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Internet Explorer 10 & 11 Users : If you have been using Internet Explorer 10 or 11 to view the site (or did at one point anyway) then you know that the equations were not properly placed on the pages unless you put IE into "Compatibility Mode". I beleive that I have partially figured out a way around that and have implimented the "fix" in the Algebra notes (not the practice/assignment problems yet). It's not perfect as some equations that are "inline" (i.e. equations that are in sentences as opposed to those on lines by themselves) are now shifted upwards or downwards slightly but it is better than it was.

If you wish to test this out please make sure the IE is not in Compatibility Mode and give it a test run in the Algebra notes. If you run into any problems please let me know. If things go well over the next week or two then I'll push the fix the full site. I'll also continue to see if I can get the inline equations to display properly.
Graphing and Functions Previous Chapter   Next Chapter Polynomial Functions
Parabolas Previous Section   Next Section Hyperbolas


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.

(a)    [Solution]

(b)    [Solution]

(c)    [Solution]


(a) So, the center of this ellipse is  and as usual be careful with signs here!  Also, we have  and .  So, the points are,



Here is a sketch of this ellipse.


[Return to Problems]


(b)  The center for this part is  and we have  and .  The points we need are,


Here is the sketch of this ellipse.


[Return to Problems]


(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,


Here is this ellipse.


[Return to Problems]


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.

Parabolas Previous Section   Next Section Hyperbolas
Graphing and Functions Previous Chapter   Next Chapter Polynomial Functions

Algebra (Notes) / Common Graphs / Ellipses    [Notes] [Practice Problems] [Assignment Problems]

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