposite ways: y = x³ starts low on the left and ends high on the right, because a negative number cubed is negative. So a cubic always crosses the x-axis at least once, and can bend to cross it three times.

Why does y = 1/x have asymptotes?

Because 1 ÷ x is never 0, however large x is, the curve gets closer and closer to the x-axis without reaching it. And 1 ÷ 0 has no value, so near x = 0 the curve shoots up (or down) beside the y-axis without touching it. A short table with x = 0.5, 0.25 and 0.1 shows students this before any rule.

Is this cubic and reciprocal graphs lesson suitable for Foundation GCSE?

Yes. The core, recognising and plotting y = x³ and y = 1/x and reading roots from a graph, a table or brackets (as students do for quadratics), is for Foundation and Higher. A repeated root, a minus in front of x³ and y = k/x with a negative k are Higher, and the lesson can leave them out at its 30-minute length.

Why do students join the two branches of y = 1/x?

Usually because they plot from a table that jumps from x = −1 to x = 1 and join the dots. Ask for x = 0.5, 0.25 and −0.5 too: the values grow quickly near 0, and they see that the curve can't cross from one branch to the other.

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Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.

Curriculum reference: AQA GCSE Mathematics 8300: A12 (recognising, sketching and reading the graphs of cubic functions and the reciprocal function y = k/x). The repeated root, a minus in front of x³ and a negative k are Higher.

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