1. 5.1 Graphing Sine and Cosine Functions
When Function values repeat at regular intervals the function could be referred to as a cyclic function or periodic function.
You can model these types of behaviour with sine or cosine functions.
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2. Math 30-1

3. Functions that repeat themselves over a particular interval
Periodic Functions
Functions that repeat themselves over a particular interval
of their domain are periodic functions. The length of the interval of repeat is called the period of the function.
One cycle
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4. Characteristics of a Periodic Function Graph y = sin x , radians
The amplitude of a periodic function is one half the distance
between the maximum and minimum values. 1
Period: 2p
Domain: all real numbers
Range: {y| -1 ≤ y ≤ 1}
Amplitude: 1
y-intercept: 0
zeros : 0, ±, ±2 , ...
General expression for zeros
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5. Graphing a Periodic Function Graph y = cos x, radians1
Period: 2p
Domain: all real numbers
Range: {y| -1 ≤ y ≤ 1}
Amplitude: 1
y-intercept: 1
zeros:
General expression
Math 30-1

6. y = af(b(x - c)) + d y = asin[b(x - c)] + d
Parameters that affect the graphs of sine and cosine.
Vertical stretch factor |a|
5.1 Transformations abcd
y = af(b(x - c)) + d
Horizontal and vertical translation
y = asin[b(x - c)] + d
amplitude
|a|
phase shift
displacement
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7. Effect of parameter a in y = a sin x
Graph y = 2sin x and y = 0.5sin x.
y = 2sin x
y = sin x
y = sin x
y = 0.5sin x
Math 30-1

8. Comparing the Graphs of y = a sin x
y = sin x
y = 2sin x
y = 0.5sin x
2 
2 
2 
Period
Amplitude
Domain
Range
Zeros 1 2
0.5
All real numbers
All real numbers
All real numbers
{-1 ≤ y ≤ 1}
{-2 ≤ y ≤ 2}
{-0.5 ≤ y ≤ 0.5}
The amplitude of the graph of y = a sin x is | a |.
When a > 0, there is a vertical stretch by a factor of |a|.
When a < 0, there is a vertical stretch by a factor of |a| and a reflection in the x-axis.
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9. Determining the Period for y = sin bx, b > 0
Graph y = sin 2x
y = sin x
y = sin 2x
y = sin x
transformed graph
Length of period of parent
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10. Determining the Period for y = sin bx, b > 0
Graph
y = sin x
y = sin x
transformed graph
Length of period of parent
Math 30-1

11. Comparing the Graphs of y = sin bx
y = sin x
y = sin 2 x
y = sin 0.5 x
2  
4 
Period
Amplitude
Domain
Range
Zeros 1 1 1
All real numbers
All real numbers
All real numbers
{-1 ≤ y ≤ 1}
{-1 ≤ y ≤ 1}
{-1 ≤ y ≤ 1}
The period for y = sin bx is
When b > 0, there is a horizontal stretch by a factor of 1/|b|.
When b < 0, there is a reflection and a horizontal stretch of 1/|b|.
Math 30-1

12. Determining the Period and Amplitude of y = a sin bx
Given the function y = 3sin 4x, determine the period
and the amplitude.
The period of the function is .
Therefore, the period is .
The amplitude of the function is | a |.
Therefore, the amplitude is 3.
y = 3sin 4x
Math 30-1

13. Determining the Period and Amplitude of y = a sin bx
Determine the characteristics of y = -3sin 3x.
The period is
The amplitude is 3.
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14. Writing the Equation of the Periodic Function
Amplitude
Period 
= 2
b = 2
Therefore, the equation as a function of sine is
y = 2sin 2x.
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15. Writing the Equation of the Periodic Function
Amplitude
Period
4 
= 3
b = 0.5
Therefore, the equation as a function of cosine is
y = 3cos 0.5x.
Math 30-1

16. Assignment Suggested Questions: Pages 233 1, 2, 4, 5, 6, 8, 10, 11b, c
14, 17a, 20, C4
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