Galileo, the first to prove a thrown stone flies along a parabola.
Why does a thrown ball trace exactly that shape of curve through the air?
The rate of change itself changes at a constant rate — the quadratic function.
Experiment
Hands-on experiment
Predict first — with the same throwing force, what angle sends the ball farthest?
Adjust the angle and throw (from a height of 2m, same force every time)
Distance flown
21.3m
Peak height (vertex)
7.9m
My best throw
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The orange dot is the vertex — the curve is symmetric about it.
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
In the world of linear functions, change was always constant. But a thrown ball is different — it slows as it rises, and past the peak it falls faster and faster.
Motion whose speed changes at a constant rate — because gravity pulls with the same force at every instant. The simplest function that captures this structure is the quadratic, the one with x² in it.
And the quadratic's real treasure is the vertex. The highest point, the cheapest point, the biggest profit — it's where humanity's question 'what is the best?' first gets an answer.
Insight
Insights from the video
“The quadratic is the first vessel that holds 'the change of change'.”
Linear functions live in a world of constant change; quadratics live where the rate of change itself changes steadily. Constant acceleration rather than constant speed — that's why every trajectory under gravity is a parabola.
“At the vertex, everything pauses for an instant.”
As the ball passes its peak, its vertical speed is exactly 0. The observation 'the maximum sits where change hits zero' later becomes the key to solving optimization with derivatives.
Misconception
Common misconceptions
Throwing at 45° always goes farthest.
45° is only optimal with no air resistance and a landing at the same height. When you release above the landing point (a basketball shot, shot put), the optimal angle drops below 45°. Checking a model's assumptions is what mathematical thinking looks like.
The vertex of a parabola is a concept for graph exercises.
The vertex is the prototype of optimization. What price maximizes profit? When does revenue turn downward? — for every problem modeled by a quadratic, the answer sits at the vertex.
Formula
Writing it as math
Write the curve from the shooting experiment as a formula, and the things you were manipulating reveal their identities.
General form of a quadratic
The sign of a sets the curve's direction (∪/∩), the size of a sets its width, and c is the y-intercept (launch height).
Location of the vertex
Where the maximum or minimum happens. It's also the axis of symmetry — the reason time up equals time down.
Projectile motion
The height of a thrown ball. −(1/2)gt² is gravity's share, v₀t is the throw's share, h₀ is the launch height — physics and math write the same equation.
In Real Life
Where you meet it in real life
Trajectories in sports
Basketball shots, golf drives, long passes — every ball in the air draws a quadratic. An athlete's 'feel' is really parabola computation.
Pricing decisions
Raise the price and profit per unit grows, but sales shrink. The profit curve is ∩-shaped — the optimal price sits at the vertex.
Bridges and architecture
Suspension-bridge cables, arch bridges — the curve that spreads force evenly is the parabola, making it a fundamental curve of structural design.
Braking distance
The distance a car skids after braking grows with the square of its speed. Double the speed = four times the distance — the mathematical reason speeding is dangerous.
Connection
Concepts connect
Previous concept
Linear Functions
See the straight-line world of constant change first, and you'll understand why a world of changing change is special.
← Linear Functions labLeads to next
Derivatives
If you noticed that 'the instantaneous change is 0 at the vertex', you're already standing at the door of calculus. Meet the tool that measures the instantaneous slope at every point of a curve.
Go to the Derivatives lab →