Increasing and decreasing functions

Consider the curve with equation , shown below. The gradient of the tangent at any point, is given by , i.e. . So the gradient at the point is .

At the point , the gradient is 2.

When the tangent has a positive gradient, the graph is going up the page. We say that the function is increasing.

At the point , the gradient is -2.

When the tangent has a negative gradient, the graph is going down the page. We say that the function is decreasing.

At the point , the gradient is 0.

That is, the tangent is parallel to the -axis and the graph is at this instant going straight across the page. It is increasing immediately before and decreasing immediately after . This is called a maximum turning point. It's an example of a stationary point.

There are 4 types of stationary point:

Maximum turning point

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6. Next

f is increasing at

f is decreasing at

f is stationary at

Have a go at Wave Rider. This game has been designed to help you practise your ability to recognise and manipulate the components of a cosine wave.