Graphing_hand.jpgThe gradient of a function is a measure of the rate of change of the y-value compared to the x-value.

Increasing Functions

If the gradient is positive over a range of values then the function is said to be increasingi.e the value of the y is increasing as x increases. On the graph the curve will be sloping up from left to right. e.g. Y11_Stationary_Points_01.gif For an increasing function f '(x) > 0

Decreasing Functions

If the gradient is negative over a range of values then the function is said to be decreasingi.e the value of the y is decreasing as x increases. On the graph the curve will be sloping down from left to right. e.g.Y11_Stationary_Points_02.gifFor a decreasing function f '(x) < 0

Stationary Points

If the gradient of a curve at a point is zero, then this point is called a stationary point. This can be a maximum stationary point or a minimum stationary point. They are also called turning points. For a stationary point f '(x) = 0

Stationary points are often called local because there are often greater or smaller values at other places in the function.

Another type of stationary point is called a point of inflection. With this type of point the gradient is zero but the gradient on either side of the point remains either positive or negative.


The diagram below shows examples of each of these types of points and parts of functions.


Finding Stationary Points

To find the coordinates and the type (nature) of the stationary points:



Find the coordinates and nature of the
stationary point(s) of the function
f(x) = x2 − 6x − 7

Step 1
Differentiate the function to find f '(x) f '(x) = 2x − 6
Step 2
Put f '(x) = 0 as the gradient at the stationary points is zero. 2x − 6 = 0
Step 3
Solve this equation to find the x-value of the stationary point. 2x = 6
x = 3
Step 4
Substitute this value into the original function to find the y-value of the stationary point.

When x = 3

y = 32 − 6 x 3 − 7 = -16

Coordinates of the turning point are (3, -16)

Step 5
Examine the gradient on either side of the stationary point to find its nature.

Choose values either side of the turning point

When x = 2 When x = 3
When x = 4
f ' (2) is negative f '(3) is zero  f '(4) is positive

This shows a minimum point.