Carisa Lindsay

Assignment 10

Parametric Equations Exploration

 

Suppose we have the following equations:

 

 

For simplicity, let’s investigate the graph for values of a = b  and t-values between zero and 2p.

 

For values of a = b greater than the absolute value of 1, we will get a circle of radius 1 (not =0).  For instance, the following graph represents a = b = 3.  Because we are dealing with a circle, it will not matter if we use t-values larger than 2p since the graph will continue along the circle.

 

Now we can consider rational values of a and b.

 

For a = b =.5 and 0<t<2p, we only see the top half of circle located in quadrants I and II.

 

However, if we consider larger values of t, we will get slightly different results:

For a = b =.5 and 0<t<3p

For a = b =.5 and 0<t<4p

As you can see, as we increase the range of t-values to include larger multiples of Pi, we will again arrive at a circle.  If we try larger values of Pi, we will continue to get circles because the graph will repeatedly create circles.

 

For a = b =-.5 and 0<t<2p, we see only the bottom half of the circle in quadrants III and IV.

Again, if we increase the t-values to larger intervals of Pi, we will again arrive at a circle.

a = b=-.5 and 0<t<4p

 

For a = b =.25 and 0<t<2p, we only see ¼ of the circle in quadrant 1

Likewise, for a = b = -.25, we see ¼ of the circle in quadrant IV.

 

This time, it will require us to use 0<t<8p to get a full circle:

 

Quite predictably, for a = b =.75  we see ¾ of circle in quadrants I, II, and III.

Finally, if a = b = -.75, our resulting circle resides in quadrants II, III, and IV.

 

Lastly, we can use 0<t<3p to get a full circle for the above values of a = b = .75 or -.75:

 

 

 

 

For a ¹ b, the graphs become quite interesting.  However, the t-values seemingly do not affect how many “loops” each of the following graphs. For simplicity’s sake, we will maintain the values of t to remain between 0 and 2pi.

 

For a = 1 and b any natural number, the number of “loops” corresponds to the value of b.  For instance, in the following graph, for a = 1 and b = 4, there are 4 “loops”.

 

 

As b increases, so do the number of “loops”.

 

Suppose we consider large values of b.  For instance, let a= 1 and b = 15.  According to the pattern seen thus far, we should see 15 “loops”.

 

As predicted, we see 15 “loops”.  What about a = 1 and b = 50?

 

 

 

Please note that for all of these values, the graph oscillates between -1<x<1 and -1<y<1.

 

However, if we explore negative values of b, we see the same picture.

 

 

What about rational values of b?  For values between 0 and 1, the graph appears to behave much like a slinky.  Again, we have oscillatory motion for -1<y<1 and -1<x<1.  As b approaches 1, the graph appears more like a circle and will eventually be a circle for b = 1. 

 

 

Click here for animation

 

 

 

Suppose we change the values of a and keep b = 1.  For a values between 0 and 5, the graph follows a similar oscillatory pattern except in a vertical fashion.  These graphs also have a domain of -1<x<1 and a range of -1<y<1.  Obviously, the values of a and b control which direction the graph oscillates.  Unlike the values of b, the number of waves also corresponds to the value of a for odd values.  For even values of a, we see the period corresponds to half the value of a. 

 

For instance, suppose a = 5 and b = 1.  We can predict a graph with vertical oscillations and 5 “loops”.

 

 

Now suppose a = 10 and b =1.  We can predict another graph with vertical oscillations and 5 periods.

 

To see other values of a, Click here for animation.  This animation has 0<a<6, b=1,  and 0<t<2Pi. 

Perhaps some of the more interesting observations are when a is a whole number, the graph is continuous.  When a=1, we again see a circle; if a =2, we see a sideways parabola; if a=3, we see a vertical “spring” with 3 “loops”; if a=4, it resembles a=2 with an extra curve; and a=5 has 5 “loops”.  It appears that for odd-values of a, we will see a figure that is “closed” meaning that it will continue along that path without having to move backwards.  For instance, an object can continue to travel along a circular path forever. 

However, for even values of a, we will result in a graph that is not “closed”:

 

 

 

When we use negative values of a, we get similar results as when b was negative.  In fact, the graph is identical to the resultant graph from positive values of a. Essentially, we are only interested in the absolute value of a since the graph is the same for negative values of a. 

 

Click here for animation

 

 

 

For rational values of a, we see similar oscillatory motions except that the movements are horizontal.  Furthermore, all non-whole values of a will result in a graph that is not “closed”:

 

 

Click here for animation

 

 

 

 

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