Remind me of the rule for differentiating powers ...
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If y = kxn then dy/dx = knxn-1
What is a turning point?
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The easiest way to think of a turning point is that it is a point at which a
curve changes from moving upwards to moving downwards, or vice versa
Turning points are also called stationary points
Ensure you are familiar with Differentiation – Basics before moving on
At a turning point the gradient of the curve is zero.
o If a tangent is drawn at a turning point it will be a horizontal line
o Horizontal lines have a gradient of zero
This means at a turning point the derived
function (aka gradient function or derivative) equals zero
How do I know if a curve has turning points?
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You can see from the shape of a curve whether it has turning points or not
At IGCSE, two types of turning point are considered:
o Maximum points – this is where the graph reaches a “peak”
o Minimum points – this is where the graph reaches a “trough”
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These are sometimes called local maximum/minimum points as other parts of
the graph may still reach higher/lower values
How do I find the coordinates of a turning point?
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STEP 1 Solve the equation of the derived function (derivative) equal to zero
ie. solve dy/dx = 0
This will find the x-coordinate of the turning point
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STEP 2 To find the y-coordinate substitute the x-coordinate into the equation of
the graph
ie. substitute x into “y = ...”
How do I know which point is a maximum and which is a minimum?
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There are several ways to do this
o Exam questions may ask you to “give reasons” or “justify” why you think a
particular turning point is, say, a maximum
The easiest way to do this is to recognise the shape of the curve
o ... either from a given sketch of the curve
o ... a sketch of the curve you can quickly draw yourself
(You may even be asked to do this as part of a question)
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o ... the equation of the curve
For parabolas (quadratics) it should be obvious ...
o ... a positive parabola (positive x2 term) has a minimum point
o ... a negative parabola (negative x2 term) has a maximum point
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Cubic graphs are also easily recognisable ...
o ... a positive cubic has a maximum point on the left, minimum on
the right
o ... a negative cubic has a minimum on the left, maximum on the right