📐 Vertex Form Calculator
Convert between vertex form and standard form
Vertex Form: y = a(x - h)² + k
Cannot be zero
How to Use This Calculator
Enter Coefficient a
Input the coefficient a in y = a(x - h)² + k. a cannot be zero (not a parabola if a = 0).
Enter Vertex Coordinates
Input h (vertex x-coordinate) and k (vertex y-coordinate). The vertex is at (h, k).
Calculate
Press "Convert to Standard Form" to get the standard form y = ax² + bx + c.
Formula
Vertex Form: y = a(x - h)² + k
Standard Form: y = ax² + bx + c
where b = -2ah and c = ah² + k
Example: y = 2(x - 3)² + 5
a = 2, h = 3, k = 5
b = -2 × 2 × 3 = -12
c = 2 × 3² + 5 = 18 + 5 = 23
Standard Form: y = 2x² - 12x + 23
Vertex: (3, 5)
About Vertex Form Calculator
The Vertex Form Calculator converts parabolas from vertex form y = a(x - h)² + k to standard form y = ax² + bx + c. Vertex form reveals the vertex (h, k) directly.
When to Use This Calculator
- Algebra: Convert between parabola equation forms
- Mathematics: Find vertex and standard form of parabolas
- Education: Learn parabola transformations
Frequently Asked Questions
What is vertex form?
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex. It directly shows the vertex coordinates, axis of symmetry (x = h), and direction (upward if a > 0, downward if a < 0).
How do I convert vertex form to standard form?
Expand: y = a(x - h)² + k = a(x² - 2hx + h²) + k = ax² - 2ahx + ah² + k. So b = -2ah and c = ah² + k.
What does the vertex represent?
The vertex (h, k) is the parabola's turning point: minimum if a > 0, maximum if a < 0. It's where the parabola changes direction.