IGCSEMathsCambridge (CIE)International MathsExtendedRevision NotesCoordinate Geometry & GraphsTypes of GraphsQuadratic Graphs

Quadratic Graphs (Cambridge (CIE) IGCSE International Maths): Revision Note

Exam code: 0607

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Edexcel Maths A: HigherCambridge (CIE) Maths: Extended

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 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 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

Worked Example

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.

How do I find the coordinates of the vertex?

  • 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 vertex but it will change the shape of the graph

    • If it is positive, a greater than 0, 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

    • If it is negative, a less than 0, 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

Sketch the graph of y equals open parentheses x minus 3 close parentheses squared plus 4 showing the y-intercept and the coordinates of the vertex.

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

The vertex 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)

To find the y-intercept, substitute in x equals 0

table row y equals cell open parentheses 0 minus 3 close parentheses squared plus 4 end cell row y equals cell 9 plus 4 end cell row y equals 13 end table

y-intercept: (0, 13)

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.

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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    Author
    Reviewer
    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.