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Further Investigation of Parametric Curves

By

Tonya DeGeorge

 


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:

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

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

Š      Radius of the circle = |a| = |b|

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

 

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

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

Š      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:

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

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

Š      Radius of the circle = |a| = |b|

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

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

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

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

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

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

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

 


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