Graphing a Circle

Date: 04/14/98 at 15:02:55
From: Philip J. Peisch
Subject: algebra

I would like to know how to graph the equation of a circle. I have a 
big test coming up - can you please help me?

Thanks -
Philip J. Peisch

Date: 04/14/98 at 15:48:57
From: Doctor Kate
Subject: Re: algebra

The equation of a circle is usually:

   (x-a)^2 + (y-b)^2 = r^2

(Here "^2" means "squared").

If your equation isn't in that form, you will have to use algebraic 
techniques to get it into that form.  I won't go into that in too much 
depth if you don't need it, but here's an example:

   x^2 + y^2 + 2y + 1 = 36

You've probably done some factoring, so you'll notice that
y^2 + 2y + 1 is just (y+1)^2, and also that 36 = 6^2.

So your equation is then:

   (x-0)^2 + [y-(-1)]^2 = 6^2

(here a = 0, b = -1 and r = 6)

Let's go back to (x-a)^2 + (y-b)^2 = r^2

To graph this, you notice two things:  

   - The origin (centre) of your circle is at (a,b). (Notice that in    
     the example I gave above, b is -1 not 1. Don't make that mistake
     on your test). 

   - The radius of your circle is r. 

So draw your circle with that information. Pick the centre and then 
pick some points that are r units away from it and draw the circle 
through those (so you might draw (a, b+r), (a, b-r), (a+r, b) and 
(a-r, b) as four points and draw a circle that passes through all 
four).

So in the example above, the centre of your circle is (0, -1) and it 
has a radius (not diameter) of 6 units. So it will pass through the 
points (0, 5), (0, -7), (6, -1) and (-6, -1).

By the way, why is the centre (a,b) and the radius r?  Well in 
x^2 + y^2 = r^2, it's just the Pythagorean Theorem. And then going to 
(x-a) and (y-b) is just a translation of your graph. These are 
interesting ideas; please think about them (they are more important 
things to know than just 'how to graph a circle'). If you have more 
questions, please write us, we're happy to help.

Hope this helps and good luck on your test.

-Doctor Kate,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/