1. 5. FUNCTIONS Straight Lines Parabolas Hyperbolas Exponential Graphs
Average Gradient
Trig Graphs:
Recap
Amplitude changes
Period changes
Horizontal shifts

2. STRAIGHT LINES
y = m x + c
m is the gradient
c is the y - intercept

3. Example
Find the equation of this straight line graph.
the y intercept is – 3
so c = -3
the gradient is
6 down and 3 to the left so m = −𝟔 −𝟑 = 2
y = 2x – 3

4. Example
Find the equation of this straight line graph.
a = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏
= −𝟖−𝟎 𝟎 −𝟐
= −𝟖 −𝟐 =𝟒
y-intercept = -8
y = 4x – 8

5. Example
Sketch the graph of y = -3x – 3.
c = -3 so the
y - intercept is – 3
x –intercept (y=0)
0 = -3x – 3
3x = -3
x = -1

6. Standard form: y = ax2 + bx + c
PARABOLAS
Standard form: y = ax2 + bx + c
a: a > 0: arms go up (smile)
a < 0: arms go down (frown)
b: b > 0: graph shifts to the left
b < 0: graph shifts to the right
c: c > 0: positive y-intercept
c < 0: negative y-intercept
Investigating the effects of b in a Parabola
Finding the roots and vertex of a parabola
Effects of a, b & c of Standard form of a Parabola

7. Example
Sketch the graph of y = 2x² + 5x + 2
y-intercept: read off std. form
c = 2
x-intercepts: y = 0
y = 2x² + 5x + 2
0 = (2x + 1) (x + 2)
x = - ½ or x = - 2
Turning Point: Use formula …

8. … Sketch: y = 2x² + 5x + 2 …
(Turning-point formula)

9. Example
Find the equation of the parabola.
(Given x-intercepts and 1 other point)
y = a(x – root1)(x – root2)
y = a(x – (-3))(x – 8)
y = a(x + 3)(x – 8)
Subst. pt: y-int (0;-24)
y = a(x + 3)(x – 8)
-24 = a(0 + 3)(0 – 8)
-24 = -24a
1 = a

10. Finding the Equation of a Parabola
… Find the equation …
Found …
y = a(x + 3)(x – 8)
a = 1
Equation in Std. Form:
y = a(x + 3)(x – 8)
y = 1(x + 3)(x – 8)
y = x² - 5x - 24
Finding the Equation of a Parabola

11. Effects of p of the Turning Point form of a Parabola
PARABOLAS
Turning - point form: y = a(x - p)2 + q
a: a > 0: arms go up (smile)
a < 0: arms go down (frown)
(p;q) is the co-ordinate of the Turning Point
p: p > 0: graph shifts to the left
p < 0: graph shifts to the right
q: q > 0: graph shifts up
q < 0: graph shifts down
Effects of p of the Turning Point form of a Parabola

12. Example
Sketch the graph of y = 2(x – 1)2 – 18
y-intercept: x = 0
y = 2(0 – 1)2 – 18
y = - 8
TP (p; q)
TP [-(-1); -18]
TP (1;-18)
x-intercepts: y = 0
y = 2(x – 1)2 – 18
0 = 2(x² - 2x + 1) – 18
0 = 2x² - 4x + 2 – 18
0 = 2x² - 4x – 16
0 = x² - 2x – 8
0 = (x - 4)(x + 2)
x = 4 or x = - 2

13. Example
Find the equation of the parabola.
(Given the turning-point and 1 other point)
TP (p;q) ∴ TP [-(1);-2]
y = a(x - p)2 + q
y = a(x + 1)² - 2
Subst. pt: y-int (0;-3)
y = a(x + 1)² - 2
-3 = a(0 + 1)² - 2
-3 = a - 2
-1 = a

14. Finding the Turning Point Formula of a Parabola
… Find the equation …
Found …
y = a(x + 1)² - 2
a = -1
Equation in Std. form:
y = a(x + 1)² - 2
y = -1(x + 1)² - 2
y = -x² - 2x - 3
Finding the Turning Point Formula of a Parabola

15. REFLECTING PARABOLAS Example: y = x2 + 8x – 2
* Reflect in the x-axis: every y swops signs
– y = x2 + 8x – 2
y = – x2 – 8x + 2
* Reflect in the y-axis: every x swops signs
y = (– x)2 + 8(– x) – 2
y = x2 – 8x - 2
* Reflect in the line y = x: swop x and y
x = y2 + 8y – 2
Reflecting Parabolas
Reflecting Lines

16. HYPERBOLAS
RECAP! 𝒚= 𝒂 𝒙 +𝒒
Sketch the following graphs and write down the equation of the asymptotes: 1) 2)
What are the equations of the asymptotes for …
Vertical asymptote: x = 0
Horizontal asymptote: y = -1
2) Vertical asymptote: x = 0
Horizontal asymptote: y = 4

17. Standard form of a Hyperbola:
𝒚= 𝒂 𝒙−𝒑 +𝒒
a determines the quadrants
a > 0 => Q 1 & 3
a < 0 => Q 2 & 4
q determines the horizontal asymptote
i.e. vertical translation OR up/down shifts
q > 0 => graph shifted up
q < 0 => graph shifted down

18. Standard form of a Hyperbola:
𝒚= 𝒂 𝒙−𝒑 +𝒒
p determines the vertical asymptote
i.e. horizontal translation OR left/right shifts
p > 0 => graph shifted left
p < 0 => graph shifted right

19. Example: Sketch the graph of:
a (quadrants):
Q 1 & 3
q (horizontal asymptote):
y = 4
p (vertical asymptote):
p<0 so graph shifted to the right
x = 3
Now, what about the x- and y-intercepts?

20. y – intercept (x=0):
x – intercept (y=0):

21. Sketching Hyperbolas

22. Example: Sketch the graph of:
a (quadrants):
Q 2 & 4
q (horizontal asymptote):
y = -4
p (vertical asymptote):
p>0 so graph shifted to the left
x = -3
Now, what about the x- and y-intercepts?

23. y – intercept (x=0):
x – intercept (y=0):

25. Example: Find the equation of the following graph:

26. 𝒚= 𝒂 𝒙−𝒑 +𝒒 Substitute in the asymptotes …
Substitute a point that lies on the graph …
Subst: (6;0)
The Hyperbola

27. EXPONENTIAL GRAPHS
RECAP! y = a.bx + q
Sketch the following graphs and write down the equation of the asymptote:
1) y = 5 x ) y = 5 –x ) y = ½ x - 2
What is the equation of the asymptote for …
1) y = 0 (i.e. x-axis)
2) y = 2
3) y = -2
Cell Division

28. Example: Sketch the graph of: y = - (3)x
How will it differ from y = 3x ?
What are the intercepts?
What is the equation of the asymptote ?

29. y = 3x
y = - (3)x

30. Standard form of an Exponential:
y = a.b x − p + q
q is the horizontal asymptote
i.e. represents a vertical shift (up/down shift)
q > 0 => graph shifted up
q < 0 => graph shifted down
p represents a horizontal shift (left/right shift)
p > 0 => graph shifted left
p < 0 => graph shifted right

31. Standard form of an Exponential:
y = a.b x − p + q
b determines the shape of the graph
b > 0 => increasing function
0 < b < 0 (a fraction) => decreasing function
a determines where the graph lies
a > 0 => graph lies above the x-axis
a < 0 => graph lies below the x-axis
Investigating the Exponential Graph

32. Example: Sketch the graph of: y = 5.5 x + 1
Asymptote:
y = 1
(no x-intercept)
y-intercept:
y = =
= 6

33. Example: Sketch the graph of: y = -3.3-x - 3
Asymptote:
y = - 3
(no x-intercept)
y-intercept:
y = – 3 =
= 6

34. Example: Sketch the graph of: y = 4.2x+1 - 2
Asymptote:
y = - 2
(no y-intercept)
x-intercept:
0 = 4.2x = 4.2x+1
½ = 2x+1
2-1 = 2x+1
-1 = x + 1
x = -2

35. Example: Find the equation of the graph if y = a.b x + q :
Subst. in asymptote:
y = a.b x + 1
Subst. y-intercept:
3 = a.b 0 + 1
2 = a
Subst. in pt (1;5):
5 = 2.b 1 + 1
4 = 2b
b = 2
y = 2.2x + 1

36. Example: Determine the values of a and q, given
Summary of Exponential Transformations
Example: Determine the values of a and q, given
y = a.3 x+1 + q :
Subst. in asymptote:
y = a.3 x+1 + 1
q = 1
Subst. x-intercept:
0 = a.3 −
-1 = a. 3 −1
a = -3
Finding the Equations of Exponential Graphs

37. Exponential Function in Life
Exponential growth of a bacterial culture
Exponential Growth
Exponential decay of radioactive material
Exponential Decay

38. AVERAGE GRADIENT B(x1;y 1) 𝑮𝒓𝒂𝒅𝒊𝒆𝒏𝒕= 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 A(x2;y2)
𝑮𝒓𝒂𝒅𝒊𝒆𝒏𝒕= 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙
A(x2;y2)
y2 – y1
x2 – x1
B(x1;y 1)

39. Calculating the Gradient
Example
Calculate the gradient of AB.
m(DE) = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1
= 8−(−1) 5−(−4)
= 9 9 =1
D(5;8)
E(-4;-1)
Calculating the Gradient

40. Example:
Find the average gradient between x = 1 and x = 2, given y = x2.

41. When x = 1, y = 1 therefore f(1) = 1.

42. m
f(x) = x2
f(2)=4
f(1)=1
= 3

43. Average grad between x = 1 and x = 2 is 3
NB The gradient of a straight line is a constant

44. Average Gradient to Gradient at a Point
In general …
f(x2)
f(x1)
Grad = x1 x2
Average Gradient to Gradient at a Point

45. 𝑦 = 𝑠𝑖𝑛𝑥 𝑦 = 𝑐𝑜𝑠𝑥 𝑦 = 𝑡𝑎𝑛𝑥 TRIG GRAPHS: RECAP
Let’s investigate this change by plotting the graph of y = sin 2x …

46. y = sin x Domain: x ϵ [00 ; 3600] Range: - 1 ≤ y ≤ 1 Period: 3600
Amplitude: 1
The Sin Graph

47. y = cos x Domain: 00 ≤ x ≤ 3600 Range: y ϵ [- 1; 1] Period: 3600
Amplitude: 1
The Cos Graph

48. y = tan x

49. The graph of y = tan x does not go through x = 900 or x = 2700
Take note!
The graph of y = tan x does not go through x = 900 or x = 2700
These lines i.e. x = 900 and x = 2700
are called asymptotes.
y = tan x is undefined at these points.
The Tan Graph
Trig Graphs and the Unit Circle

50. Sketches of Trig Graphs between ±1080º
Summary of Trig Graphs
The key characteristics of each trig graph:
y = sinx …. starts at (0º; 0); “wave” curve
y = cosx …. starts at (0º; 1); “bell” curve
y = tanx …. starts at (0º; 0); “escalator” curve
asymptotes at 90º and 270º
Sketches of Trig Graphs between ±1080º

51. TRIG GRAPHS: AMPLITUDE CHANGES
𝑦 =𝑎𝑠𝑖𝑛𝑥 𝑦 =𝑎𝑐𝑜𝑠𝑥
𝑦 =𝑎𝑡𝑎𝑛𝑥
Let’s investigate this change by plotting the graph of y = 2 sinx …

52. Use your calculator to determine y …x 30 60 90
120
150
180
y = sin x
0.5  
0.87 1
0.5 
y = 2 sin x
210
240
270
300
330
360
-0.5 
-0.87 -1
 -0.5

53. Now plot the graph of y = 2 sinx for x ϵ [ 00; 3600 ]
Values of y = 2 sinx are … x 30 60 90
120
150
180
y = sin x
0.5
0.87 1
0.5 
y = 2 sin x  1
1.73 2 1 
210
240
270
300
330
360
-0.5 
-0.87 -1
-1 
-1.73 -2
Now plot the graph of y = 2 sinx for x ϵ [ 00; 3600 ]

54. Amplitude changes to y=asinx
y = 2 sinx
Domain: ≤ x ≤ 3600
Range: y ϵ [- 2; 2]
Period:
Amplitude: 2
Amplitude changes to y=asinx

55. Sketch the graph of y = 2 cosx and write down
Example:
Sketch the graph of y = 2 cosx and write down
the amplitude, period, domain and range.
Amplitude: 2
Period:
3600
Domain:
00 ≤ x ≤ 3600
Range:
y ϵ [- 2; 2]
y = cos x
y = 2 cos x

56. Summary: Amplitude changes in trig graphs
Example:
Sketch the graph of y = ½ tanx and write down
the amplitude, period, domain and range.
Summary: Amplitude changes in trig graphs
Amplitude:
Period:
1800
Domain:
00 ≤ x ≤ 3600
Range:
y ϵ R
y = tan x
y = ½ tan x

57. TRIG GRAPHS: PERIOD CHANGES
𝑦 = 𝑠𝑖𝑛𝑘𝑥 𝑦 = 𝑐𝑜𝑠𝑘𝑥
𝑦 = 𝑡𝑎𝑛𝑘𝑥
Let’s investigate this change by plotting the graph of y = sin 2x …

58. Use your calculator to determine y …x 00
150
300
450
600
750
900
2 x
 300
y = sin 2x
0.5
1050
1200
1350
1500
1650
1800

59. Now plot the graph of y = sin 2x for x ϵ [ 00; 3600 ]
Values of y = sin 2x are … x 00
150
300
450
600
750
900
2 x
 300
1200
1500 
1800
y = sin 2x
0.5
0.87 1
1050
1200
1350
1500
1650
1800
2100 
2400
2700
3000
3300
3600
-0.5 
-0.87 -1
Now plot the graph of y = sin 2x for x ϵ [ 00; 3600 ]

60. Graph of y = sin 2x
Amplitude: 1 Domain: ≤ x ≤ 3600
Period: Range: y ϵ [- 1; 1]

61. Let’s compare … y = sin x y = sin 2x
y = sin 2x means there are 2 sin graphs with 3600

62. Sketch the graph of y = sin 4x
Example:
Sketch the graph of y = sin 4x
y = sin x
y = sin 4x
What is its period?
90◦
Investigating the effects of k in y = sinkx

63. Example:
Use the table below to sketch the graphs of: x 00
900
1800
2700
3600
y = cos 2x
y = cos 3x
y = cos x y = cos 2x y = cos 3x

64. Summary: Period changes in trig graphs
Example:
Sketch the graph of y = tan 2x for x ϵ [ 0 ; 3600 ]
and state the period of the graph.
Period = 90◦
Summary: Period changes in trig graphs

65. TRIG GRAPHS: VERTICAL CHANGES
𝑦 = sin 𝑥 +𝑞 𝑦 = cos 𝑥 +𝑞
𝑦 = tan𝑥+𝑞
Recap: Vertical translations

66. TRIG GRAPHS: HORIZONTAL CHANGES
𝑦 = sin⁡(𝑥+𝑝) 𝑦 = cos⁡(𝑥+𝑝)
𝑦 = tan⁡(𝑥+𝑝)
Let’s investigate this change by plotting the graph of y = sin (x + 30◦) …

67. Use your calculator to determine y …x 00
300
600
900
1200
1500
1800
x + 300
y=sin(x+30)
0.5
2100
2400
2700
3000
3300
3600
-0.87

68. Values of y = sin (x + 30◦) are …00
300
600
900
1200
1500
1800
x + 300
 600
2100
y=sin(x+30)
0.5
0.87 1
-0.5
2100
2400
2700
3000
3300
3600
3900
-0.87 -1
-0.5
0.5
Now plot the graph of y = sin (x+30◦) for x ϵ [-300; 3600 ]

69. Amplitude: 1 Domain: 00 ≤ x ≤ 3600 Period: 3600 Range: y ϵ [- 1; 1]
Graph of y = sin(x + 30◦)
y=sin(x+300)
y = sin x
Amplitude: 1 Domain: ≤ x ≤ 3600
Period: Range: y ϵ [- 1; 1]

70. Shifting trig graphs horizontally
Example:
Sketch y = sin(x – 300 ) for x ϵ [ 0 ; 4000 ]
Graph of y = a sin b (x – p)
Shifting trig graphs horizontally

71. Trig graphs: Period changes and Horizontal Shifts
Example:
Sketch y = cos(x – 300 ) for x ϵ [ 0 ; 4000 ]
y = cos x
y = cos (x-30◦)
Trig graphs: Period changes and Horizontal Shifts

72. Example:
Sketch y = tan (x + 100) for x ϵ [ 0 ; 3600 ]

73. Summary of Sketching Trig Graphs: y = a sin/cos/tan (kx + p) + q
a = amplitude changes
k = period changes
p = horizontal shifts
q = vertical shifts
Graph of y = asinkx + q
Graph of y = acoskx + q
Revision: Sin & Cos Graphs

74. Determining the Equation of Trig Graphs y = a sin/cos/tan (kx + p) + q
Determine the “resting position of the graph” => this allows us to determine a i.e. the height amplitude of the graph
Determine q => how far up/down the
graph has shifted i.e. the vertical shift

75. 4. Determine k => how many complete
Determine p => by how many degrees has the graph shifted left or right from it’s usual starting position i.e. the horizontal shift of the graph
4. Determine k => how many complete
graphs within the “normal” 360º / 180º
i.e. the period of the graph
Finding the Equation of a y=sinx Graph
Match the Sin or Cos Graph with its Equation