IGCSEMathsEdexcelMaths AHigherRevision NotesSequences, Functions & GraphsDifferentiationFinding Stationary Points & Turning Points

Finding Stationary Points & Turning Points (Edexcel IGCSE Maths A) : Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Edexcel Maths A: HigherEdexcel Maths A (Modular): Higher Unit 2Cambridge (CIE) Maths: Extended

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Finding Stationary Points & Turning Points

What is a stationary point?

  • A stationary point is a point on the graph at which the gradient is zero (a tangent drawn at this point will be horizontal)

    • These include peaks (maximum points) and troughs (minimum points)

  • Maximum points and minimum points are collectively know as turning points

    • At a turning point a curve changes from moving upwards to moving downwards, or vice versa

A curve on a graph with the direction of the graph at different points highlighted and the stationary points labelled where the direction changes.

  • At a turning point the gradient of the curve is zero

    • If a tangent is drawn at a turning point it will be a horizontal line

    • Horizontal lines have a gradient of zero

  • Because the gradient at a turning point is zero

    • Substituting the x-coordinate of a turning point into the gradient function fraction numerator straight d y over denominator straight d x end fraction (the derivative) will give an output of zero 

Graph of a curve with the stationary points highlighted. Tangents are drawn at the stationary point, which are horizontal lines.

How do I find the coordinates of a turning point?

  • STEP 1
    Find the gradient function (derivative) fraction numerator straight d y over denominator straight d x end fraction of the original equation

    • E.g. Find the coordinates of the turning point of the equation y equals 4 x squared minus 8 x plus 9 space

    • fraction numerator straight d y over denominator straight d x end fraction equals 8 x minus 8

  • STEP 2
    Set the gradient function (derivative) equal to zero and solve for x

    • This will find the x-coordinate of the turning point

    • E.g. 0 equals 8 x minus 8, so x equals 1

  • STEP 3
    Substitute the x-coordinate into the original equation of the graph

    • This will find the y-coordinate of the turning point

    • E.g. y equals 4 open parentheses 1 close parentheses squared minus 8 open parentheses 1 close parentheses plus 9 equals 5

  • STEP 4

    Write out the coordinates of the turning point

    • E.g. The turning point is at open parentheses 1 comma space 5 close parentheses

Examiner Tips and Tricks

  • Remember to read the questions carefully

    • Sometimes only the x-coordinate of a turning point is required

Worked Example

Find the coordinates of the turning point on the curve with equation y equals 2 x squared plus 8 x minus 9.

Use differentiation to find the gradient function (derivative) of the equation

fraction numerator straight d y over denominator straight d x end fraction equals 4 x plus 8

Set the derivative equal to 0 and solve for x

table row 0 equals cell 4 x plus 8 end cell row cell negative 8 end cell equals cell 4 x end cell row cell negative 2 end cell equals x end table

Substitute the x-coordinate into the origination equation of the curve and solve for y

y equals 2 open parentheses negative 2 close parentheses squared plus 8 open parentheses negative 2 close parentheses minus 9 y equals 8 minus 16 minus 9 y equals negative 17

Form a set of coordinates

The turning point has coordinates stretchy left parenthesis negative 2 comma space minus 17 stretchy right parenthesis

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Author
Reviewer
Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths 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.

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