We all know that the all trigonometric functions are periodic functions i.e. the values of the functions repeat at regular intervals.
Period: The range of values of x for which the function repeats itself.
Amplitude: The range of the values of the functions divided by 2 i.e. (Max – Min)/2
Sine Function
Below is a graph of the sine function. It is a periodic function with the functions repeating every 2π radians.
As can be seen, the values of sine vary between -1 and +1
Period of the function = 2π
Amplitude = (Max – Min)/2 = (1-(-1))/2 = 1
Sine Function Transformations:
Adding a Constant to the Sine Function:
Graph the Function f(θ) = sin(θ)+1. What happens to the period and the amplitude and how is the function transformed?
Below is a graph showing the comparison of sinθ and sinθ+1
As can be seen from the graph, the graph of sinθ+1 shifts upwards by 1 unit.
Period: Adding a constant to the sine function does not change the period of the function
Amplitude: Max = 2 Min = 0. So Amplitude = (2-0)/2 = 1. Adding a constant to the sine function does not change the amplitude of the function
Transformation: Adding a constant(i.e. 1 in this case ) shifts the functions vertically. The vertical shift is equal to the value of the constant
Exercise: Graph the Function f(θ) = sin(θ)+1. What happens to the period and the amplitude and how is the function transformed?
Multiple the Sine Function by a Positive Constant:
Graph the Function f(θ) = 2sin(θ). What happens to the period and the amplitude and how is the function transformed?
Below is a graph showing the comparison of sinθ and 2sinθ
As can be seen from the graph, the function 2sinθ varies between -2 and +2.
Period: The period of the new function 2sinθ does not change and is still 2π
Amplitude: The amplitude of the function changes.
Max = 2, Min = -2
Amplitude = (Max-Min)/2 = (2-(-2))/2 = 2. So amplitude increases from 1 to 2 (i.e. by a factor of the constant)
Transformation: Multiplying the sine function by a positive constant results in the value of the function changing by the value of the constant.
Exercise: What happens to the sine function when you multiple it by a negative constant?
Adding a Constant to the Input of the Sine Function:
Graph the Function f(θ) = sin(θ+π/4). What happens to the period and the amplitude and how is the function transformed?
Below is a graph showing the comparison of sinθ and sin(θ+π/4)
As can be seen, adding a constant π/4 to the input of the sine functions shifts the function to the left by π/4.
Period: The period of the new function sin(θ+π/4) is still 2π. The function shifts to the left by π/4 units
Amplitude: The amplitude of the function does not change and is still 1
Transformation: Adding a constant to the input of the sine function results in the function shifting to the left by the constant
Exercise: What happens when you subtract a constant from the input to the sine function?
Multiple the Input to the Sine Function by a Positive Constant Greater than 1:
Graph the Function f(θ) = sin(2θ). What happens to the period and the amplitude and how is the function transformed?
Below is a graph showing the comparison of sinθ and sin2θ
As can be seen, multiplying the input to the sine function by a positive constant “greater then 1” causes the function to compress horizontally. In the above example, multiplying in the input by 2 causes the function to compress by a factor of 2
Period: The period of the function is π (i.e. period is reduced by a factor of 2). As can be observed from the graph above, there are two complete periods of the function sin(2θ) between θ = 0 and θ = 2π.
Amplitude: The amplitude of the function does not change
Transformation: Multiplying the input to the sine function by a positive constant greater then 1 causes the function to compress horizontally. The period of the function is reduced by the value of the constant
Exercise: What happens when the input to the function is multiplied by a positive constant < 1?
What happens when the input to the function is multiplied by a negative constant?
All Things Mathematics 
Posted on July 31, 2014
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