Algebra 2 Graphing Ellipses
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Graphing Ellipses

 

 So the equation of a circle looks something like:

 

 X2 + Y2 = 52

 
 If the numbers next to the X 2 and Y 2 are different from each other,
 the circle gets stretched ...

 

 
 To see how this stretching works,
 let's start with:

 4X2 + Y2 = 100

 
 The coefficient on the X 2 (4) 
 is different from the coefficient on the Y 2 (1)
 and both terms are positive.
 
 That means we have an ellipse
 
 But we can't tell yet how big it is,
 or which direction it is stretched.
 
 To work these out,
 first divide both sides by the number on the right side
 and simplify ...

 

 
 The main point is to get the right side of the equation to be 1.
 Next, we need to get the left side of the equation to look like:
 

 

 
 To do that, we need to figure out the square roots
 of the numbers in the denominators.
 
 Sometimes these work out to be whole numbers
 sometimes they don't.
 To keep things a bit cleaner in this example, they do ...
 

 

 
 So wonderful,
 now that we've gone through all of these shenanigans,
 what have we got?
 
 Well now we know that the ellipse goes to + 5 and - 5 in the X direction,
 and out to + 10 and - 10 in the Y direction.
 

 

 
 OH YEAH?
 Just how do we know that?
 
 We know that because those are the X and Y term denominators.
 

 

 
 When X is zero the whole first term on the left side is zero.
 The only way to make the left side equal to 1
 is for Y to equal 10 or -10.
 
 When Y is zero the whole second term on the left side is zero.
 The only way to make the left side equal to 1
 is for X to equal 5 or -5.
 
 That ellipse was centered at the origin (0,0).
 Moving the ellipse center away from the origin
 works just like it did with a circle.
 
 Subtract something from X moves it that far right
 Add something to X moves it that far left.
 Subtract something from Y moves it that far up.
 Add something to Y moves it that far down.
 
 Example:
 Say you wanted the equation 
 of an ellipse that was the same size as the one in the last example
 and centered at X = 2, Y - 1.
 

 

 

   copyright 2005 Bruce Kirkpatrick

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