Parametric Equations

by Jamila K. Eagles

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A parametric curve in plane is a pair of functions :

where the two continuous functions define the ordered pairs (x,y). The two equations are usualy called the parametric equations of a curve. The extent of the curve will depend on the range of t.
This exploration looks at the parametric equation:

where a and b vary. We begin selecting values for a and b such that a>b. Let's choose a = 4 and b = 2. Th graph of this functions gives us an ellipse with endpoint on the x axis at x = 4 and x = -4. The enpoints on the y axis are at y = 2 and y = -2. Recall from the study of ellipses that x is considered the major axis in this case.

Next, let's choose a = 3 and b = 1. The graph shows gives us an ellipse with endpoints on the x axis at x = 3 and x = -3. The endpoints on the y axis are at y = 1 and y = -1. Once again the x axis is the major axis.

Before drawing any conclusions, let's try this case with negative values. Choose a = -2 and b = -3. From the graph below, we get an ellipse with endpoints at 3 and -3 on the y axis and 2 and negative 2 on the x axis. Here our major axis is the y axis. What is happening here? It follows all the examples above and a is greater than b. We will leave this to explore these equations further.

Let's explore the case where a < b. We choose a = 2 and b = 4. This gives us an ellipse with 4 and -4 as the endpoints on the y axis, and 2 and -2 as the endpoints on the x axis. The major axis in this case is the y axis.

Now let's choose a = 1 and b = 3. This gives us an ellipse with endpoints 3 and -3 on the y axis, and 1 and -1 on the x axis. The major axis of this graph is the y axis.

Once again let's try using negative values. Let's choose a = -3 and b = -2. The example using negative values again does not give us the expected graph. Here the major axis is the x axis. This example follows the case a < b.

Taking a closer look at negative values, let's choose a = -2 and b = -3.
Now overlaying a new graph with a = 2 anb b = 3, we see that we get the same ellipse with end points at 2 and -2 on the x axis and 3 and -3 on the y axis. The y axis is the major axis.







This tells us that replacing positive values with negative values gives us the same graph. We know that the only time 2 and -2 are equal is when we take the absolute value of -2. Therefore we can conclude that when |a| > |b|, we get an ellipse with the x axis as its major axis. When |a|< |b|, we get an ellipse with the y axis as its major axis.

Now what happens when |a| = |b|? We begin this exploration by choosing |a| = |b| = 3. This gives us the figure below. The graph is a special ellipse known as circle. In this case the major axis and the minor axis are equal. This gives us the fact that the radius is equal to |a|=|b|, and in this case our radius is 3.

In the parametric equation:

when |a|>|b|, we have a graph of an ellipse with the x axis as its major axis.
When |a|<|b|, we have a graph of an ellipse with the y axis as its major axis.
When |a| = |b|, we have a graph of a circle with radius |a| = |b|.


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