Parabola Calculator

What this calculator does

So, you have a quadratic f(x) = ax² + bx + c, and you want more than its solutions, you want the whole shape it draws. This tool lays out the full parabola: its vertex, its axis of symmetry, its focus, its directrix, its y-intercept, and its zeroes. It is the complete geometric picture of the curve, all from the three coefficients.

You enter a, b, and c, and it reports every one of those features at once.

How to use it

1. Enter coefficient a, then b and c.
2. Press Calculate.

The coefficient a cannot be zero. With no x² term there is no curve, just a straight line, so the parabola is not defined.

The parabola as a curve, not just an equation

A parabola is the U-shaped curve a quadratic draws, and it has a geometry that goes well beyond its formula. The ancient Greeks studied it, and it was given the name parabola by Apollonius. Its defining property is a lovely one: every single point on the curve is exactly the same distance from a fixed point, called the focus, as it is from a fixed line, called the directrix. This tool works out both of those, along with the more familiar features, so you can see the curve as a shape rather than only a rule.

Vertex and axis of symmetry

The vertex is the turning point, the very bottom of an upward parabola or the top of a downward one. It sits at x = -b / (2a), and the tool finds its height by putting that x back into the equation. Running vertically through the vertex is the axis of symmetry, the line x = -b / (2a). The parabola is a perfect mirror image across that line, so whatever happens on one side is matched on the other.

Focus and directrix

These two are what make a parabola more than a graph. The focus is a point inside the curve, and the directrix is a line outside it, placed so that every point on the parabola is equidistant from the two. For y = ax² + bx + c, the focus sits a distance 1/(4a) above the vertex, and the directrix lies the same distance below it. This is not a curiosity: it is the property behind satellite dishes, headlight reflectors, and telescope mirrors, where rays arriving parallel to the axis all bounce off the curve and gather at the focus.

Zeroes and the y-intercept

The zeroes are where the parabola crosses the x-axis, the solutions of ax² + bx + c = 0. The tool solves that with the discriminant, so you get two zeroes, one repeated zero, or none on the real axis, depending on whether the curve cuts through, just touches, or floats clear of the x-axis. The y-intercept, where the curve meets the y-axis, is the easiest feature of all: it is simply c, the value when x is zero. If you want just the zeroes worked through, the quadratic formula calculator focuses on those.

A worked example, step by step

Take f(x) = x² - 4x + 3, with a = 1, b = -4, c = 3.

• Vertex: x = -b / (2a) = 4/2 = 2, and y = 4 - 8 + 3 = -1, so the vertex is (2, -1).
• Axis of symmetry: the line x = 2.
• Zeroes: discriminant 16 - 12 = 4, so x = (4 ± 2)/2, giving 3 and 1.
• Y-intercept: (0, 3). Focus: 1/(4a) = 0.25 above the vertex, at (2, -0.75). Directrix: y = -1.25.

That single quadratic, fully mapped, is what the tool hands back in one go.

Questions people ask

What are the vertex and axis of symmetry?

The vertex is the turning point at x = -b / (2a). The axis of symmetry is the vertical line through it, and the parabola mirrors across that line.

What are the focus and directrix?

A fixed point and a fixed line that define the curve: every point on the parabola is the same distance from both. The focus is 1/(4a) above the vertex, the directrix the same below.

Why does a parabola gather rays at the focus?

Its reflection property: any ray coming in parallel to the axis reflects off the curve and passes through the focus. That is why dishes and reflectors are parabolic.

What are the zeroes?

The points where the parabola crosses the x-axis, found by solving the quadratic. There may be two, one, or none on the real axis.

Why can't a be zero?

Because without an x² term the graph is a straight line, not a parabola, so none of these features exist.

References

On the parabola and its features. A parabola is the set of points equidistant from a focus and a directrix, with a vertex and axis of symmetry, and it is the graph of a quadratic, so its zeroes are that quadratic's solutions.

1. Eric W. Weisstein, "Parabola," from MathWorld, a Wolfram resource, on the focus, directrix, vertex, and the naming of the curve by Apollonius.
2. Eric W. Weisstein, "Quadratic Equation," from MathWorld, a Wolfram resource, on the quadratic whose graph is the parabola and whose solutions are its zeroes.