We are given the cubic function y equals x cubed. Let's consider a point P with coordinates (a, a cubed) on this curve.
To find the tangent line at point P, we need the derivative of y equals x cubed. The derivative is dy/dx equals 3x squared. At point P with x-coordinate a, the slope is 3a squared.
The tangent line intersects the curve again at point Q. To find Q's coordinates, we need to solve the system of equations formed by the cubic function and the tangent line equation. After algebraic manipulation, we find that if P has coordinates (a, a cubed), then Q has coordinates (-2a, -8a cubed).
Let's verify our result with a specific example. If point P has coordinates (1, 1), then according to our formula, point Q should have coordinates (-2, -8). We can see this is indeed the case where the tangent line intersects the curve again.
In general, for any point P with coordinates (a, a cubed) on the curve y equals x cubed, the tangent line at P intersects the curve again at point Q with coordinates (-2a, -8a cubed). This relationship holds for all points on the cubic curve.