Circle Equation
circles, graphing
The general equation for a circle is given as(x–h)2 + (y-k)2 = r2,
where (h,k) is the center of the circle and r is the radius. Breaking this down into its components leads to a much better understanding of exactly what is going on.
Let's start with a circle centered at the origin with a radius of 1. The equation for this circle would be
x2 + y2 = 1.
This equation is directly related to the Pythagorean Theorem. You can think of my x-value as one leg of the triangle, the y-value as the other leg, and the radius as the hypotenuse.
As we can see, since the center is (0,0), there is no need to include these values inside of our parentheses. Now let's assume that I want to move my circle 1 unit to the right. Now my x values will all be 1 bigger. To compensate for this increase I have to decrease them all by 1. This will keep the leg of the triangle the same length. Note, I am not changing the triangle, just moving it around.
(x–1)2 + y2 = 1
If I wanted to move my center up by 2 units, I would have to again compensate by subtracting 2 from my y variable.
(x-1)2 + (y-2)2 = 1
Again, and this can be really tricky, one always wants to subtract the center values in order to compensate for the increase. If the center was (-2, -3), the equation would look like
(x+2)2 + (y+3)2 = 1
Finally, referring back to the fact that this equation is related to the Pythagorean Theorem, if the radius is increased from 1 to 3, the hypotenuse of the effective triangle is increased from 1 to 3, so the equation would become
(x–1)2 + (y–2)2 = 32, or
(x–1)2 + (y–2)2 = 9
circles, graphing