Canonical Form Calculator

Enter Quadratic Coefficients

This calculator converts a quadratic equation in standard form, y = ax² + bx + c, into canonical form, y = a(x - h)² + k.

Coefficient a

Coefficient b

Coefficient c

Graph start x

Graph end x

Graph samples

Optional x value to evaluate

Decimal precision

Example Data Table

Standard Form	Canonical Form	Vertex	Axis	Roots
y = x² - 6x + 5	y = (x - 3)² - 4	(3, -4)	x = 3	1 and 5
y = 2x² + 8x + 6	y = 2(x + 2)² - 2	(-2, -2)	x = -2	-1 and -3
y = -x² + 4x - 1	y = -(x - 2)² + 3	(2, 3)	x = 2	2 + √3 and 2 - √3

Formula Used

Standard form: y = ax² + bx + c

Canonical form: y = a(x - h)² + k

Vertex x-coordinate: h = -b / (2a)

Vertex y-coordinate: k = c - b² / (4a)

Discriminant: Δ = b² - 4ac

Roots: x = (-b ± √Δ) / (2a)

Focus: (h, k + 1 / 4a)

Directrix: y = k - 1 / 4a

These formulas let you rewrite the parabola around its vertex, analyze symmetry, inspect roots, and view the exact graph from the same input set.

How to Use This Calculator

Enter the coefficients a, b, and c from your quadratic equation.
Set a graph range to control the visible x-values.
Choose graph samples for smoother or lighter plotting.
Optionally add a specific x-value to evaluate one point.
Select decimal precision for cleaner or more detailed output.
Press the convert button to show the result above the form.
Review the summary table, worked steps, and Plotly graph.
Use the CSV and PDF buttons to save your result.

Frequently Asked Questions

1. What is canonical form for a quadratic equation?

Canonical form writes a quadratic as y = a(x - h)² + k. It reveals the vertex immediately and makes symmetry, minimum or maximum, and graphing much easier.

2. How does the calculator find h?

It uses h = -b / (2a). This value gives the x-coordinate of the vertex and the axis of symmetry for the parabola.

3. How does the calculator find k?

After finding h, it computes k using k = c - b² / (4a). That value is the y-coordinate of the vertex.

4. Why must a not equal zero?

If a equals zero, the equation becomes linear, not quadratic. Canonical form in this calculator is specifically for parabolas.

5. Do the roots change after conversion?

No. Standard form and canonical form describe the same parabola. Only the appearance changes, while roots, vertex, and intercepts remain mathematically consistent.

6. What does the graph help me see?

The graph shows the parabola, its vertex, and any evaluated point. It helps you verify direction, width, symmetry, and whether real roots exist.

7. Can I convert a quadratic with no real roots?

Yes. Canonical form still exists even when the discriminant is negative. The graph simply does not cross the x-axis.

8. When is the vertex a minimum or maximum?

If a is positive, the parabola opens upward and the vertex is a minimum. If a is negative, the vertex is a maximum.