Assignment 10: Parametric Equations

by

Melissa Silverman

Similar to the first few investigations, graphing calculator will be used to explore various parametric equations and their graphs. We begin with a simple parametric equation and its graph.

The graph is a circle centered at (0,0). The radius of the circle is 1.

How can we alter the parametric equation? Do these alterations change the graph? We can begin by adding a coefficient in front of the cosine function.

Putting a coefficient in front the of cosine function changes the width of the graph. The figure is still centered at (0,0). The size of the coefficient determines the width of the enclosed shape. In this example, the coefficient is 2 and the width of the shape is four units, 2 units on either side of the origin.

From the above example, one can infer that putting a coefficient in front of the sine function must change the height of the figure, while the center remains at (0,0).

The height of the figure is now 4 units, 2 on either side of the origin.
Combining coefficients results in larger enclosed figures. When the coefficients are equal, the shape is a circle whose radius is equal to the coefficient. Both graphs are still centered at (0,0).

What if the sine and cosine functions are squared? The result is a line segment whose endpoints are the coefficients of the parametric equation. The graph is no longer an enclosed figure. In this example, the coefficients are both 1, so the segment spans from (0,1) to (1,0).

Changing the coefficients change the span of the line. The coefficient of the cosine function determines the endpoint of the segment on the x-axis while the coefficient of the sine function determines the endpoint of the segment on the y-axis. For example, when the coefficients are 2 and -2 for the cosine and sine function (respectively), the line segment spans from (0,-2) to (2,0).

What about cubing the parametric equation? What would the graph look like?

The graph has a similar shape to a diamond. The vertices of the diamond are all 1 unit away from the origin because the coefficients are 1. Changing the coefficients of the parametric equation will alter the placement of the vertices of the diamond.

If the graph is raised to an even power higher than 2, the graph is a curve, where the coefficients of the sine and cosine function determine the y and x coordinates, respectively.

And the graphs continue in this pattern. For all odd powers, the shape of the graph is a diamond. For all even powers (greater than 2), the graph is a portion of a curve.

Combining even and odd powers results in a "v" shaped curve, for all cases of even and odd exponents. If cosine is raised to an odd power and sine to an even power, then the graph is symmetric about the y-axis. If cosine is raised to an even power and sine to an odd power, then the graph is symmetric about the x-axis.

Like Investigation 1, explorations such as these are important in understanding parametric equations and their graphs. In this investigation, the primary focus were the sine and cosine functions. By manipulating coefficients and exponents, much was learned about how changing the parametric equation affects its graph. This allows one to know what the graph of a parametric equation will look like just by looking at its form. And by looking at a graph, one can predict the graph's parametric equation. This flexibility is important to mathematical understanding. In the first investigation, much of the experimenting could have been done with a graphing calculator or pencil and paper sketches because parabolas are more simple to draw. Using computer technology to investigate parametric equations allows one to do things that could not easily be done on paper. This assignment is certainly a good example of where technology allows one to do things not easily done on paper.
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