Assignment 10:

Parametric Curves

by: Katie Gilbert

A parametric curve in the plane is a pair of functions:

where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t . In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.

In this write-up we are going to explore the following parametric equations:

Investigate each of the following for

Describe each when a = b, a < b, and a > b.

First we will take a look at the graph of the functions above where we set a=b, and see what that gives us.


First let's explore the graph above, where a=1 and b=1. We can see that for our function when the exponent is even, the graph is only located in the first quadrant. We also see that our X and Y intercepts are at our +/- A and +/-B respectively.


Now let's take a look at our next graph, where a>b, and see if we can further our findings.


It seems that when we set our a=1.5, and our b = .5 we change our x and y intercepts. The graph when a>b appears to distort our graph into an ellipse . Again we see that the functions with even graphs only graph in the first quadrant.

Now let's take a look at when our a<b, will it be similar to what we just found above??


Our suspisions are correct, our graph does respond like our previous investigation, when we change b=1.5 and a=.5 we find that our x-intercept becomes +/- .5 and our y-intercept becomes +/- 1.5. Again causing our graph to take an elliptical shape, but this time vertically. We also find that our functions with even exponents again only graph in the first quadrant.

I find the functions with even exponents interesting and nextwe will explore this a little further. Let's take our a= -1, and leave b=1 and see what happens to our graph.

It is what I thought, but let's investigate a little further before we conclude anything. Let's take a look at when a=1, and b = -1.

I think you get the idea, but let's take a look at a= -1, and b = -1, and confirm what we believe to be true.........

As you have probably already figured out when we vary our a and b from positive to negative we change the quadrant in which our even functions graph. We can see from above that when our value for a is negative, the even functions will graph in Quadrant II or III, depending on the sign of b; likewise, if our b value is negative the even functions will graph in either quadrant III or IV, depending on if a is positive or negative. With those familiar with the functions of sine and cosine, these finding make perfect sense. We know that in graphing our sine function that the sine function is negative in quadrants II and III (which is dependent on our b value), likewise our graph of the cosine function is negative in the III and IV quadrants (which is dependent on our a value).

We also can conclude that it "makes sense" that our even functions only graph in one quadrant, (which quadrant depends on the sign our our a and b values), because when you raise a positive or negative number to an even power you will always get a positive number.

Click here if you would like to explore different values for our function above.