Graphs

1. Graphs

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

3. Function -5 -1 3

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

5. (1.31, -24.6) (-3.31, 24.6)

6. (1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

7. (1.31, -24.6) (-3.31, 24.6) Gradient function is positive i.e. Function is increasing

8. (1.31, -24.6) (-3.31, 24.6) Gradient function is negative i.e. Function is decreasing

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

11. is a maximum is positive is negative is a minimum

12. is concave down is negative

13. is positive is concave up

14. Concave Up - 2nd derivative positive • Concave Down - 2nd derivative negative

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

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

17. x = -3.63

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

19. -3, 28 0, 1 -1, 0

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

21. 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

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

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

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

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

27. x = -3

28. x = -1

29. x = 0 Let’s take a closer look!

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

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

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

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

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

35. Practice: Find where the function is increasing Draw the graph