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📘 About the Quadratic Grapher Calculator
A quadratic function y = ax² + bx + c draws a parabola. Its vertex is the highest or lowest point, the axis of symmetry passes through it, and the roots are where it crosses the x-axis. Parabolas model projectiles, profit curves, satellite dishes and bridges.
Enter a, b and c. The calculator draws the graph and gives the vertex (−b/2a, f(−b/2a)), the direction it opens, the discriminant and roots, the y-intercept, the vertex and factored forms, the focus and directrix, the range and a table of values. Fractions and square roots are kept exact.
🧮 Formula
✏️ Worked examples
y = x² − 4x + 3
You get| Equation | y = x² − 4x + 3 |
| Vertex | (2, −1) |
| Opens | Upwards (∪) — the vertex is a minimum |
| Axis of symmetry | x = 2 |
| y-intercept | (0, 3) |
| Discriminant b² − 4ac | 4 |
| x-intercepts (roots) | x = 1 and x = 3 |
| Vertex form | y = (x − 2)² − 1 |
| Factored form | y = (x − 1)(x − 3) |
| Focus | (2, −3/4) |
| Directrix | y = −5/4 |
| Range | y ≥ −1 |
Vertex (2, −1).
y = 2x² + 3x + 5 (no real roots)
You get| Equation | y = 2x² + 3x + 5 |
| Vertex | (−3/4, 31/8) ≈ (−0.75, 3.875) |
| Opens | Upwards (∪) — the vertex is a minimum |
| Axis of symmetry | x = −3/4 |
| y-intercept | (0, 5) |
| Discriminant b² − 4ac | −31 |
| x-intercepts (roots) | None — the parabola does not cross the x-axis |
| Vertex form | y = 2(x + 3/4)² + 31/8 |
| Focus | (−3/4, 4) |
| Directrix | y = 15/4 |
| Range | y ≥ 31/8 |
Vertex (−3/4, 31/8) ≈ (−0.75, 3.875).
🧭 How to use this calculator
What you enter
- a (x² coefficient)
- b (x coefficient)
- c (constant)
What you get
- Equation
- Vertex
- Opens
- Axis of symmetry
- y-intercept
- Discriminant b² − 4ac
- x-intercepts (roots)
- Vertex form
- Factored form
- Focus
- Directrix
- Range
❓ Frequently asked questions
How do I find the vertex of a parabola?
x = −b/(2a); substitute it to get y. For y = x² − 4x + 3 the vertex is (2, −1).
What does the discriminant tell me?
b² − 4ac > 0: two roots; = 0: one repeated root (the vertex touches the x-axis); < 0: no real roots.
How do I know which way it opens?
Upward (a minimum) if a > 0, downward (a maximum) if a < 0.
