Adjust the launch angle and speed and fire the ball. Its trajectory is the graph of a quadratic function of the form y = ax² + bx. The highest point is the vertex of the graph, and where the ball lands is the root (x-intercept).
As the ball is thrown, gravity slows its upward motion and then pulls it back down, creating a y = ax² + bx relationship between horizontal distance x and height y. Here, a is determined by gravity and the launch speed when air resistance is ignored, while b is related to the launch angle through tan.
The vertex of this graph is the point where the ball reaches its maximum height. Of the two points where the graph meets the x-axis (the roots), one is the starting point (x=0), and the other is where the ball lands (the range). The axis of symmetry, x = -b/2a, corresponds exactly to the moment when the ball reaches its highest point.