Examining the Sine Function
By Cara Haskins, Robin Kirkham, and Matt Tumlin
We will examine the effect that changing the coefficient values of the sine function have on the graph.
Examine graphs of y = a sin(bx + c) for different values of a, b, and c.
Our first step will be 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 2p. 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 b in conjunction with adjusting the period. In this case 2p.
In our example, the sine wave phase is controlled through variable ÔcÕ in this first case c=0.
Continue 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 using different values for a, remembering to use positive and negative values.
The blue graph is 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=sinx original graph as demonstrated above. This is known as a vertical stretch. Similarly, when ÔaÕ is negative, the amplitude is still increased by a factor of 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 b. Our new equation is now y = sin (bx).
Notice that the amplitude of the graphs does not change even though the value for b was varied. When ÔbÕ >1, the period of the graph is changed to 2p/b, resulting in a horizontal shrinking of the graph. When 0 < b < 1, then the period is still changed to 2p/b however the graph is now stretched.
This leaves us with the question, what happens when negative values are substituted for ÔbÕ?
By substituting negative values for 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 ÒcÓ 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 Ò-cÓ. To be sure, let us check that with a change to variable ÒbÓ simultaneously.
This shows that the phase shift is effected by ÔbÕ. Thus, the phase shift is actually Ô-c/bÕ.
In summary, given the equation y = a sin (bx +c) the following are true:
Changes in the value of ÔaÕ effects the altitude of the sine graph.
Changes in the value of ÔbÕ effects the period of the graph.
Changes in the value of ÔcÕ effects in conjunction with ÔbÕ together effect the phase shift of the graph.