Worked example
Projectile Motion: Graphing a Thrown Ball
The height of a ball thrown straight up follows a quadratic function of time: gravity pulls with a constant acceleration, so height is a downward-opening parabola. Here, -4.9x² + 20x + 1.5 models a ball launched at 20 meters per second from a height of 1.5 meters (the 4.9 comes from half of Earth's gravitational acceleration, 9.8 m/s²).
The vertex of the parabola is the moment the ball reaches its highest point — the instant its velocity is zero before it starts falling. Because the parabola is symmetric, the ball lands as long after the peak as it took to climb to it. The two x-intercepts mark launch (near x = 0) and landing; only the positive root is physically meaningful.
Plotted expressions
- y = -4.9*x^2 + 20*x + 1.5
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- The vertex of the parabola gives the maximum height and the time it is reached.
- The positive x-intercept is when the ball hits the ground — find it with the root finder.
- The coefficient -4.9 controls how "wide" the flight is; a larger launch speed (the 20x term) stretches the flight longer.