Quadratic Function Visualizer

Visualize and analyze quadratic functions interactively

Quick answer

Quadratic Function Visualizer

Quadratic functions form parabolas. The coefficient 'a' determines the direction and width, 'b' affects the horizontal position, and 'c' is the y-intercept.

Formula

y = ax² + bx + c

📊 Properties

Equation
y = x²
Vertex
(0.00, 0.00)
Axis of Symmetry
x = 0.00
Y-Intercept
(0, 0)
Direction
Opens Upward ⬆️

🔍 Roots Analysis

Discriminant
Δ = 0.00
One real root
X-Intercepts (Roots)
x = 0.00

Try These Examples

📈Formula

y = ax² + bx + c

💡How it works

Quadratic functions form parabolas. The coefficient 'a' determines the direction and width, 'b' affects the horizontal position, and 'c' is the y-intercept.

ℹ️ What is Visual Quadratic Solver?

A visual quadratic solver shows the parabola of ax² + bx + c = 0 alongside the algebraic solution, making it easier to understand how the equation's roots correspond to where the parabola crosses the x-axis. It bridges the gap between algebraic manipulation and geometric intuition.

📐 Formula

y = ax² + bx + c | Vertex: (−b/2a, c − b²/4a)
a— Coefficient of x² — determines parabola opening direction and width
b— Coefficient of x — determines horizontal position of vertex
c— Constant term — y-intercept of the parabola
Vertex— Minimum (a>0) or maximum (a<0) point of the parabola

✏️ Worked Example

a: 1
b: -2
c: -3
  1. 1Vertex x = −(−2)/(2×1) = 1
  2. 2Vertex y = 1² − 2(1) − 3 = −4 → vertex at (1, −4)
  3. 3Discriminant = (−2)² − 4(1)(−3) = 4 + 12 = 16
  4. 4Roots = (2 ± 4) / 2 → x = 3 or x = −1
  5. 5y-intercept: set x=0 → y = −3
✅ Result: Parabola opens up, vertex (1,−4), roots at x=3 and x=−1

💡 How to Interpret Results

  • ▸If a > 0: parabola opens UP (U-shape) — vertex is a minimum.
  • ▸If a < 0: parabola opens DOWN (∩-shape) — vertex is a maximum.
  • ▸The roots are symmetric about the vertex x-coordinate (axis of symmetry).
  • ▸The vertex y-value is the minimum or maximum value of the quadratic.
  • ▸Changing c shifts the parabola vertically; changing b shifts the vertex horizontally.

❓ Frequently Asked Questions

Related tools for this workflow

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