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Further Investigation of Parametric Curves

By

Tonya DeGeorge

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In this investigation, we will be looking at the following equations:

We will begin this investigation by looking at various values of a and b. There are three cases to consider: a < b, a = b, a > b. This part of the investigation was done in a previous assignment. To see the results in full detail, please click here.

The conclusions from the last investigation showed the following:

Ø When the value of a is equal to the value of b, we get a circle with the center at the origin.

· The size of the circle is dependent upon the value of a and b.

· Radius of the circle = |a| = |b|

Ø When a < b or when a > b, the circle changes into an ellipse.

Ø The size of the ellipse depends on the values of a and b:

· a determines the length of the ellipse along the x-axis (or 2a = length along x-axis)

· b determines the length of the ellipse along the y-axis (or 2b = length along y-axis)

We will continue this investigation by considering what happens to the graphs when we add in another term. For instance the equations then become:

Cases to consider here are for values of a and b, as well as for values of h. We have the following to investigate:

a < b	a = b	a > b
h < 0	h < 0	h < 0
h = 0	h = 0	h = 0
h > 0	h > 0	h > 0

However, we already know what happens to the graphs when h = 0; we get a circle when a = b, and an ellipse when a < b or when a > b (see above).

So let’s investigate the other situations.

When a < b, and when h < 0:

Let’s use a = 2, b = 4, h = -2, we get:

When a < b, and when h > 0:

Let’s use a = 2, b = 4, h = 2, we get:

When a = b, and when h < 0:

Let’s use a = 4, b = 4, h = -2, we get:

When a = b, and when h > 0:

Let’s use a = 4, b = 4, h = 2, we get:

When a > b, and when h < 0:

Let’s use a = 4, b = 2, h = -2, we get:

When a > b, and when h > 0:

Let’s use a = 4, b = 2, h = 2, we get:

From observing these graphs, it appears that the value of h changes the direction of the ellipse (or circle). For example, when h < 0, the ellipse appears to be “slanted” down (from left to right), “moving” from quadrant II to quadrant IV. However, when h > 0, the ellipse appears to be “slanted up” (from left to right), “moving” from quadrant III to quadrant I.

Final Conclusions:

Ø When the value of a is equal to the value of b, we get a circle with the center at the origin.

· The size of the circle is dependent upon the value of a and b.

· Radius of the circle = |a| = |b|

Ø When a < b or when a > b, the circle changes into an ellipse.

Ø The size of the ellipse depends on the values of a and b:

· a determines the length of the ellipse along the x-axis (or 2a = length along x-axis)

· b determines the length of the ellipse along the y-axis (or 2b = length along y-axis)

Ø The value of h determines the “direction” of the ellipse

· when h < 0, the ellipse “slants” down from quadrant II to quadrant IV

· when h > 0, the ellipse “slants” up from quadrant III to quadrant I

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