6/24 Lesson Overview

Circles: General equation is (x-h)^2+(y-k)^2=r^2, where (h, k) are the coordinates of the center, and r is the radius.  Keep track of the negatives!  For (x+4)^2, that means that h=-4, because x+4 is actually x-(-4).  Sometimes the equation you have will look a little bit different from the general form given, and you will have to complete the square to get a more convenient form.

Graphing a circle: Plot the center, then plot 4 points up, down, left, and right the distance of the radius away from the center, and then draw your circle.

Ellipses:  General forms of the equation (there are two, one horizontal and one vertical).  Remember, a^2 is always the larger denominator.

Screen Shot 2013-06-24 at 4.42.16 PM Screen Shot 2013-06-24 at 4.42.35 PM

 

Just like in circles, (h,k) is the center of the ellipse.  Then find a, b, and c.  Remember the other important equation: c^2=a^2-b^2.

a tells you how far away the vertices are from the center, and 2a is the length of the Major axis.

b tells you how far away the covertices are from the center, and 2b is the length of the minor axis.

c tells you how far away the foci are from the center.

To determine which form your using, I think of the denominators as “stretching” certain variables.  The bigger the denominator, the more it stretches the variable it is beneath.  If an ellipse is stretched more horizontally, then it is a horizontal ellipse, and so the foci and vertices are left and right of the center, where as if the ellipse is stretched more vertically, then it is a vertical ellipse, and so the foci and vertices are up and down from the center.

If an ellipse is stretched the same amount horizontally and vertically, what shape is it?

The equation you’re given will not always be in the convenient form, and you will need to rearrange it and do some completing the square.  Group your x terms together, factor out the leading coefficient, and complete the square with what’s left, remembering to multiply whatever you complete the square with by the coefficient before you add it to the other side.  Do the same thing with the y’s.  After all this, the equation will be set equal to some number.  The general form is set equal to 1!  Whatever # your equation is equal to, if it’s not 1, divide everything by it so that it is 1, and what’s left should look like the general form of an ellipse.

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