Final answer:
When the center of a circle crosses the y-axis or the x-axis, only the h or k values in the circle's equation change, respectively. When the radius of a circle changes, the r^2 value in the equation is adjusted to reflect the new radius.
Step-by-step explanation:
The equation of a circle is typically written in the form (x - h)^2 + (y - k)^2 = r^2, where r is the radius of the circle, and (h, k) is the center. When the center crosses the y-axis, only the h value is affected, which represents the x-coordinate of the center. When the center crosses the x-axis, only the k value changes, which represents the y-coordinate of the center. These transitions do not involve a change in signs of x and y or a multiplication by a constant factor; only the specific values of h and k are adjusted.
As for changes in the radius, r in the equation (x - h)^2 + (y - k)^2 = r^2 reflects the length of the radius. When the radius changes, r is affected; if the radius increases or decreases, r is squared to produce the new constant term. This also does not involve interchanging signs or multiplying coefficients of x and y by a constant factor. Therefore, the correct answer to how the equation changes when the radius changes is that the squared radius value is adjusted accordingly in the equation.