The document provides guidance on sketching curves. It explains that to sketch a curve defined by y = f(x), one should look for discontinuities, asymptotes, stationary points where the derivative f'(x) = 0, maximum/minimum turning points where f''(x) changes sign, and points of inflection where f''(x) = 0. It also defines what it means for a curve to be increasing, decreasing, concave up or down based on the signs of f'(x) and f''(x). An example of sketching the curve y = x3 - 6x2 + 9x - 5 is then worked through step-by-step.

1. Curve Sketching

2. Curve Sketching
Look for;

3. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x

4. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x

5. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 

6. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0

7. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0

8. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0

9. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0

10. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0
A curve is;

11. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0
A curve is;  increasing if f  x   0

12. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0
A curve is;  increasing if f  x   0
 decreasing if f  x   0

13. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0
A curve is;  increasing if f  x   0  concave up if f  x   0
 decreasing if f  x   0

14. Curve Sketching
Look for; 1
 points of discontinu ity e.g. y  , x  0, y  0
x
1
 asymptotes e.g. y  x  , x  0, y  x
x
On the curve y  f  x 
1 stationary points occur when f  x   0
2 maximum turning point if f  x   0, f  x   0
3 minimum turning point if f  x   0, f  x   0
4 point of inflection if f  x   0 and there is a change in
concavity  f  x   0
A curve is;  increasing if f  x   0  concave up if f  x   0
 decreasing if f  x   0  concave down if f  x   0

15. e.g. Sketch the curve y  x 3  6 x 2  9 x  5

16. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx

17. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx

18. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx

19. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx

20. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx
i.e. 3x 2  12 x  9  0

21. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx
i.e. 3x 2  12 x  9  0
x2  4x  3  0
 x  1 x  3  0

22. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx
i.e. 3x 2  12 x  9  0
x2  4x  3  0
 x  1 x  3  0
x  1 or x  3

23. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx
i.e. 3x 2  12 x  9  0
x2  4x  3  0
 x  1 x  3  0
2
x  1 or x  3
d y
when x  1, 2  61  12
dx
 6  0

24. e.g. Sketch the curve y  x 3  6 x 2  9 x  5
dy
 3x 2  12 x  9
dx
d2y
2
 6 x  12
dx
d3y
3
6
dx dy
Stationary points occur when  0
dx
i.e. 3x 2  12 x  9  0
x2  4x  3  0
 x  1 x  3  0
2
x  1 or x  3
d y
when x  1, 2  61  12
dx
 6  0
 1,  1 is a maximum turning point

25. d2y
when x  3, 2  63  12
dx
60

26. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point

27. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point
d2y
Possible points of inflection occur when 2  0
dx

28. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point
d2y
Possible points of inflection occur when 2  0
dx
i.e. 6 x  12  0

29. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point
d2y
Possible points of inflection occur when 2  0
dx
i.e. 6 x  12  0
x2

30. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point
d2y
Possible points of inflection occur when 2  0
dx
i.e. 6 x  12  0
x2
d3y
when x  2, 3  6  0
dx

31. d2y
when x  3, 2  63  12
dx
60
 3,  5 is a minimum turning point
d2y
Possible points of inflection occur when 2  0
dx
i.e. 6 x  12  0
x2
d3y
when x  2, 3  6  0
dx
 2,  3 is a point of inflection

32. y
x

33. y
x
1,  1

34. y
x
1,  1
5
3,  5

35. y
x
1,  1
2,  3
3,  5

36. y
x
1,  1
2,  3
5
3,  5

37. y
x
1,  1
2,  3 y  x3  6 x 2  9 x  5
5
3,  5

38. y
x
1,  1
Exercise 10F;
1bdf, 2, 4b, 2,  3 y  x3  6 x 2  9 x  5
5ace etc, 6bd
Exercise 10G
1, 2ace, 4b, 5c
5
3,  5