The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type it provides the standard form of the equation and notes on identifying features like intercepts, vertices, and behavior as powers increase. It emphasizes using standard forms, intercepts, and factoring to determine a curve's shape.
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*