## AN EXAMPLE OF AUTOMATED EXPLORATIONS IN MATHEMATICS

### EXPLORATIONS IN MATHEMATICS

by Paul W. Godfrey
for Dr Jim Wilson
EMT 668, Fall 1995

Algebra Xpresser was used to explore, understand and extend the following function..

y = a sin( bx + c)
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The parameters a, b, and c of the above equation are first described in terms of the cyclical sine function graphed below. Then, the function is related to something commonly familiar in everyday life. The parameters a, b, and c are varied and the results graphed. These graphs along with discussion of their results, is presented below.

First, lets see the graph of this equation with a and b equal to 1, and c equal to 0. This is the basic sinusoidal function of trigonometry.
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As we see from the graph, this function varies smoothly above and below the y axis. It goes from a maximum y value of 1 to a minimum y value of -1. We can also note that when x has value 0, so does y. We see that this curve repeats itself, again and again regardless of the direction we go on the x axis. It appears that the complete curve is a repetition of that section between the -3 and 3 values of the x axis. The shape of the graph itself resembles a wave in the ocean, and, in fact, that is the name we give to this particular graph. It is called a sine wave.Next Page
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We can give names to the various sinusoidal properties of the sine wave. The maximum to minimum value of the graph on the y axis is known as the amplitude of the wave. The location that the graph crosses the x axis from a negative y value to a positive y value determines the phase of the wave. The one we see above has phase equal to 0 because the graph passes through the origin as it goes from negative to positive. Finally, the distance from one positive peak of the wave to the next determines the frequency of the wave. This frequency is often measured in the number of cycles the wave completes in a unit of measurement along the x axis. A cycle occurs when the wave has passed through every possible y value it can have exactly once. These properties of the sine wave determine to a great extent how it is used.

Many things in everyday life are represented by a sine wave. Electricity found in the home is one thing that can be so represented. It has a sine wave output, an amplitude of around 120 in units called volts. The current from the wall outlet is called single phase and characteristically its phase is said to be 0. The sine wave operates at frequency of 60 cycles per second. Now, let’s see what happens as we change the a , b, and c parameters. First we change the a parameterNext Page

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The original sine wave is in red. When we reduce a to 1/4, we get the green sine wave. For values of a of 1/2, 2, and 4 we get blue, olive, and purple sine waves respectively. When we make a smaller, we shrink the distance between the peaks of the sine wave. When we increase the value of a the distance between the peaks gets greater. We conclude from this that the amplitude of the sine wave is directly related to a, that is, as agets bigger or smaller the amplitude gets bigger or smaller. View Graph

Next, we vary the b parameter. Next Page

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Once again, the original sine wave is in red. When we reduce b to 1/4, we get the purple sine wave. For values of b of 1/2, 2, and 4 we get olive, blue, and green sine waves respectively. It is clear that when we make b smaller, we shrink the distance between the positive peaks of the sine wave, and when we increase the value of b the distance between the peaks gets greater. We conclude from this that the frequency of the sine wave is directly related to b, that is, as b gets bigger or smaller the frequency gets bigger or smaller.View Graph

Now, we vary the c parameter.Next Page

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The original sine wave is in red. When we reduce c to 1/4, we get the blue sine wave. For values of c of 1/2, 2, and 4 we get green, purple, and olive sine waves respectively. It is clear that when we make c smaller, the point the wave crosses the x axis from negative to positive moves left. When we increase the value of c this point moves to the right. We conclude from this that the phase of the sine wave shifts to the left or right (i.e., negative or positive with respect to the origin) as c gets smaller or larger. So, we see that the phase of the sine wave is directly related to c View Graph
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Now that we've seen the affects the basic parameters have on the wave, let's explore some combinations of these. In particular, lets look at something called the harmonics. The harmonics of a particular sine wave at a certain frequency are sine waves whose frequencies vary from that of the particular sine wave by whole multiples and whose amplitudes are less. We know from above that if sin x represents the particular sine wave, we can change the frequency by whole multiples with the b parameter and we can make the amplitude less by reducing a. We end the discussion with a look at what happens when we combine sine waves and their even and odd harmonics. Next Page

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This is a graph of even harmonics. The basic sine wave is graphed in red as y = sin x. As these are even harmonics, the green line is the equation y = 1/2 sin 2x. Also, the blue is y = 1/4 sin 4x, the olive is y = 1/6 sin 6x, and finally, the purple is y = 1/8 sin 8x.Next Page ```

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When we add the even harmonics together, this is what we get. As you can see, it is a somewhat distorted sine wave. Next Page ```

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The following graph places a sine wave (red) and some of it's odd harmonics on the same graph. Red is the graph of y = sin x, the basic sine wave. The green line graphs y = 1/3 sin 3x, blue y = 1/5 sin 5x, olive y = 1/7 sin 7x; and finally purple graphs y = 1/9 sin 9x.Next Page ```

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When we add these odd functions together, we get the following result. This is very close to something called a square wave. This is what the signals a computer sends to a modem look like.Next Page ```

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There are many more ways one can combine the various sine wave functions. The Algebra Xpresser is an excellent tool to facilitate these explorations. Using it, we have been able to visually relate the sine function to something we run across in our every day experiences. These concepts apply not only to electricity, but to most sinusoidal vibrating things . . . from springs on a car to strings on a guitar. I believe this visualization is a most helpful tool in appreciating these concepts and that this little foray into the sine wave has been worth while. "Return to Home Page".
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