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      Key features of a quadratic graph

      Lesson details

      Learning outcome

      I can identify the key features of a quadratic graph.

      Key learning points

      1. A quadratic graph is a parabola.
      2. The roots of a quadratic graph are where the graph intersects with the x-axis.
      3. The turning point is the maximum or minimum point of the graph.
      4. The coordinates of the turning point can be found by completing the square.

      Keywords

      • 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

      Parabolas are 'upwards' or 'downwards'.

      Encourage use of language such as "The turning point of this parabola is a maximum/minimum value" and "As the absolute values of $$x$$ increase, the $$y$$ values decrease/increase".

      Teacher tip

      Model good technical language and get pupils to repeat it to you. This encourages use of the correct mathematical language.

      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.
      What shape is the graph of the equation $$y = 6x^2-3x+9$$?

      A linear graph
      Correct answer: A parabola
      An upward curve
      A vertical line

      Q2.
      What is the $$y$$-intercept of this linear graph?

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

      Q3.
      Which of these are key features of the graph of the equation $$3x+y=6$$?

      An image in a quiz
      Gradient of 3
      Correct answer: $$x$$-intercept at (2, 0)
      Correct answer: Gradient of -3
      Correct answer: Linear graph
      Correct answer: $$y$$-intercept at (0 ,6)

      Q4.
      Factorise $$x^2-8x+7$$.

      $$(x-8)(x+7)$$
      $$(x-7)(x+1)$$
      Correct answer: $$(x-7)(x-1)$$
      $$(x+4)(x+3)$$
      $$(x+7)(x+1)$$

      Q5.
      Factorise $$x^2-8x+16$$.

      $$(x+4)(x-4)$$
      $$(x+4)^2$$
      Correct answer: $$(x-4)^2$$
      $$(x-8)(x+16)$$

      Q6.
      Expand and simplify $$(x+5)^2-10$$

      $$x^2+15$$
      $$x^2+5x+15$$
      $$x^2+25x+15$$
      Correct answer: $$x^2+10x+15$$
      $$x^2+10x+25$$

      6 Questions

      Q1.
      $$x=-2$$ and $$x=3$$ are __________ of the equation shown in this graph.

      An image in a quiz
      intercepts
      intersects
      Correct answer: roots
      Correct answer: solutions

      Q2.
      (3, 1) is the __________ of this quadratic graph.

      An image in a quiz
      bottom
      end
      lowest solution
      Correct answer: minimum point
      Correct answer: turning point

      Q3.
      What are the roots of this equation?

      An image in a quiz
      (2, 0)
      Correct answer: $$x$$ = 2
      $$x$$ = 3
      Correct answer: $$x$$ = 4
      $$x$$ = 8

      Q4.
      What is the turning point of this graph?

      An image in a quiz
      $$x$$ = 2
      $$x$$ = 3
      (0, 8)
      (2, 0)
      Correct answer: (3, -1)

      Q5.
      Factorise $$y=x^2-4x-12$$ to find the roots of the equation.

      $$(x-3)(x-4)$$, therefore roots at $$x=3$$ and $$x=4$$
      $$(x+6)(x-2)$$, therefore roots at $$x=-6$$ and $$x=2$$
      Correct answer: $$(x-6)(x+2)$$, therefore roots at $$x=6$$ and $$x=-2$$
      $$(x+3)(x+4)$$, therefore roots at $$x=-3$$ and $$x=-4$$

      Q6.
      The quadratic equation $$y=x^2+14x+49$$ has __________.

      no roots
      one root
      Correct answer: one repeated root
      two roots

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