Parabola and Quadratic Equations — Explained, With a Free Game 7+

What is a parabola?

A parabola is the graph of a quadratic function y = ax² + bx + c with a ≠ 0. It is a smooth curve with a single turning point and two identical branches: draw a vertical line through that turning point and the left half mirrors the right exactly. That line is the axis of symmetry.

Which way the branches point is decided by a:

• a > 0 — branches open upward, the curve is a bowl with a lowest point;
• a < 0 — branches open downward, the curve is an arch with a highest point (this is how a thrown ball flies);
• the larger |a| is, the narrower and steeper the curve; the smaller it is, the wider and flatter.

The coefficient c is the y-intercept: put x = 0 and you get y = c. In a throwing problem that is simply the height the stone left from.

The vertex of a parabola and how to find it

The vertex is the one point where the curve stops rising and starts falling (or the reverse). Its x-coordinate comes from a short formula:

x = −b / (2a), and then y is found by substituting that x back into the function.

Example: y = x² − 6x + 5. Here a = 1 and b = −6, so x = 6 / 2 = 3. Substituting: y = 9 − 18 + 5 = −4. The vertex is (3, −4) and the axis of symmetry is the line x = 3.

There is a second route with no formula at all: the roots always sit symmetrically around the vertex, so the vertex x is the average of the two roots. If the roots are 1 and 5, the vertex sits at x = 3.

The same information is packed into vertex form, y = a(x − h)² + k, where (h, k) is the vertex outright — handy when you are asked for the maximum or minimum value rather than for a graph.

Roots of a quadratic equation are crossing points

Solving ax² + bx + c = 0 means finding the x values where the parabola crosses the horizontal axis. Geometrically those are the places where the arc lands on the line — in the game they are the burning marks on the ground, and on a rope strung with lanterns there are exactly two of them.

Hence the headline fact: a quadratic has two roots, one root, or none — as many as the times the arc met the line.

The discriminant tells you how many

The discriminant is the number D = b² − 4ac. You compute it before solving, so you know what to expect:

• D > 0 — two distinct roots, the arc crosses the line twice;
• D = 0 — one root, the arc touches the line with its vertex;
• D < 0 — no real roots, the arc never reaches the line.

The roots themselves come from the quadratic formula x = (−b ± √D) / (2a). That plus-or-minus is precisely the two crossings: one to the left of the axis of symmetry, one to the right.

Worked example: solving a quadratic equation

Take x² − 6x + 5 = 0, so a = 1, b = −6, c = 5.

1. Discriminant: D = (−6)² − 4 · 1 · 5 = 36 − 20 = 16.
2. D > 0, so there are two roots. √16 = 4.
3. x₁ = (6 + 4) / 2 = 5 and x₂ = (6 − 4) / 2 = 1.

Check by substituting 5 and 1 back in — both give zero. And notice that the vertex (3, −4) found earlier sits exactly halfway between the roots 1 and 5.

A one-root case: x² − 4x + 4 = 0 gives D = 16 − 16 = 0, so x = 4 / 2 = 2 and the parabola touches the axis at a single point. And x² + x + 3 = 0 has D = 1 − 12 = −11, negative: the graph floats entirely above the axis and never crosses it.

When the numbers are friendly, factoring is faster: x² − 6x + 5 = (x − 1)(x − 5), and a product is zero only when one of its factors is. The discriminant is the method that always works, factoring is the shortcut worth spotting.

How to graph a parabola

Five points and a few minutes are enough:

1. Find the vertex with x = −b / (2a) and mark it.
2. Draw the axis of symmetry — a vertical line through the vertex.
3. Mark the y-intercept, the point (0, c).
4. Find the roots if D ≥ 0 — those are the x-intercepts.
5. Take a couple of points on one side of the vertex and mirror them across the axis instead of recalculating.

Join the points with a smooth curve, never with straight segments — a parabola has no corners. Then check the branch direction against the sign of a; that is the most common slip in a quick sketch.

Why real things fly along parabolas

Anything thrown at an angle travels a parabola: a ball, a jet of water, a catapult stone. It happens because the object moves forward at a steady rate while gravity pulls it down at a constant acceleration — combine "even sideways, accelerating downward" and the height depends on the distance quadratically. The same shape gives us parabolic mirrors and dish antennas, because a parabola gathers every parallel incoming ray into one point.

How the game drills all of this

In the game above the child stands behind a fortress wall at a catapult, pulls the winch, turns the frame, picks a ball — and sees a glowing ghost arc of the flight to come. That arc is the parabola: its top is labelled the vertex, the marks on the ground are the roots, and the letter D turns up exactly when the question becomes "will it even get there?"

The hundred levels follow the same order as this page:

• 1–20 — intuition, no formulas at all: pull harder for a longer arc, steepen the angle for a higher one;
• 21–40 — a wall appears and the vertex starts to matter, because it has to clear the merlons;
• 41–60 — a rope of lanterns is strung between two towers and one stone takes both, because the arc crosses that height twice — that is what two roots means;
• 61–75 — the discriminant: touch the beacon with the very top of the arc, i.e. reach D ≈ 0;
• 76–90 — the knobs are labelled a, b and c: steepness, launch slope, platform height;
• 91–100 — the predicted arc is hidden and shots are scarce, so the shot has to be computed first.

The maths is verified automatically: every build re-fires thousands of shots across all one hundred levels and compares the algebraic roots and vertex against where the simulated stone actually lands. A disagreement of more than one percent and the build does not ship.