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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...