The general form of a circle is:
where r is the radius of the circle and the point (h, k) is the center of the circle. The equation of a circle with its center at the origin and a radius of 1 is:
with a graph of:
We will discuss transformations of the circle in terms of this equation and its graph.
The first transformation to discuss is the h. Since the h is subracted from the x, it moves the center of the circle left or right from the origin. The one thing to remember is that since the equation uses a minus sign that the movement is actually counter-intuitive to the actual movement.
For example: (x - 2) would appear to translate the circle to the in the negative direction, left. Actually, this moves the circle to the right two places. This can be seen below.
Thus, when we had (x + 2) this would translate the circle to the left. This can be seen below.
Click Here to see a movie of this where h has values of -10 to 10.
The second transformation is that of the k. This works just like the h, except it moves the center of the circle up or down, vertically. Just like the h, it is counter-intuitive since the formula uses a minus sign.
For example: (y - 2) would seem to move the equation of the circle downward when in reality it moves the equation upward. See the graph below.
Likewise, (y + 2) would seem to move the center upwards when in reality it moves the center down.
The center may have both vertical and horizontal translations at the same time like so:
Click Here to see a movie where k takes vales from -10 to 10.
The other transformation occurs when the radius is not equal to 1. This will either dilate or expand the graph of the circle. It dilates when r is less than 1 and expands when r is greater than 1. See the picture below for some examples of this.
Click Here if you would like to see a movie of the circle going through dilations for values of the radius from -10 to 10.
Just for fun, if you would like to see what happens if h, k, and r all have the same values, Click Here.
Now, one should understand the basic transformations of a circle.