The document discusses key features to notice when analyzing graphs of functions. It identifies basic curve shapes defined by common equations, such as straight lines, parabolas, cubics, and circles. It also covers concepts like odd and even functions, symmetry, and dominance - how certain terms in an equation become more prominent at higher values of x. Basic curve shapes and these analytical concepts can help recognize and sketch the shape of graphs defined by functions.

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: y  x (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: y  x (both pronumerals are to the power of one)
b) Parabolas: y  x 2 (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: y  x (both pronumerals are to the power of one)
b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is y  ax 2  bx  c

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: y  x (both pronumerals are to the power of one)
b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is y  ax 2  bx  c
c) Cubics: y  x 3 (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: y  x (both pronumerals are to the power of one)
b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is y  ax 2  bx  c
c) Cubics: y  x 3 (one pronumeral is to the power of one, the other the
power of three)
NOTE: general cubic is y  ax 3  bx 2  cx  d

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: y  x (both pronumerals are to the power of one)
b) Parabolas: y  x 2 (one pronumeral is to the power of one, the other
the power of two)
NOTE: general parabola is y  ax 2  bx  c
c) Cubics: y  x 3 (one pronumeral is to the power of one, the other the
power of three)
NOTE: general cubic is y  ax 3  bx 2  cx  d
d) Polynomials in general

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

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

12. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)

13. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two,
coefficients are NOT the same)

14. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)

15. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: y  log a x

16. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: y  log a x
j) Trigonometric: y  sin x, y  cos x, y  tan x

17. 1
e) Hyperbolas: y  OR xy  1
x
(one pronomeral is on the bottom of the fraction, the other is not OR
pronumerals are multiplied together)
f) Exponentials: y  a x (one pronumeral is in the power)
g) Circles: x 2  y 2  r 2 (both pronumerals are to the power of two,
coefficients are the same)
h) Ellipses: ax 2  by 2  k (both pronumerals are to the power of two,
coefficients are NOT the same)
(NOTE: if signs are different then hyperbola)
i) Logarithmics: y  log a x
j) Trigonometric: y  sin x, y  cos x, y  tan x
1 1 1
k) Inverse Trigonmetric: y  sin x, y  cos x, y  tan x

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
a) ODD : f  x    f  x  (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
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (symmetric about the y axis)

22. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (in other words, the curve is its own inverse)

24. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
e.g. y  x 4  3 x 3  2 x  2, x 4 dominates

27. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
e.g. y  x 4  3 x 3  2 x  2, x 4 dominates
b) Exponentials: e x tends to dominate as it increases so rapidly

28. (2) Odd & Even Functions
These curves have symmetry and are thus easier to sketch
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
e.g. y  x 4  3 x 3  2 x  2, x 4 dominates
b) Exponentials: e x tends to dominate as it increases so rapidly
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
a) ODD : f  x    f  x  (symmetric about the origin, i.e. 180 degree
rotational symmetry)
b) EVEN : f  x   f  x  (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
e.g. x 3  y 3  1 (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
e.g. y  x 4  3 x 3  2 x  2, x 4 dominates
b) Exponentials: e x tends to dominate as it increases so rapidly
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
sin x
Remember the special limit seen in 2 Unit i.e. lim 1
x0 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
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.
(1) Critical Points

39. (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.
(1) Critical Points
dy
When dx is undefined the curve has a vertical tangent, these points
are called critical points.

40. (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.
(1) Critical Points
dy
When dx is undefined the curve has a vertical tangent, these points
are called critical points.
(2) Stationary Points

41. (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.
(1) Critical Points
dy
When dx is undefined the curve has a vertical tangent, these points
are called critical points.
(2) Stationary Points
dy
When dx  0 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
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx

44. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx

45. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx
dy
NOTE: testing either side of for change can be quicker for harder
dx
functions

46. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx
dy
NOTE: testing either side of for change can be quicker for harder
dx
functions
(4) Inflection Points

47. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx
dy
NOTE: testing either side of for change can be quicker for harder
dx
functions
(4) Inflection Points
d2y d3y
a) When 2  0 and 3  0, the point is called an inflection point
dx dx

48. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx
dy
NOTE: testing either side of for change can be quicker for harder
dx
functions
(4) Inflection Points
d2y d3y
a) When 2  0 and 3  0, the point is called an inflection point
dx dx
d2y
NOTE: testing either side of 2
for change can be quicker for harder
dx
functions

49. (3) Minimum/Maximum Turning Points
dy d2y
a) When  0 and 2  0, the point is called a minimum turning point
dx dx
dy d2y
b) When  0 and 2  0, the point is called a maximum turning point
dx dx
dy
NOTE: testing either side of for change can be quicker for harder
dx
functions
(4) Inflection Points
d2y d3y
a) When 2  0 and 3  0, the point is called an inflection point
dx dx
d2y
NOTE: testing either side of 2
for change can be quicker for harder
dx
functions
dy d2y d3y
b) When  0, 2  0 and 3  0, the point is called a horizontal
dx dx dx
point of inflection

50. (5) Increasing/Decreasing Curves

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

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

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

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

55. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1

56. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1
Note:• the curve has symmetry in y = x

57. y
y=x
x

58. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)

59. y
y=x
1
1 x

60. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1
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
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
 y 3  1  x3
i.e. y 3   x 3
y  x

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

63. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1 On differentiating implicitly;
Note:• the curve has symmetry in y = x
• it passes through (1,0) and (0,1)
• it is asymptotic to the line y = -x
 y 3  1  x3
i.e. y 3   x 3
y  x

64. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1 On differentiating implicitly;
2 dy
Note:• the curve has symmetry in y = x 3x  3 y
2
0
• it passes through (1,0) and (0,1) dx
• it is asymptotic to the line y = -x dy  x 2
 2
 y  1 x
3 3 dx y
i.e. y 3   x 3
y  x

65. (5) Increasing/Decreasing Curves
dy
a) When  0, the curve has a positive sloped tangent and is
dx thus increasing
dy
b) When  0, the curve has a negative sloped tangent and is
dx thus decreasing
(6) Implicit Differentiation
This technique allows you to differentiate complicated functions
e.g. Sketch x 3  y 3  1 On differentiating implicitly;
2 dy
Note:• the curve has symmetry in y = x 3x  3 y
2
0
• it passes through (1,0) and (0,1) dx
• it is asymptotic to the line y = -x dy  x 2
 2
 y  1 x
3 3 dx y
dy
i.e. y 3   x 3 This means that  0 for all x
dx
y  x Except at (1,0) : critical point &
(0,1): horizontal point of inflection

66. y
y=x
1
x3  y 3  1
1 x
y=-x