We can transform the graphs of circles in the coordinate plane using translations and dilations. 

An original circle is centered at with a radius of 1. Its image is centered at with a radius of 3. What transformations were needed to get from the original to the image? In this case, there was a translation right 3 and down 2, with a dilation by a factor of 3. This makes sense because a translation of the circle will translate the center in the same way, and dilating the circle by a factor of 3 also dilates the radius by a factor of 3.

We can also see these transformations in the equations for the circles. We know from studying circles that the equation of any circle with center and radius can be written . Therefore, the original circle has an equation , and the image or transformed circle has an equation .

Since the translation of the circle can be determined using the centers, that means that the transformed circle has been translated horizontally by units and translated vertically by units. Since the radius of the transformed circle is the same as the scale factor of dilation, we can say that the transformed circle has been dilated by a factor of .

This means we can tell what transformations have been applied to a circle in the coordinate plane without graphing, given the original circle defined by . For example, if a circle has an equation , we can tell that it was translated right 12 units, translated down 8 units, and dilated by a factor of 9.