1. (A) Features You Should
Notice About A Graph

2. (A) Features You Should
Notice About A Graph(1) Basic Curves

3. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;

4. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines: xy  (both pronumerals are to the power of one)

5. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines:
b) Parabolas:
xy  (both pronumerals are to the power of one)
2
xy  (one pronumeral is to the power of one, the other
the power of two)

6. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines:
b) Parabolas:
xy  (both pronumerals are to the power of one)
2
xy  (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is cbxaxy  2

7. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines:
b) Parabolas:
c) Cubics:
xy  (both pronumerals are to the power of one)
2
xy  (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is cbxaxy  2
3
xy  (one pronumeral is to the power of one, the other the
power of three)

8. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines:
b) Parabolas:
c) Cubics:
xy  (both pronumerals are to the power of one)
2
xy  (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is cbxaxy  2
3
xy  (one pronumeral is to the power of one, the other the
power of three)
NOTE: general cubic is dcxbxaxy  23

9. (A) Features You Should
Notice About A Graph(1) Basic Curves
The following basic curve shapes should be recognisable from the
equation;
a) Straight lines:
b) Parabolas:
c) Cubics:
xy  (both pronumerals are to the power of one)
2
xy  (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is cbxaxy  2
3
xy  (one pronumeral is to the power of one, the other the
power of three)
NOTE: general cubic is dcxbxaxy  23
d) Polynomials in general

10. e) Hyperbolas: 1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)

11. e) Hyperbolas:
f) Exponentials:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)

12. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)

13. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: kbyax  22
(both pronumerals are to the power of two,
coefficients are NOT the same)

14. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: kbyax  22
(both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)

15. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: kbyax  22
(both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: xy alog

16. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: kbyax  22
(both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: xy alog
j) Trigonometric: xyxyxy tan,cos,sin 

17. e) Hyperbolas:
f) Exponentials:
g) Circles:
1OR
1
 xy
x
y
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
x
ay  (one pronumeral is in the power)
222
ryx  (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: kbyax  22
(both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: xy alog
j) Trigonometric: xyxyxy tan,cos,sin 
k) Inverse Trigonmetric: xyxyxy 111
tan,cos,sin 


18. (2) Odd & Even Functions

19. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch

20. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)

21. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)

22. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x

23. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)

24. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance

25. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance
As x gets large, does a particular term dominate?

26. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance
As x gets large, does a particular term dominate?
a) Polynomials: the leading term dominates
dominates,223e.g. 434
xxxxy 

27. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance
As x gets large, does a particular term dominate?
a) Polynomials: the leading term dominates
dominates,223e.g. 434
xxxxy 
b) Exponentials: tends to dominate as it increases so rapidlyx
e

28. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance
As x gets large, does a particular term dominate?
a) Polynomials: the leading term dominates
dominates,223e.g. 434
xxxxy 
b) Exponentials: tends to dominate as it increases so rapidlyx
e
c) In General: look for the term that increases the most rapidly
i.e. which is the steepest

29. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
   xfxf :ODDa) (symmetric about the origin, i.e. 180 degree
rotational symmetry)
   xfxf :EVENb) (symmetric about the y axis)
(3) Symmetry in the line y = x
If x and y can be interchanged without changing the function, the curve is
relected in the line y = x
1e.g. 33
 yx (in other words, the curve is its own inverse)
(4) Dominance
As x gets large, does a particular term dominate?
a) Polynomials: the leading term dominates
dominates,223e.g. 434
xxxxy 
b) Exponentials: tends to dominate as it increases so rapidlyx
e
c) In General: look for the term that increases the most rapidly
i.e. which is the steepest
NOTE: check by substituting large numbers e.g. 1000000

30. (5) Asymptotes

31. (5) Asymptotes
a) Vertical Asymptotes: the bottom of a fraction cannot equal zero

32. (5) Asymptotes
a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the
fraction cannot equal zero

33. (5) Asymptotes
a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the
fraction cannot equal zero
NOTE: if order of numerator order of denominator, perform a
polynomial division. (curves can cross horizontal/oblique asymptotes,
good idea to check)


34. (5) Asymptotes
a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the
fraction cannot equal zero
NOTE: if order of numerator order of denominator, perform a
polynomial division. (curves can cross horizontal/oblique asymptotes,
good idea to check)

(6) The Special Limit

35. (5) Asymptotes
a) Vertical Asymptotes: the bottom of a fraction cannot equal zero
b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the
fraction cannot equal zero
NOTE: if order of numerator order of denominator, perform a
polynomial division. (curves can cross horizontal/oblique asymptotes,
good idea to check)

(6) The Special Limit
Remember the special limit seen in 2 Unit 1
sin
limi.e.
0

 x
x
x
, it could come in handy when solving harder graphs.

36. (B) Using Calculus

37. (B) Using Calculus
Calculus is still a tremendous tool that should not be disregarded when
curve sketching. However, often it is used as a final tool to determine
critical points, stationary points, inflections.

38. (B) Using Calculus
(1) Critical Points
Calculus is still a tremendous tool that should not be disregarded when
curve sketching. However, often it is used as a final tool to determine
critical points, stationary points, inflections.

39. (B) Using Calculus
(1) Critical Points
dx
dy
When is undefined the curve has a vertical tangent, these points
are called critical points.
Calculus is still a tremendous tool that should not be disregarded when
curve sketching. However, often it is used as a final tool to determine
critical points, stationary points, inflections.

40. (B) Using Calculus
(1) Critical Points
(2) Stationary Points
dx
dy
When is undefined the curve has a vertical tangent, these points
are called critical points.
Calculus is still a tremendous tool that should not be disregarded when
curve sketching. However, often it is used as a final tool to determine
critical points, stationary points, inflections.

41. (B) Using Calculus
(1) Critical Points
(2) Stationary Points
dx
dy
When is undefined the curve has a vertical tangent, these points
are called critical points.
Calculus is still a tremendous tool that should not be disregarded when
curve sketching. However, often it is used as a final tool to determine
critical points, stationary points, inflections.
0
dx
dy
When the curve is said to be stationary, these points may be
minimum turning points, maximum turning points or points of
inflection.

42. (3) Minimum/Maximum Turning Points

43. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point

44. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point

45. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point
functions
harderforquickerbecanchangeforofsideeithertesting
dx
dy
NOTE:

46. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point
functions
harderforquickerbecanchangeforofsideeithertesting
dx
dy
NOTE:
(4) Inflection Points

47. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point
functions
harderforquickerbecanchangeforofsideeithertesting
dx
dy
NOTE:
(4) Inflection Points
,0and0Whena) 3
3
2
2

dx
yd
dx
yd
the point is called an inflection point

48. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point
functions
harderforquickerbecanchangeforofsideeithertesting
dx
dy
NOTE:
(4) Inflection Points
,0and0Whena) 3
3
2
2

dx
yd
dx
yd
the point is called an inflection point
functions
harderforquickerbecanchangeforofsideeithertesting 2
2
dx
yd
NOTE:

49. (3) Minimum/Maximum Turning Points
,0and0Whena) 2
2

dx
yd
dx
dy
the point is called a minimum turning point
,0and0Whenb) 2
2

dx
yd
dx
dy
the point is called a maximum turning point
functions
harderforquickerbecanchangeforofsideeithertesting
dx
dy
NOTE:
(4) Inflection Points
,0and0Whena) 3
3
2
2

dx
yd
dx
yd
the point is called an inflection point
,0and0,0Whenb) 3
3
2
2

dx
yd
dx
yd
dx
dy
the point is called a horizontal
point of inflection
functions
harderforquickerbecanchangeforofsideeithertesting 2
2
dx
yd
NOTE:

50. (5) Increasing/Decreasing Curves

51. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing

52. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing

53. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation

54. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions

55. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx

56. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x

57. x
y
y=x

58. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)

59. x
y
y=x
1
1

60. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x

61. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
xy
xy
xy



33
33
i.e.
1

62. x
y
y=x
1
1
y=-x

63. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
xy
xy
xy



33
33
i.e.
1
On differentiating implicitly;

64. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
xy
xy
xy



33
33
i.e.
1
On differentiating implicitly;
2
2
22
033
y
x
dx
dy
dx
dy
yx




65. (5) Increasing/Decreasing Curves
,0Whena) 
dx
dy
the curve has a positive sloped tangent and is
thus increasing
,0Whenb) 
dx
dy
the curve has a negative sloped tangent and is
thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
1Sketche.g. 33
 yx
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
xy
xy
xy



33
33
i.e.
1
On differentiating implicitly;
2
2
22
033
y
x
dx
dy
dx
dy
yx



This means that for all x
Except at (1,0) : critical point &
(0,1): horizontal point of inflection
0
dx
dy

66. x
y
y=x
1
1
y=-x
133
 yx