Many equation that cannot be easily graphed in Cartesian coordinates using x and y as variables can be easily graphed using parametric equations. Typical of the type of equation we are talking about is the following:
Which has the graph shown below for 0 < t < 4PI.
What is interesting is the affect of varying the value of the numerator and denominator of the argument of 4 sin ((a/b)t).
First, let's look at some basic graphs and see what happens. By basic graphs, I mean that we will vary the denominator while leaving the numerator equal to 1 and see what happens to the shape of the graphs. When the denominator is 2, as shown above, the graph looks like a bow-tie that is 4 units wide by 3 units high. The height and width should not have been a surprise since 4 is the amplitude of the x function and 3 is the amplitude of the y function in our parametric equations.
Changing the denominator to 3 has the effect shown below.
The graph is similar to the previous graph when the denominator what 2 except that we no longer have a closed circuit. What exactly do I mean by similar? Well, the graph intercepts the lines y = 3 and y = -3 twice, and the lines x = 4 and x = -4 once. Let's call these lines the y-boundaries and the x-boudaries respectively.The number of interceptions with these particular lines turns out to be a point of interest as we continue to increase the value of the denominator. Also of interest, is whether or not the graph is a closed circuit.
Let's try a denominator of 4:
Now we again have a closed loop. The graph intercepts each y-boundary four times and each x-boundary once. By the way, we had to increase the range of t to 0 < t < 8PI to completely draw this graph. This is 2PI times the value of the denominator. Is this always true?
Let's try a denominator of 5 and see.
The graph above was drawn using the same range as the previous graph. Increasing the range of t will not add to the graph. Can we conclude anything from this? Maybe when we go from and even number to and odd number for the denominator without increasing the numerator, we don't have to increase the range of t. We'll check that the next time when we go from 6 to 7. The number of intercepts with each y-boundary is now 3, or an increase of one from when the denominator was 3. The number of intercepts with each x-boundary is still one. Is the number of intercepts with the x-boundary determined by the numerator and the number of intercepts with the y-boundary determined by the denominator and whether it is and odd or even integer?
Let's increase the denominator to 6 and see if we can answer any of our questions.
There are 6 intercepts with each y-boundary, which matches the value of the even valued denominator. The number of intercepts with each x-boundary. We should be tempted to make a conjecture about this now, but let's wait. The graph is also a closed curve again. The range of t regquired to complete the graph is 0 < t < 12PI. We should also be almost ready to conjecture about the effect of the value of the denominator on the range of t.
First let's look at the graphs when the denominator has values of 7, 8 and 9.
Denominator = 7:
Denominatro = 8:
Denominator = 9:
At least for these three graphs, what we noticed about the changes from the previous graphs holds true. I conjecture that the number of intercepts with the y-boundaries is determined as follows: If the denominator is even then the number of intercepts is equal to the denominator. If the denominator is odd, then the number of intercepts equals the position of the denominator in the ordered set of odd integers. The range of t was not increased to go from 6 to 7, but was increased to 0 < t < 16PI to go from 7 to 8. So, I conjecture that the range for t needed to complete the graphs is 0 < t < 2PI*denominator for even values of the denominator and 0 < t < 2PI*(denominator -1) for odd values of the denominator, as long as the numerator equals one. Finally, I conjecture that the graph will always be a closed curve when the denominator is odd valued and the numerator is one.
Will these conjectures hold true when we vary the numerator?
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Let's again begin with the denominator as 2 and the numerator as 1.
What we want to observe now are the changes as we vary the numerator. First let's make the numerrator 3. (It doesn't make sense to use a numerator of 2 because we would just get the straight line y = 3/4x. Try it for yourself using Graphing Calculator 2.2.)
Well... We now have 3 x-boundary intercepts and 2 y-boundary intercepts per boundary. Is this always true?
Let's increase the value of the numerator to 4 and see.
Obviously not! We now have two intercepts with each x-boundary and only one intercept with each y-boundary. Is this an odd-even situation again?
Let's increase the value of the numerator to 5 and see if we are led to believe so.
It appears that the answer to the last question is yes. We now have 5 intercepts per x-boundary and 2 intercepts per y--boundary. By the way, the range for t has been 0 < t < 4PI for all of these graphs. Also, notice that all of these graphs are closed curves.
Let's look at the graph when the numerator has a value of 6 before we move on.
Whoops! Just when we thought we had a pattern identified, we get a surprise. However, 6/2 = 3. It turns out that this graph is identical to the graph where x = 4 sin 3t. (Check it for yourself) Further investigation shows that the same thing will happen whenever the denominator is a divisor of the numerator. (Look at the graph for x = 4 sin (4/2 t) which we viewed two graphs previously. This is identical to the graph for x = 4sin 2t. Try it yourself and see.) What this tells us is that the only interesting graphs will be when the denominator is not a divisor of the numerator. So, the pattern we saw developing when the numerator was 5 and the denominator was 2 holds for odd valued numerators. Again, try it yourself and see.
To wrap up this part of the investigation, lets look at a denominator of 3 and see what happens as we vary the numerator.
Start with a numerator of 2.
This is pretty. Now we have 3 intercepts at each y - boundary and 2 intercepts at each x - boundary. We also had to increase the range of t to 0 < t < 6PI to complete the graph. This suggests that the numerator determines the number of intercepts on each x - boundary and the denominator determines the number of intercepts on each y - boundary. Of course there are interactions to consider here. That is what we will do in the next section.
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From the previous section we know that when the denominator is a divisor of the numerator, we get the same graph as if we had entered the quotient d in x = 4 sin dt. What happens when the numerator and denominator are relatively prime but have a common divisor?
Look at the graph for the fraction 9/6. The numerator and denominator are relatively prime. But, they have a common divisor of 3. (Actually, the reduced fraction is 3/2, which is significant.)
We have seen this graph before in the previous section of this investigation. It is the same graph that we got when the coefficient of t was 3/2. Further investigation leads me to conclude that the graph will always be the graph using the reduced fraction for the coefficient of t in the x equation of the parametric equations. Further, the range necessary for t also coincides with this. Otherwise, the number of intercepts at each y - boundary is determined by the denominator and the number of intercepts at each x - boondary is determined by the numerator by the patterns identified in the previous sections.
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