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      Key features of a cubic graph

      Lesson details

      Learning outcome

      I can identify the key features of a cubic graph.

      Key learning points

      1. A cubic graph has a distinct shape.
      2. The roots of a cubic graph are where the graph intersects with the x-axis.
      3. The turning points are the local maximum and local minimum points of the graph.

      Keywords

      • Cubic - A cubic is an equation, graph, or sequence whereby the highest exponent of the variable is 3

      • Roots - When drawing the graph of an equation, the roots of the equation are where its graph intercepts the x-axis (where y = 0).

      • Turning point - The turning point of a graph is a point on the curve where, as x increases, the y values change from decreasing to increasing or vice versa.

      Common misconception

      $$y=x^3$$ has one root so all cubic graphs have one root.

      Make sure pupils see a wide variety of cubic graphs; ones with one root, two roots and three roots and link every one back to its equation and that the highest exponent of the variable is $$3$$ so they are all cubic graphs.

      Teacher tip

      Have Desmos.com on a screen visible to pupils during the lesson and get them to suggest cubic equations to graph. Get pupils to pick out the key features of their equations.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      A cubic is an equation, graph, or sequence where the highest __________ of the variable is 3.

      base
      coefficient
      Correct answer: exponent
      factor
      multiple

      Q2.
      How many solutions can a quadratic equation have?

      1
      1 or 2
      2
      0 or 1
      Correct answer: 0, 1 or 2

      Q3.
      What is $$y$$-intercept of this quadratic graph?

      An image in a quiz
      8
      $$y$$ = 8
      Correct answer: (0, 8)
      (4, 0)
      (2, 0)

      Q4.
      What are the roots of this quadratic equation?

      An image in a quiz
      (2, 0)
      (4, 0)
      Correct answer: $$x$$ = 2
      Correct answer: $$x$$ = 4
      $$y$$ = 8

      Q5.
      The point (3, -1) is the point of this quadratic curve.

      An image in a quiz
      Correct Answer: minimum, turning

      Q6.
      Factorise $$x^2-6x$$.

      $$(x+1)(x-6)$$
      $$(x-1)(x-6)$$
      Correct answer: $$x(x-6)$$
      $$(x-1)(x+6)$$
      $$(x+1)(x+6)$$

      6 Questions

      Q1.
      The point on the graph of a curve where, as $$x$$ increases, the $$y$$ values change from decreasing to increasing (or vice versa) is called the __________.

      root
      Correct answer: turning point
      $$x$$-intercept
      $$y$$-intercept

      Q2.
      How many roots can the graph of a cubic equation have?

      1
      3
      1 or 3
      Correct answer: 1, 2 or 3
      0, 1, 2 or 3

      Q3.
      This is the graph of $$y=x^{3}-9x^{2}+24x-16$$. Which $$y$$ values make this statement true? $$y=x^{3}-9x^{2}+24x-16$$ has two solutions.

      An image in a quiz
      5
      Correct answer: 4
      2
      Correct answer: 0
      -3

      Q4.
      This is the equation $$y=x^{3}-9x^{2}+24x-16$$. The root $$x=4$$ is a __________.

      An image in a quiz
      Correct answer: repeated root
      minimum value
      Correct answer: turning point
      Correct answer: local minimum
      local maximum

      Q5.
      The equation $$y=x^3-21x+20$$ factorises to $$y=(x-1)(x-4)(x+5)$$. What are the roots of the equation?

      Correct answer: $$x=1$$
      Correct answer: $$x=4$$
      $$x=-1$$
      $$x=-4$$
      Correct answer: $$x=-5$$

      Q6.
      $$x=-5$$ and $$x=0$$ are two roots of $$y=x^3+2x^2-15x$$. The third root is $$x=$$ .

      Correct Answer: 3, x=3

      To help you plan your 10 maths lesson on: Key features of a cubic graph, download all teaching resources for free and adapt to suit your pupils' needs...