Finding Stationary Points & Turning Points (Cambridge (CIE) IGCSE Maths)

Revision Note

Written by: Jamie Wood

Reviewed by: Dan Finlay

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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

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 $\frac{d y}{d x}$ (the derivative) will give an output of zero

STEP 1
Find the gradient function (derivative) $\frac{d y}{d x}$ of the original equation
- E.g. Find the coordinates of the turning point of the equation $y = 4 x^{2} - 8 x + 9$
- $\frac{d y}{d x} = 8 x - 8$
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. $0 = 8 x - 8$ , so $x = 1$
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. $y = 4 {(1)}^{2} - 8 (1) + 9 = 5$
STEP 4
Write out the coordinates of the turning point
- E.g. The turning point is at $(1, 5)$

Remember to read the questions carefully
- Sometimes only the x-coordinate of a turning point is required

Find the coordinates of the turning point on the curve with equation $y = 2 x^{2} + 8 x - 9$ .

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

$\frac{d y}{d x} = 4 x + 8$

Set the derivative equal to 0 and solve for x

$\begin{array}{rcl} 0 & = & 4 x + 8 \\ - 8 & = & 4 x \\ - 2 & = & x \end{array}$

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

$y = 2 {(- 2)}^{2} + 8 (- 2) - 9 y = 8 - 16 - 9 y = - 17$

Form a set of coordinates

The turning point has coordinates $(- 2, - 17)$

Last updated: 17 October 2024

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