GCSEMathsEdexcelHigherRevision Notes2. AlgebraGraphs of FunctionsQuadratic Graphs

Quadratic Graphs (Edexcel GCSE Maths): Revision Note

Naomi C

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Naomi C

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Edexcel Maths: HigherOCR Maths: HigherAQA Maths: Higher

Quadratic Graphs

What is a quadratic graph?

  • A quadratic graph has the form y equals a x squared plus b x plus c 

    • where a is not zero

What does a quadratic graph look like?

  • A quadratic graph is a smooth curve with a vertical line of symmetry

    • A positive number in front of x squared gives a u-shaped curve

    • A negative number in front of x squared gives an n-shaped curve

  • The shape made by a quadratic graph is known as a parabola

  • A quadratic graph will always cross the y-axis

  • A quadratic graph crosses the x-axis twice, once, or not at all

    • The points where the graph crosses the x-axis are called the roots

  • If the graph is a u-shape, it has a minimum point

  • If the graph is an n-shape, it has a maximum point

  • Minimum and maximum points are both examples of turning points

    • A turning point can also be called a vertex

Diagram showing a positive quadratic curve with a minimum point and a negative quadratic curve with a maximum point.

How do I sketch a quadratic graph?

  • It is important to know how to sketch a quadratic curve

    • A simple drawing showing the key features is often sufficient

    • (For a more accurate graph, create a table of values and plot the points)

  • To sketch a quadratic graph:

    • First sketch the x and y-axes

    • Identify the y-intercept and mark it on the y-axis

      • The y-intercept of y equals a x squared plus b x plus c will be open parentheses 0 comma space c close parentheses

      • It can also be found by substituting in x equals 0

    • Find all root(s) (0, 1 or 2) of the equation and mark them on the x-axis

      • The roots will be the solutions to y equals 0; a x squared plus b x plus c equals 0

      • You can find the solutions by factorising, completing the square or using the quadratic formula

    • Identify if the number a in a x squared plus b x plus c is positive or negative

      • A positive value will result in a u-shape

      • A negative value will result in an n-shape

    • Sketch a smooth curve through the x and y-intercepts

      • Mark on any axes intercepts

      • Mark on the coordinates of the maximum/minimum point if you know it

How do I find the coordinates of the turning point by completing the square?

  • The coordinates of the turning point (vertex) of a quadratic graph can be found by completing the square

  • For a quadratic graph written in the form y equals a open parentheses x minus p close parentheses squared plus q

    • the minimum or maximum point has coordinates open parentheses p comma space q close parentheses

  • Beware: there is a sign change for the x-coordinate

    • A curve with equation y equals open parentheses x minus 3 close parentheses squared plus 2, has a minimum point at open parentheses 3 comma space 2 close parentheses

    • A curve with equation y equals open parentheses x plus 3 close parentheses squared plus 2, has a minimum point at open parentheses negative 3 comma space 2 close parentheses

  • The value of a does not affect the coordinates of the turning point but it will change the shape of the graph

    • If it is positive, the graph will be a u-shape

      • The curve y equals 5 open parentheses x minus 3 close parentheses squared plus 2 has a minimum point at open parentheses 3 comma space 2 close parentheses

    • Of it is negative, the graph will be an n-shape

      • The curve y equals negative 8 open parentheses x minus 3 close parentheses squared plus 2 has a maximum point at open parentheses 3 comma space 2 close parentheses

Worked Example

(a) Sketch the graph of y equals x squared minus 5 x plus 6 showing the x and y intercepts clearly.

The plus c at the end is the y-intercept

y-intercept: (0, 6)

Factorise the quadratic expression

y equals open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses

Solve y equals 0

open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses equals 0 comma space so space x equals 2 space or space x equals 3

So the x-intercepts are given by the coordinates

(2, 0)  and (3, 0)

It is a positive quadratic graph, so will be a u-shape

Graph of y=x²-5x+6 with y-intercept (06) and roots (2, 0), (3, 0) marked on.

 

(b) Sketch the graph of y equals x squared minus 6 x plus 13 showing the y-intercept and the coordinates of the turning point.

It is a positive quadratic, so will be a u-shape
The turning point will therefore be a minimum

The plus c at the end is the y-intercept

y-intercept: (0, 13)

Find the minimum point by completing the square

For example, complete the square by writing the equation in the form a open parentheses x minus p close parentheses squared plus q (you may need to look this method up)

table row cell x squared minus 6 x plus 13 end cell equals cell open parentheses x minus 3 close parentheses squared minus 9 plus 13 end cell row blank equals cell open parentheses x minus 3 close parentheses squared plus 4 end cell end table

The turning point of y equals a open parentheses x minus p close parentheses squared plus q has coordinates open parentheses p comma space q close parentheses
The minimum point is therefore

(3, 4)

As the minimum point is above the x-axis, and the curve is a u-shape, this means the graph will not cross the x-axis (it has no roots)

Graph of y=x²-6x+13, with y-intercept (0,13) and minimum point (3, 4) marked on.

 

(c) Sketch the graph of y equals negative x squared minus 4 x minus 4 showing the root(s), y-intercept, and the coordinates of the turning point.

It is a negative quadratic, so will be an n-shape
The turning point will therefore be a maximum

The plus c at the end is the y-intercept

y-intercept: (0, -4)

Find the minimum point by completing the square

y equals negative open parentheses x squared plus 4 x close parentheses minus 4 y equals negative open square brackets open parentheses x plus 2 close parentheses squared minus 2 squared close square brackets minus 4 y equals negative open square brackets open parentheses x plus 2 close parentheses squared minus 4 close square brackets minus 4 y equals negative open parentheses x plus 2 close parentheses squared plus 4 minus 4 y equals negative open parentheses x plus 2 close parentheses squared

The turning point of y equals a open parentheses x minus p close parentheses squared plus q has coordinates open parentheses p comma space q close parentheses
The minimum point is therefore

Minimum = (-2, 0)

As the maximum is on the x-axis, there is only one root

Graph of y=-x²-4x-4 with root (-2, 0) and y-intercept (0, -4) marked on.

How do I find the equation of a quadratic from its graph?

  • If the vertex and one other point are known

    • Use the form y equals a open parentheses x minus p close parentheses squared plus q to fill in p and q

      • The vertex is at open parentheses p comma space q close parentheses

    • Then substitute in the other known point open parentheses x comma space y close parentheses to find a

  • If the roots (x-intercepts) and one other point are known

    • Use the form y equals a open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 2 close parentheses to fill in x subscript 1 and x subscript 2

      • The roots are at open parentheses x subscript 1 space comma space 0 close parentheses and open parentheses x subscript 2 space comma space 0 close parentheses

    • Then substitute in the other known point open parentheses x comma space y close parentheses to find a

  • If a equals 1 then you only need either the vertex or the roots

Worked Example

(a) Find the equation of the graph below.

Positive u-shaped curve with roots at (2,0) and (3,0) and y-intercept of (0,24)

The graph shows the roots and a point on the curve (in this case the y-intercept)

Use the form y equals a open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 2 close parentheses to fill in x subscript 1 and x subscript 2 by inspection

The roots are at open parentheses x subscript 1 space comma space 0 close parentheses and open parentheses x subscript 2 space comma space 0 close parentheses

y equals a open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses

Substitute in the other known point (0, 24) to find a

table row 24 equals cell a open parentheses 0 minus 2 close parentheses open parentheses 0 minus 3 close parentheses end cell row 24 equals cell a open parentheses negative 2 close parentheses open parentheses negative 3 close parentheses end cell row 24 equals cell 6 a end cell row 4 equals a end table

Write the full equation

Error converting from MathML to accessible text.

You could also write this in expanded form: y equals 4 x squared minus 20 x plus 24

(b) Find the equation of the graph below.

Positive u-shaped curve with vertex at (9,-16) and a point on the curve at (2,82)

The graph shows the vertex and a point on the curve

Use the form y equals a open parentheses x minus p close parentheses squared plus q to fill in p and q by inspection

The vertex is at open parentheses p comma space q close parentheses

y equals a open parentheses x minus 9 close parentheses squared minus 16

Substitute in the other known point (2, 82) to find a

table row 82 equals cell a open parentheses 2 minus 9 close parentheses squared minus 16 end cell row 82 equals cell a open parentheses negative 7 close parentheses squared minus 16 end cell row 82 equals cell 49 a minus 16 end cell row 98 equals cell 49 a end cell row 2 equals a end table

Write the full equation

Error converting from MathML to accessible text.

You could also write this in expanded form: y equals 2 x squared minus 36 x plus 146

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Naomi C

Author: Naomi C

Expertise: Maths Content Creator

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

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