Extremes and Intercepts of a Path

A parametric path is really two separate stories, x(t) and y(t), moving in sync. Since x(t) controls left-right position and y(t) controls up-down position, the highest and lowest points of the whole path depend only on y(t): you find where y(t) hits its max or min, using whatever technique fits that function, a vertex formula for a parabola, or the fact that sine and cosine max out at 1 and bottom out at -1. The horizontal extremes, leftmost and rightmost points, come the same way from x(t) alone. You never need both equations at once to find an extreme, just the one function that matches the direction you care about.

Intercepts work by a similar split, but the logic flips. A point sits on the y-axis when x(t) = 0, so you solve x(t) = 0 and plug those t-values into y(t) to get the y-intercepts. A point sits on the x-axis when y(t) = 0, so you solve that instead and plug into x(t). Each equation controls the axis it does NOT belong to.

This connects directly to projectile motion, where y(t) is a downward parabola like -gt² + vt. The peak height is just the vertex of that one equation, found with t = v/(2g) and substituted back in. The horizontal distance traveled at that same instant comes from plugging that t into x(t) separately. Unit 4 keeps reusing this trick: treat each coordinate function on its own terms, then recombine only when the question asks for a full point.