The word "intercept" means crossing point, and the y-intercept of a graph refers to the point at which the equation crosses the y-axis of the coordinate plane. When a point is on the y-axis, it is neither to the left nor the right of the origin. Therefore, it is located at the spot in the equation where x equals zero. Because a circle is round, it can cross the y-axis twice and have up to two y-intercepts. However, you find the y-intercept(s) of a circle the same way you would for any other equation - by substituting "0" for x.
If you end up having to take the square root of negative number, this means that there are no y-intercepts.
Substitute "0" in for x in the standard form of the equation of a circle -- (x-h)^2 + (y-k)^2 = r^2, where h and k are integers and r stands for the radius of the circle. For example, (x-3)^2 + (y+4)^2 = 25 becomes (0-3)^2 + (y+4)^2 = 25 when plugging "0" in for x.
Square the part of the equation that used to have the x, the h value. Then, subtract that from both sides. Here, you will get 9 + (y+4)^2 = 25, then (y+4)^2 = 16.
Take the positive and negative square root of both sides to create two linear equations. For instance, in the example above, you will have y + 4 = 4 and y + 4 = -4.
Solve each equation for y to get your y-intercepts. In this case, you subtract 4 from both sides in both equations to end up with (0, -8) and (0, 0).
About the Author
Kathryn White has over 11 years of experience tutoring a range of subjects at the kindergarten through college level. Her writing reflects her instructional ability as well as her belief in making all concepts understandable and approachable. White earned a Bachelor of Arts in history from Illinois Wesleyan University.