Changing the Equation, Moving the Graph

A circle's equation is really just a locked-in description of every point sitting the same distance from one fixed center. That form, (x - h)² + (y - k)² = r², is built so the center and the size each live in their own separate slot. The numbers h and k are the coordinates of the center, hiding inside subtraction, and r² on the right is the radius squared. Because the roles are separated, editing one slot changes exactly one feature of the picture. Move the center and the circle slides without changing shape. Change r² and the circle grows or shrinks without moving.

This split is why shifting a graph feels like simple arithmetic on h and k. Shifting right by p means the new center's x-coordinate is h + p, so the equation becomes (x - (h+p))² since x is measured from that new center. Shifting down by q lowers the y-coordinate of the center to k - q. The right side never changes during a slide, because sliding doesn't touch the radius at all. Only when you edit the number standing alone on the right does the radius change, and even then the center stays exactly where it was. That number is r², not r, so doubling it does not double the radius: the radius scales by the square root of that ratio.

This same h, k, r pattern shows up again with parabolas, ellipses, and absolute value graphs, where shifting the input shifts the graph and scaling the output resizes or stretches it. Recognizing which number controls position and which controls size is the real skill behind every question that asks what happens to a graph after one small edit.