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
12. Basic Curves
(1) Straight Lines
y yx
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 yx
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 yx
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 yx
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.
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
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
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*