R.2 Circles
The graph of an equation
 the graph of the function y = x2
 is the set of all points (x, y) that satisfy y = x2
 e.g. (2, 4), (-3, 9), etc.
 we can take any equation and graph it
 by placing all the points that satisfy it
 on a coordinate system
Watch while we graph x2 + y2 = 16.
It’s a circle! (but it’s not a function. Why?)
If we graph (x - 2)2 + (y - 1)2 = 4, we get the graph:
y



(2, 1)




x



Notice that the center is at (2, 1), and the radius is 2.
If we expand our equation, we get it in another form:
x2 + y2 - 4x - 2y + 1 = 0 (same circle!)
R.2-1
Circles: Four views and twelve skills
General form:
2
2
Ax + Ay + Bx + Cy + D = 0
(A>0)
complete the square
expand
Standard form:
( x - h )2 + ( y - k )2 = r2
read off std. form
substitute in
std form
Parameters:
Center: (h, k)
Radius: r
draw the picture
read off picture
Graph:
R.2-2
Example: Find the standard form of a circle with center
(5, -1) and radius 2.
Example: Find the standard form of the circle with general
form x2 + y2 - 4x - 2y + 1 = 0, and graph it.
Example: Find the equation of the circle with the graph:
y


(-1, 3)









x

Example: Find a circle with center at (0, 5) and the point
(2,5) on the circle.
R.2-3