Calculator Input
Example Data Table
| Point 1 | Point 2 | Point 3 | Equation | Vertex |
|---|---|---|---|---|
| (0, 1) | (1, 4) | (2, 9) | y = x² + 2x + 1 | (-1, 0) |
| (-1, 6) | (0, 3) | (2, 7) | y = 2x² - x + 3 | (0.25, 2.875) |
Formula Used
The calculator fits the vertical parabola y = ax² + bx + c.
For three points, it solves this system:
y1 = ax1² + bx1 + c
y2 = ax2² + bx2 + c
y3 = ax3² + bx3 + c
Then it calculates h = -b / 2a and k = f(h).
The focus is (h, k + 1 / 4a). The directrix is y = k - 1 / 4a.
How to Use This Calculator
- Enter three points from the curve.
- Make sure all x-values are different.
- Select the decimal precision.
- Choose a graph range around the vertex.
- Press the calculate button.
- Review the equation, vertex, roots, focus, and directrix.
- Use CSV or PDF buttons to save results.
Parabola From Points Guide
What This Tool Solves
A parabola from points calculator finds the equation of a curve from three coordinate pairs. It works with vertical parabolas. These curves use the form y equals ax squared plus bx plus c. The three points must have different x-values. This gives enough information to solve the three unknowns. The calculator returns a complete curve model. It also gives helpful geometry values.
Why Three Points Are Needed
A quadratic equation has three coefficients. These are a, b, and c. Each point creates one equation. Three points create three equations. Solving those equations gives one possible parabola. When the x-values repeat, the system fails. The calculator checks this before solving.
Understanding the Output
The standard form is useful for direct substitution. The vertex form shows the turning point. The vertex is the lowest point for an upward curve. It is the highest point for a downward curve. The axis of symmetry passes through the vertex. It divides the curve into matching halves. The focus and directrix describe the geometric definition of the parabola.
Practical Uses
This calculator is useful for modeling paths, charts, trend curves, and classroom examples. It can help with projectile shapes, design curves, and algebra problems. The graph makes the result easier to inspect. The export tools help keep records. Use the example table to test the page quickly.
FAQs
1. What does this calculator find?
It finds a vertical parabola equation from three coordinate points. It also gives vertex, roots, focus, directrix, symmetry axis, and graph values.
2. Why do x-values need to be different?
Repeated x-values can break the equation system. A vertical parabola gives only one y-value for each x-value.
3. Can it solve sideways parabolas?
No. This version solves vertical parabolas only. Sideways parabolas require a different model using x as a function of y.
4. What is vertex form?
Vertex form is y = a(x - h)² + k. It shows the vertex directly as the point (h, k).
5. What does the focus mean?
The focus is a fixed point used in the geometric definition of a parabola. It helps describe the curve’s shape.
6. What is the directrix?
The directrix is a fixed line. Every point on the parabola is equally distant from the focus and the directrix.
7. Why are CSV and PDF exports useful?
CSV helps with spreadsheets. PDF helps with reports, assignments, records, and easy sharing.
8. Can the graph range be changed?
Yes. Increase the range to view a wider curve. Reduce it to inspect the vertex area closely.