Examining the Sine Function
Let us examine the sine function as the coefficient values change to see the effects these changes have on the various graphs.
Given the graph y = a sin (bx + c) with variables of a, b, and c.
Our first step is to :
Look at the basic sine graph when a=1, b=1, and c=0
Notice that the Domain is the set of real numbers, and the Range is [-1,1]. The Period for the sine function is 2¹. The graph of the sine function continues indefinitely.
The amplitude is the distance from the axis to the highest or lowest point. Or it is half the distance from the highest to the lowest point. In this case the amplitude is 1. Let us use during this example the variable ÔaÕ in demonstrating the amplitude. Currently a = 1.
The period is the time it takes for the graph to make one complete cycle or in other words, the amount of time it takes for the graph to begin repeating. Let us use the variable ÔbÕ in conjunction with the adjusting the period. In this case 2¹.
In our example the sine wave phase is controlled through variable ÔcÕ, initially let c = 0.
Continue to use the basic sine graph as our frame of reference. Let us examine what happens to the graph under the following guidelines.
Step 1: a sin (bx +c)
Let b=1, c=0, and vary the values of a. Our new equation becomes y=a sin(x).
Let us use the graphing calculator to examine the effects of varying the values for ÔaÕ, remembering to use both positive and negative values.
The blue graph is y=sin x. Y = sin (x), the basic sine graph will always be in blue in future examples for comparison purposes.
Notice that when the value for variable ÔaÕ is positive, the amplitude increases by a factor of the absolute value of ÔaÕ, and the graph emulates the y=sin x original graph demonstrated above. This is known as a vertical stretch. Similarly, when the variable ÔaÕ is negative, the amplitude still increases by the same factor, the absolute value of ÔaÕ.
However, the negative value of ÔaÕ causes the graph to be a reflection across the x-axis.
Step 2: Now we are examining the effects of variable ÔbÕ. Let a=1, and c=0 and change the values for variable ÔbÕ. Our new equation is now y = sin (bx).
Notice that the amplitude of the graphs does not change even though the value of variable ÔbÕ was varied. When b>1 the period of the graph is changed to 2¹/b resulting in a horizontal shrinking of the graph. When 0 < b < 1 then the period is still changed to 2¹/b, however, the graph is now stretched.
This leaves the question: What happens when negative values are substituted for variable ÕbÕ?
By substituting negative values for variable ÔbÕ, notice there is a reflection across the x-axis for our two graphs as well as horizontal change of the basic sine graph.
Step 3: Let us again start with equation y= asin(bx+c). Let a=1, b=1, and vary c, resulting in y = sin(x+c)
The value of variable ÕcÕ moves the sine graph to the right or the left. When c > 0, the graph moves to the left. When c < 0, the graph moves to the right.
This horizontal movement is called the phase shift. The phase shift appears to be equal to the value of ÔÐcÕ. To be sure let us check what with a change to variable ÔbÕ simultaneously.
This shows that the phase shift is effected by ÔbÕ. Thus, the phase shift is actually the result of Ô-c/bÕ.
In summary, given the equation y = a sin (bx +c) the following are true: