Online Notes / Algebra (Notes) / Common Graphs / Ellipses

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In a previous section we looked at
graphing circles and since circles are really special cases of ellipses we’ve
already got most of the tools under our belts to graph ellipses. All that we really need here to get us
started is then standard form of the
ellipse and a little information on how to interpret it.

Here is the standard form of an ellipse.

Note that the right side MUST be a 1 in order to be in
standard form. The point is called the center of the ellipse.

To graph the ellipse all that we need are the right most,
left most, top most and bottom most points.
Once we have those we can sketch in the ellipse. Here are formulas for finding these points.

Note that a is the
square root of the number under the x
term and is the amount that we move right and left from the center. Also, b
is the square root of the number under the y
term and is the amount that we move up or down from the center.

Let’s sketch some graphs.

Example 1 Sketch
the graph of each of the following ellipses.

(c) Now with this ellipse we’re going to have to be a little
careful as it isn’t quite in standard form yet. Here is the standard form for this ellipse.

Note that in order to get the coefficient of 4 in the
numerator of the first term we will need to have a in the denominator. Also, note that we don’t even have a
fraction for the y term. This implies that there is in fact a 1 in
the denominator. We could put this in
if it would be helpful to see what is going on here.

So, in this form we can see that the center is and that and . The points for this ellipse are,

Finally, let’s address a comment made at the start of this
section. We said that circles are really
nothing more than a special case of an ellipse.
To see this let’s assume that . In this case we have,

Note that we acknowledged that and used a
in both cases. Now if we clear
denominators we get,

This is the standard form of a circle with center and radius a. So, circles really are special cases of
ellipses.