Robin Kirkham, Cara Haskins , and Matt Tumlin

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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 2π. 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 ‘a’ 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 ‘b’ in conjunction with the adjusting the period. In this case 2π.

In our example the sine wave phase is controlled through variable ‘c’, 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 ‘a’, 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 ‘a’ is positive, the amplitude increases by a factor of the absolute value of ‘a’, and the graph emulates the y=sin x original graph demonstrated above. This is known as a vertical stretch. Similarly, when the variable ‘a’ is negative, the amplitude still increases by the same factor, the absolute value of ‘a’.

However, the negative value of ‘a’ causes the graph to be a reflection across the x-axis.

Step 2: Now we are examining the effects of variable ‘b’. Let a=1, and c=0 and change the values for variable ‘b’. Our new equation is now y = sin (bx).

Notice that the amplitude of the graphs does not change even though the value of variable ‘b’ was varied. When b>1 the period of the graph is changed to 2π/b resulting in a horizontal shrinking of the graph. When 0 < b < 1 then the period is still changed to 2π/b, however, the graph is now stretched.

This leaves the question: What happens when negative values are substituted for variable ’b’?

By substituting negative values for variable ‘b’, 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 ’c’ 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 ‘–c’. To be sure let us check what with a change to variable ‘b’ simultaneously.

This shows that the phase shift is effected by ‘b’. Thus, the phase shift is actually the result of ‘-c/b’.

In summary, given the equation y = a sin (bx +c) the following are true:

Changes in the value of variable ‘a’ effects the altitude of the sine graph.

Changes in the value of variable ‘b’ effects the period of the graph.

Changes in the value of variable ‘c’ in conjunction with the value of variable ‘b’ together effects the phase shift of the graph.

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