Graphs

What can we find out from the function itself? Take the function To find the roots

Function -5 3

Stationary Points Find where the first derivative is zero Substitute x-values to find y-values (1.31, -24.6), (-3.31, 24.6)

(1.31, -24.6) (-3.31, 24.6)

(1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

(1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

(1.31, -24.6) (-3.31, 24.6) Gradient function is negative i.e. Function is decreasing

Nature of turning points Function First derivative Second derivative Substitute the x-values of the stationary points Positive indicates minimum Negative indicates maximum

is a maximum is negative is a minimum is positive

is concave down is negative

is concave up is positive

Concave Up - 2nd derivative positive Concave Down - 2nd derivative negative

has a point of inflection is zero There is a change in curvature

Example 1 Find the stationary points of the following function and determine their nature. To find the roots Roots are: (-3.63, 0) (-1, 0) Using solver on graphics calculator

x = -3.63

Example 1 To find the stationary points. Differentiate Factorise Stationary Points are: (0, 1), (-1, 0), (-3, 28)

-3, 28 -1, 0 0, 1

The first derivative tells us where the function is increasing/decreasing and where it is stationary.

Function is stationary Function is stationary Function is stationary

The first derivative tells us where the function is increasing/decreasing and where it is stationary. Gradient is positive

The first derivative tells us where the function is increasing/decreasing … Function is increasing Function is increasing Function is increasing

The first derivative tells us where the function is increasing/decreasing … Function is decreasing

To determine the nature of the turning points: Differentiate again:

x = -3

x = -1

x = 0 Let’s take a closer look!

x = 0 This means we need to look at the gradient function.

x = 0 Before ‘0’, the gradient is negative.

x = 0 After ‘0’, the gradient is positive.

To determine the nature of the turning points: Differentiate again: Gradient is negative just before “0” and positive just after “0” minimum

Practice: Concavity Find where the following function is concave down. Differentiate twice:

Practice: Find where the function is increasing Draw the graph