The document discusses how to sketch graphs based on their equations. It provides the following key points: - Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0. - Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0. - Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1. Once intercepts are found, curves can be sketched by

1. Sketching Graphs

2. Sketching Graphs
* Numbers on axes must be evenly spaced

3. Sketching Graphs
* Numbers on axes must be evenly spaced
* y intercept occurs when x = 0

4. Sketching Graphs
* Numbers on axes must be evenly spaced
* y intercept occurs when x = 0
* x intercept occurs when y = 0

5. Sketching Graphs
* Numbers on axes must be evenly spaced
* y intercept occurs when x = 0
* x intercept occurs when y = 0
Once the intercepts have been found, curves are easy to
sketch, if you know the basic shape.

6. Sketching Graphs
* Numbers on axes must be evenly spaced
* y intercept occurs when x = 0
* x intercept occurs when y = 0
Once the intercepts have been found, curves are easy to
sketch, if you know the basic shape.
If in doubt use a table of values and plot some points

7. Basic Curves

8. Basic Curves
(1) Straight Lines
y yx
x

9. Basic Curves
(1) Straight Lines
y yx
y  mx  b
x

10. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1

11. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1
(2) Parabola
y y  x2
x

12. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1
(2) Parabola
y y  x2
y  ax 2  bx  c
x

13. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1
(2) Parabola
y y  x2 power `1
y  ax 2  bx  c
power `2
x

14. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1
(2) Parabola
y y  x2 power `1
y  ax 2  bx  c
power `2
x x intercepts:
solve ax 2  bx  c  0

15. Basic Curves
(1) Straight Lines
y yx
power `1
y  mx  b
x power `1
(2) Parabola
y y  x2 power `1
y  ax 2  bx  c
power `2
x x intercepts:
solve ax 2  bx  c  0
vertex: halfway between x intercepts

16. e.g. find the x intercepts and vertex of y  x 2  4 x  3

17. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2  1
2
 vertex 2,1

18. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1

19. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
x = 1 or x = 3

20. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1

21. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1
(3) Cubic
y y  x3
x

22. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1
(3) Cubic
y y  x3
y  ax 3  bx 2  cx  d
x

23. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1
(3) Cubic
y  x3 power `1
y
y  ax 3  bx 2  cx  d
x power `3

24. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1
(3) Cubic
y  x3 power `1
y
y  ax 3  bx 2  cx  d
x power `3
If it can be factorised to y  a x  k 
3
then it will look like the basic cubic.

25. e.g. find the x intercepts and vertex of y  x 2  4 x  3
y   x  2   1 x intercepts:  x  2 2  1
2
 vertex 2,1 x  2  1
x  2 1
y x = 1 or x = 3
1 3 x
2,1
(3) Cubic
y  x3 power `1
y
y  ax 3  bx 2  cx  d Otherwise;
y
x power `3
If it can be factorised to y  a x  k 
3
x
then it will look like the basic cubic.

26. (4) Higher Powers
even
y
y  x4

x y  x6

etc

27. (4) Higher Powers
even odd
y y
y  x4 y  x5
 
x y  x6 x y  x7
 
etc etc

28. (4) Higher Powers
even odd
y y
y  x4 y  x5
 
x y  x6 x y  x7
 
etc etc
As power gets bigger, curve gets; - flatter at base
- steeper at the sides

29. (4) Higher Powers
even odd
y y
y  x4 y  x5
 
x y  x6 x y  x7
 
etc etc
As power gets bigger, curve gets; - flatter at base
- steeper at the sides
(5) Hyperbola
y
x

30. (4) Higher Powers
even odd
y y
y  x4 y  x5
 
x y  x6 x y  x7
 
etc etc
As power gets bigger, curve gets; - flatter at base
- steeper at the sides
(5) Hyperbola
1
y y
x
OR
x xy  1

31. (4) Higher Powers
even odd
y y
y  x4 y  x5
 
x y  x6 x y  x7
 
etc etc
As power gets bigger, curve gets; - flatter at base
- steeper at the sides
(5) Hyperbola
1
y y
x 1
y
OR x
x xy  1
one power on top of a fraction,
one power on bottom of fraction

32. (6) Circle
y
r
x2  y2  r 2
-r r x
-r

33. (6) Circle
y
r
x2  y2  r 2
x2  y2  r 2
-r r x
both power ‘2’
-r coefficients are the same

34. (6) Circle
y
r
x2  y2  r 2
x2  y2  r 2
-r r x
both power ‘2’
-r coefficients are the same
(7) Exponential
y y  ax
a  1
1
x

35. (6) Circle
y
r
x2  y2  r 2
x2  y2  r 2
-r r x
both power ‘2’
-r coefficients are the same
(7) Exponential
y y  ax
a  1
1 y  ax
x pronumeral in the power

36. (8) Roots
y
r y  r 2  x2
-r r x

37. (8) Roots
y y
r y  r 2  x2 y x
-r r x x

38. (8) Roots
y y
r y  r 2  x2 y x
-r r x x
to tell what the curve looks like, square both sides

39. (8) Roots
y y
r y  r 2  x2 y x
-r r x x
to tell what the curve looks like, square both sides
y  r 2  x2
y2  r 2  x2
x2  y2  r 2
 circle

40. (8) Roots
y y
r y  r 2  x2 y x
-r r x x
to tell what the curve looks like, square both sides
y  r 2  x2 y x
y2  r 2  x2 y2  x
x2  y2  r 2  parabola
 circle

41. (8) Roots
y y
r y  r 2  x2 y x
-r r x x
to tell what the curve looks like, square both sides
y  r 2  x2 y x
y2  r 2  x2 y2  x
x2  y2  r 2  parabola
 circle  means top half
 means bottom half

42. (8) Roots
y y
r y  r 2  x2 y x
-r r x x
to tell what the curve looks like, square both sides
y  r 2  x2 y x
y2  r 2  x2 y2  x
x2  y2  r 2  parabola
 circle  means top half
Exercise 2G; 1adeh, 2adgk, 3bdf,  means bottom half
5aeh, 7acegi, 8ad, 9a, 11c, 13acd,
15bd, 17*