1. FUNCTIONS Concepts in Functions Straight Line Graphs Parabolas
Hyperbolas
Exponentials
Sine Graphs
Cos Graphs
Tan Graphs

2. CONCEPTS IN FUNCTIONS Formulae and Tables x -2 1 2 3 y -4 4 6
E.g. Given the formula y = 2x, the table can be constructed: x -2 1 2 3 y -4 4 6
x in this relation is called the independent variable, since the values of x were chosen randomly.
However, it is clear that the values of y depended entirely on the values of x as well as the rule used, namely, y = 2x. In this relation, y is called the dependent variable.

3. Ordered Pairs on the Cartesian Plane (Co-ordinates)
Graphs
In order to draw graphs, we need to plot co-ordinates or ordered pairs onto the Cartesian Plane
Co-ordinate or Ordered Pair:
(independent variable; dependent variable)
(x; y)
Ordered Pairs on the Cartesian Plane (Co-ordinates)

4. Plotting points on a Cartesian Plane
The Cartesian Plane
Decode the message by writing down the letter found at each of the following coordinates: (1,5), (3,1), (3,4), (5,2), (5,4), (1,1), (4,5) and (1,2).
Plotting points on a Cartesian Plane

5. Domain and Range Domain refers to the x-values that exist
Range refers to the y-values that exist
Examples
a) Given: set of ordered pairs {(-2; 16); (0; 4); (3; 7)}
Domain = {-2; 0; 3}
Range = {16; 4; 7}
b) Given: equation
Domain = {-2; -1; 0; 1; 2; 3}
Range = {-4;-2;0;2;4;6}.

6. Finding the Domain and Range of Graphs
b) Given: graph
Domain:
Range:
Finding the Domain and Range of Graphs

7. f(x) Notation f(x) is the equivalent of y
Example: Given determine the value of:
a) f(2) … therefore, substitute x = 2
b) f(x)=0

8. EXERCISE Determine the domain and range:
a) Given the set: {(-3;-4); (2;1); (-6;-9)}
b) Given the equation: y=4x-6
c) Given the graph:

9. 2. If determine the value of:
a) f(1)
b) f(- 1)
c) f (2x)
d) f(x)=0
e) f(x)=1

10. STRAIGHT LINE GRAPHS
The graph of a linear function is a straight line.
Standard form of a straight line graph: y = m x + c
where m = gradient
and c = y-intercept

11. Effects of m and c in y = mx + c:
m represents the gradient of the straight line
If m > 0, the gradient is positive
If m < 0, the gradient is negative
c represents the y-intercept of the straight line graph and also indicates the vertical translation of the graph
If c > 0, the graph has a positive y-intercept
and shifts up by c units
If c < 0, the graph has a negative y-intercept
and shifts down by c units
Investigating the effects of m in a Straight Line
Investigating the effects of c in a Straight Line

12. Sketching Straight Line Graphs Using The Table Method
Example 1
Sketch the graph of y = 2x+4 by using the table
method. x -1 1 2 y 4 6 8
The x-values were randomly chosen.
The y-values were found by substituting the x - values into the equation y = 2x +4.

13. Note!
x-intercept is at
(-2; 0)
y-intercept is at
(0; 4)

14. Sketching Straight Lines using the Dual-Intercept Method
Sketching Straight Line Graphs Using The Dual-Intercept Method
Sketching Straight Lines using the Dual-Intercept Method
Example
Sketch the graph of y = -3x - 3 by using the dual-
intercept method.
x-intercept: y = 0 y-intercept: x = 0
0 = -3x – 3 y = -3(0) – 3
3x = y = 0 – 3
x = y = -3
(-1; 0) (0;-3)

15. The Graph of y = -3x – 3:

16. Sketching Straight Line Graphs Using The Gradient-Intercept Method
Example
Sketch the graph of by using the gradient-
intercept method.
The negative sign indicates that the line slopes to the left. The y-intercept is 0.
The numerator tells us to rise up 1 unit from the y - intercept.
The denominator tells us to run 2 units to the left
Determining the Gradient using the Gradient-Intercept Method

17. Horizontal and Vertical Lines
a) Sketch the graph of y = 2
In this relation, the x-values can vary but the
y - values must always remain 2.
Lines which cut the y - axis
and are parallel to the x – axis
have equation y = number x -1 1 2 y
Determining the Gradient of Horizontal Straight Lines
Horizontal Straight Lines

18. Vertical Straight Lines
b) Sketch the graph of x = 2
In this relation, the y-values can vary but the
x - values must always remain 2.
Lines which cut the x - axis
and are parallel to the y – axis
have equation x = number x 2 y -2 -1 1
Vertical Straight Lines

19. Straight Line Graphs Calculator
EXERCISE
1. Draw neat sketch graphs of the following
linear functions on separate axes.
(a) f (x) = 3x- 6 (f) y - 3x = 6
(b) g (x) = - 2x + 2 (g) y = 3x + 2
(c) h (x) = - 4x (h) 2x + 3y + 6 = 0
(d) 5x + 2y = 10 (i) 3x + 3y = 0
(e) y – x = 0 (j) 3x = 2y
Straight Line Graphs Calculator

20. With each of the equations below, do the following:
Write the equation in standard form
Determine the gradient
Determine the y-intercept
(a) 2y - 4x = 0 (c) 6x - 3y = 1
(b) 2y + 4x = 2 (d) x - 2y = 4
Given the following equation: f (x) = - 2x + 1
(a) Sketch the graph
(b) Draw the graph of g(x) if it is f(x) the
has been translated 2 units upwards.

21. Finding the Equations of Straight Line Graphs
Determining the Equation of a Straight Line Graph
Example 1: Determine the equation of the line:
The y-intercept is 3, so c = 3.
y = m x + 3
Substitute the point (8 ; - 1)
- l = m(8) + 3
- 1 = 8m + 3
- 8m = 4
m = -½
Therefore the equation is :
y = - ½ x +3
Calculating the Gradients of Different Staircases
Challenge!Finding the gradient when c = 0

22. EXERCISE (a) (b) (c ) (d)
1. Determine the equations of the following lines:
(a) (b)
(c ) (d)

23. 2. Determine the equation of the line:
a) passing through the point (-1; - 2) and cutting the y-axis at 1.
b) Determine the equation of the line with a gradient of - 2 and passing through the point (2; 3).
c) Determine the equation of the line which cuts the x-axis at 5 and the y - axis at - 5.
d) Determine the equation of the line which cuts the x-axis at – 3 and the y - axis at 9.

24. Parallel Lines Parallel lines have the same gradient (m)
1. Are these lines parallel? y = 2x + 4 and y – 2x – 6 = 0
y = 2x => m = 2
y – 2x – 6 = 0
y = 2x + 6 => m = 2 Yes! They are parallel.
2. Are these lines parallel? 2y = 2x + 4 and – 2x + y – 6 = 0
2y = 2x + 4
y = x + 2 => m = 1
- 2x + y – 6 = 0
y = 2x + 6 => m = 2 No! They are not parallel.
Parallel lines

25. Perpendicular Lines
Perpendicular Lines

26. Intersecting Straight Line Graphs
Point of Intersection:
The point at which both straight line graphs have the same value for x and for y
Can determine the point of intersection
a) graphically
b) algebraically (solve equations simultaneously)
Example:
Solve 3x-y=4 and 2x-y=5
a) Graphically
Point of intersection

27. Points of Intersection Calculator
Example: Solve 3x-y=4 and 2x-y-5
b) Algebraically
3x – y = 4 … (1)
-y = 4 – 3x
y = 3x – 4 … (3)
2x-y=5 … (2)
Substitute (3) into (2)
2x – (3x-4) = 5
2x – 3x + 4 = 5
-x = 1
x = -1
Substitute x = -1 into (3)
y=3(-1)-4
y =-7
Point of intersection: (-1;-7)
Points of Intersection Calculator

28. EXERCISE
1. Draw neat sketch graphs of the following lines on the same set of axes:
x + y = 3 and x - y = -1.
2. Solve x + y = 3 and x - y = - 1 using the method of simultaneous equations.
3. Determine the coordinates of the point of intersection of the following pairs of lines: x + 2y = 5 and x- y = - 1

29. Parabolas Quadratic Equations have the formula: y = ax² + q
Investigation 1
Complete the following table and then draw the
graphs of each function on the set of axes provided
below. x -2 -1 1 2 x²
2x²
3x²

30. Investigating the effects of a of a Parabola
x² 2x² 3x²
Investigating the effects of a of a Parabola

31. Investigation 2
Complete the following table and then draw the graphs of
each function on the set of axes provided below. x -2 -1 1 2
x² - 2 x²
x² + 2

32. Investigating the effects of q of a Parabola
x² x²+2 x²-2
Investigating the effects of q of a Parabola

33. Example Given: (a) Write down the coordinates of the y - intercept.
(b) Determine the coordinates of the x - intercepts.
0 = (x + 2)(x - 2)
x + 2 = 0 or x – 2 =0
x= - 2 or x = 2
x-intercepts : (- 2; 0) and (2; 0)
(c) Sketch the graph.

35. Given:
(d) Determine the turning point of the graph.
Turning point is (0; - 4)
(e) Determine the minimum value of the graph.
Minimum value = -4.
(f) Determine the domain of the graph.
(g) Determine the range of the graph.

36. Reflecting Straight Lines & Parabolas
Reflecting Straight Lines & Parabolas

37. THE HYPERBOLA Standard form of a hyperbola graph: x -6 -4 -2 2 4 6 y
Investigation 1: e.g.
Complete the table using substitution: x -6 -4 -2 2 4 6 y
- 1/ ½ undefined ½ /3

39. x -6 -4 -2 2 4 6 y Investigation 2: e.g.
Complete the table using substitution: x -6 -4 -2 2 4 6 y
2, , undefined , ,33

40. Hyperbolas

41. EXERCISE
Sketch the following, using the table method: 1) 2)

42. Standard form of a hyperbola graph:
a > 0: Quadrants 1 & a < 0: Quadrants 2 & 4
q > 0: Graph shifted up
q < 0: Graph shifted down
NOTE! Asymptote on y-axis (x = 0)

43. EXPONENTIALS Standard form of an exponential graph: y = a x + q x -3
Investigation 1: e.g. y = 3 x
Complete the table using substitution: x -3 -2 -1 1 2 3 y
0, , ,

44. y = 3x

45. y = 0 is the equation of the asymptote
y = 3x
Write down the co-ordinates of the x intercept.
X intercept :
This graph does not cut the x – axis.
y = 30 = 1
ie y
y = 0 is the equation of the asymptote

46. y = 3x
y = 0

47. Sketch the graph of: 1) y = 2 -x 2 ) y = ½ x
What do you notice?
It’s the same graph!
Why?
Because of our Exponent Laws!

48. Sketch the graph of y = 3x - 1
What do you think it will look like ?
How will it differ from y = 3x ?
What is the equation of its asymptote ?

49. Investigating the Exponential Graph
y = 3x - 1
y = -1
Investigating the Exponential Graph

50. Standard form of an exponential graph: y = a x + q
a is a whole number: e.g. y = 2 x
(increasing function)
a is a fraction: e.g. y = ½ x or y = 2 -x
(increasing function)
q > 0: Graph shifted up
q < 0: Graph shifted down
NOTE! Asymptote on x-axis (y = 0)

51. Example 3 PLEASE ANIMATE
Find the value of a and q
in the equation of the
exponential graph:
y = a.2x+1+q
q = -3
Subst. (-1;0)
0 = a
0 = a.1 – 3
3 = a

52. Example 2 PLEASE ANIMATE
Find the value of b and q in the equation of the
exponential graph: y = -3.bx+1+q
q = 1
Subst. (-2;0)
0 = -3.b
0 = -3.b-1 + 1

53. SINE GRAPHS The trig function “Sine” can be sketched Example 1:
a) Complete the table of y = sinx for x [0°; 360°] 0°
30°
45°
60°
90°
120°
135°
150°
180°
y = sin x
210°
225°
240°
270°
300°
315°
330°
360°
y = sin x

54. Link between sinx as a ratio and as a graph

55. b) Determine the following based on the graph:
The maximum value is…. 1
The minimum value is … -1
Amplitude
= (maximum - minimum value) divided by 2
= (1-(-1)) = =1
c) Determine:
The domain of the graph is …
The range of the graph is …

56. Example 2:
a) Draw the graph of y = sin x for x [-360°; 360°]
b) After how many degrees does the graph (pattern) start repeating itself? … 360°
This is called the period of the graph.

57. Exercise Graph Amplitude Maximum Minimum Period
y = sin x
y = 2 sin x
y =3 sin x
y = - sin x
y = - 2 sin x
The effects of a in y = asinx
Amplitude shifts of y = asinx

58. Vertical shifts of y = sinx + q
Exercise
Graph
Amplitude
Maximum
Minimum
Period
y = sin x
y = sin x + 1
y = sin x - 2
Vertical shifts of y = sinx + q

59. COS GRAPHS The trig function “Cosine” can be sketched Example 1:
a) Complete the table of y = cosx for x [0°; 360°] 0°
30°
45°
60°
90°
120°
135°
150°
180°
y = cos x
210°
225°
240°
270°
300°
315°
330°
360°
y = cos x

60. Link between cosx as a ratio and as a graph

61. b) Determine the following based on the graph:
The maximum value is…. 1
The minimum value is … -1
Amplitude
= (maximum - minimum value) divided by 2
= (1-(-1)) = =1
c) Determine:
The domain of the graph is …
The range of the graph is …

62. Example a) Draw the graph of y = cosx for x [-360°; 360°]
b) After how many degrees does the graph (pattern) start repeating itself? … 360°
This is called the period of the graph.

63. Exercise Graph Amplitude Maximum Minimum Period y = cos x y= 2 cos x
y = - 2 scos x

64. Exercise Graph Amplitude Maximum Minimum Period y = cos x

65. TAN GRAPHS The trig function “Tangent” can be sketched Example 1:
a) Complete the table of y = tan x for x [0°; 360°]
x is undefined at 90◦ and 270◦
Therefore, the tan graph has ASYMPTOTES at
x = 90◦ and
x = 270◦ 0°
45°
90°
135°
180°
y = tan x
225°
270°
315°
360°
y = tan x

67. b) Determine the following based on the graph:
The maximum value is…. There is none!
The minimum value is … There is none!
Amplitude - There is none!
However: y = atanx … a is the “amplitude”
c) Determine:
The domain of the graph is …
The range of the graph is …

68. Example a) Draw the graph of y = tan x for x [-360°; 360°]
b) After how many degrees does the graph (pattern) start repeating itself? … 180°
This is called the period of the graph.

69. Exercise Graph Amplitude Maximum Minimum Period y = tan x y= 2 tan x

70. Exercise Graph Amplitude Maximum Minimum Period y = tan x