Quadratic equations turn up everywhere — from the arc of a thrown object to the best price to charge for a product. Here are three examples that show why this matters beyond the classroom.
All three examples share the same underlying shape: f(x) = ax² + bx + c. What changes is what x and f(x) represent, and what question the vertex answers.
vertex at x = −b ÷ 2a → the most important point on every parabola
adjust the sliders on each example to see the vertex move
Physics — kinematics
Angry Birds — the parabola behind the game
When a bird is launched, it follows a curved path called a projectile trajectory. Game developers use the exact same quadratic equation that physicists use to model real thrown objects.
h(t) = −4.9t² + v₀t + h₀
v₀ launch speed (m/s)15
h₀ launch height (m)2
Peak height
—
Time to peak
—
Landing time
—
The vertex is the peak. The highest point the bird reaches is always at t = v₀ ÷ 9.8 seconds — the x-coordinate of the vertex. Gravity (the coefficient −4.9) never changes; only the launch conditions do.
Business — optimisation
Ice Cream Shop — finding the sweet spot
An ice cream shop tracked how many scoops they sold at different prices over a summer. Charge too little and revenue is low even with lots of customers; charge too much and sales collapse. The data points form a curve — and fitting a quadratic finds the price that maximises revenue.
R(p) = ap² + bp + c
Observed sales data
Price
Units
Revenue
Best price
$3.84
Max revenue
$982
R² of fit
0.952
The vertex is the optimal price. Revenue = price × units sold. As price rises, units fall — so revenue rises then falls, tracing a downward parabola. The vertex gives the exact price where revenue peaks. Toggle the fitted curve to see how well the quadratic captures the real data.
Optics — reflective geometry
Flashlights & Satellite Dishes — the focusing property
Parabolas have a remarkable geometric property: any ray arriving parallel to the axis reflects directly through a single point called the focus. This is why parabolic mirrors concentrate light, heat, and radio signals so effectively.
y = 1/(4p) x² focus at (0, p)
p focal length3
Focus point
—
Equation
—
Shape
—
Incoming parallel rays (blue) all reflect through the focus (orange dot). Move the focal length slider to see the dish deepen or flatten.
Smaller p = deeper dish, tighter focus. A satellite dish has a small focal length and a deep curve to concentrate weak signals. A car headlight uses the same principle in reverse — a bulb at the focus sends parallel beams outward.
Things to think about
Discussion questions
Peak height is h₀ + v₀²/(2g). Since v₀ is squared, doubling the launch speed quadruples the peak height (minus launch height). This is a signature of quadratic relationships — the output grows much faster than the input. It's also why a small increase in launch speed has a surprisingly large effect on how high something flies.
Maybe it was an unusually hot day, a local event drove extra foot traffic, a nearby competitor was closed, or there was a special promotion running. It could also be a data recording error. This is a key lesson about real data: outliers are not mistakes to discard — they're questions waiting to be answered.
They would not converge perfectly at the focus. The focus property only holds exactly for rays parallel to the axis — which is why satellite dishes work best when aimed precisely at the satellite. A source slightly off-axis (like a star other than the one a telescope is pointed at) produces a blurred image. This is the principle behind the design of radio telescopes and solar concentrators.