02 // Visual Math & Finance Solversy = ax² + bx + c ⇒ x = [−b ± √(b² − 4ac)] / (2a)
Visual Quadratic Equation & Parabola Grapher ($ax^2 + bx + c$)
A quadratic function $f(x) = ax^2 + bx + c$ ($a \ne 0$) traces a symmetric parabola with vertex at $h = -\frac{b}{2a}, k = c - \frac{b^2}{4a}$ and roots given by the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
y = ax² + bx + c ⇒ x = [−b ± √(b² − 4ac)] / (2a)Inputs & Parameters (3)
-5±0.255
-12±0.512
-15±0.515
Discriminant Δ = b² − 4ac
36
2 Distinct Real Roots
Roots (Zero Crossings)
x₁ = -2, x₂ = 4
Sum = 2 | Product = -8
Vertex (h, k)
(1, -9)
Global Minimum (Opens Up)
Axis & Y-Intercept
x = 1
Y-Intercept = (0, -8)
20-Row Comparison Table
Compare across:
| # | Quadratic Coefficient (a) | Discriminant (Δ = b² − 4ac) | Vertex (h, k) | Root Nature |
|---|---|---|---|---|
| 01 | -5 | -156 | (-0.2, -7.8) | 2 Complex |
| 02 | -4.5 | -140 | (-0.22, -7.78) | 2 Complex |
| 03 | -4 | -124 | (-0.25, -7.75) | 2 Complex |
| 04 | -3.5 | -108 | (-0.29, -7.71) | 2 Complex |
| 05 | -3 | -92 | (-0.33, -7.67) | 2 Complex |
| 06 | -2.25 | -68 | (-0.44, -7.56) | 2 Complex |
| 07 | -1.75 | -52 | (-0.57, -7.43) | 2 Complex |
| 08 | -1.25 | -36 | (-0.8, -7.2) | 2 Complex |
| 09 | -0.75 | -20 | (-1.33, -6.67) | 2 Complex |
| 10 | -0.25 | -4 | (-4, -4) | 2 Complex |
| 11 | 0.25 | 12 | (4, -12) | 2 Real |
| 12 | 0.75 | 28 | (1.33, -9.33) | 2 Real |
| 13 | 1.25 | 44 | (0.8, -8.8) | 2 Real |
| 14 | 1.75 | 60 | (0.57, -8.57) | 2 Real |
| 15 | 2.25 | 76 | (0.44, -8.44) | 2 Real |
| 16 | 3 | 100 | (0.33, -8.33) | 2 Real |
| 17 | 3.5 | 116 | (0.29, -8.29) | 2 Real |
| 18 | 4 | 132 | (0.25, -8.25) | 2 Real |
| 19 | 4.5 | 148 | (0.22, -8.22) | 2 Real |
| 20 | 5 | 164 | (0.2, -8.2) | 2 Real |
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Key Analytical Takeaways
- 01Sign of $a$ controls concavity: when $a > 0$ the parabola opens upward with a global minimum; when $a < 0$ it opens downward with a global maximum.
- 02Changing $c$ shifts the entire parabola vertically without altering the axis of symmetry $x = -b/(2a)$.
- 03Both standard form $ax^2 + bx + c$ and vertex form $a(x - h)^2 + k$ are displayed side by side.
Frequently Asked Questions
What happens if coefficient 'a' is set to zero?+
When a = 0, the x² term vanishes and the curve flattens into a straight linear equation y = bx + c.
How do I convert standard form ax² + bx + c into vertex form?+
Complete the square: y = a(x − h)² + k, where h = −b / (2a) and k = f(h) = c − b² / (4a).
How do you rewrite standard form ax² + bx + c into vertex form?+
Completing the square yields y = a(x − h)² + k, where the vertex coordinates are h = −b / (2a) and k = c − b² / (4a).