Finding Stationary Points & Turning Points (Edexcel IGCSE Maths)

Spec point — 3.4C

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

• 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$\frac{dy}{dx}$ (the derivative) will give an output of zero

How do I find the coordinates of a turning point?

• STEP 1 Find the gradient function(derivative) $\frac{dy}{dx}$ of the original equation

  • E.g. Find the coordinates of the turning point of the equation $y=4x^{2}-8x+9$
  • $\frac{dy}{dx}=8x-8$

• 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. $0=8x-8$, 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 $( 1 , 5 )$

Exam Hint

• 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 $y=2x^{2}+8x-9$.

Answer:

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

$\frac{dy}{dx}=4x+8$

*Set the derivative equal to 0 and solve for *x

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

*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

Final answer:** The turning point has coordinates $(-2,-17)$