Graphing Parabolas

Quadratic Function

y = x² − 4x + 3

x² coeff.

x coeff.

constant

y = (x − 2)² − 1

stretch

x of vertex

y of vertex

y = (x − 1)(x − 3)

stretch

root 1

root 2

Examples

Key Features

Enter values and press Graph Parabola to see the vertex, intercepts, and all key features.

Standard Form

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Vertex (h, k)

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Axis of Symmetry

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Y-Intercept

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X-Intercepts

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Domain

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Range

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Opens

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Key Features of a Parabola

y = ax² + bx + c

Vertex (h, k) — the turning point of the parabola. Found via h = −b/(2a), k = f(h). The minimum (a > 0) or maximum (a < 0).

Axis of Symmetry — the vertical line x = h that divides the parabola into two mirror halves.

Y-Intercept — where the parabola crosses the y-axis. Always at (0, c) in standard form.

X-Intercepts (roots) — where y = 0. Found using the quadratic formula. Discriminant b²−4ac determines: positive → 2 roots, zero → 1 repeated root, negative → no real roots.

Domain: all real numbers (parabolas extend infinitely left and right). Range: y ≥ k if a > 0, or y ≤ k if a < 0.

The vertex is always exactly halfway between the two x-intercepts when they exist.

The Three Forms & When to Use Each

Choose your input form based on what information you already have:

Standard form ax²+bx+c — easiest to find the y-intercept (it's just c) and to expand from other forms. Most common form you'll see on tests.
Vertex form a(x−h)²+k — immediately reveals the vertex (h, k) and the direction. Best for graphing, finding max/min values, and completing-the-square problems.
Intercept form a(x−p)(x−q) — immediately reveals both x-intercepts p and q. Best when you know the roots, such as after factoring.

All three forms represent the same parabola — this calculator converts any of them to standard form first, then extracts every feature.

Parabolas still feel confusing?