The figure shows the curve $y^2 = x^3$, which has a sharp point (a cusp) at the origin and two branches, one above the $x$-axis and one below it, mirror images of each other. A convenient way to describe its points is $x = t^2$, $y = t^3$, where $t$ is any real number. The point $P(4, 8)$ lies on the upper branch, where $t = 2$. The tangent to the curve at $P$ crosses the curve once more, and the normal at $P$ is the line through $P$ perpendicular to that tangent. Which of the following statements are true?

Fig. 1: The curve y^2 = x^3 from x = 0 to x = 5, with axes and a light grid. It has a sharp point (a cusp) at the origin, an upper branch rising to the right and a lower branch that is its mirror image in the x-axis. The point P(4, 8) is marked on the upper branch. No tangent is drawn.