Quadratic Equation Given Three Points Calculator

Calculator Inputs

Example Data Table

Example	Point 1	Point 2	Point 3	Expected Equation
Simple parabola	(1, 6)	(2, 11)	(3, 18)	y = x² + 2x + 3
Downward curve	(-1, 1)	(0, 4)	(2, 2)	y = -x² + 2x + 4
Linear special case	(0, 2)	(1, 5)	(2, 8)	y = 3x + 2

Formula Used

The calculator builds the equation y = ax² + bx + c from three ordered pairs.

It uses Lagrange interpolation:

y = y1[(x - x2)(x - x3)] / [(x1 - x2)(x1 - x3)]
  + y2[(x - x1)(x - x3)] / [(x2 - x1)(x2 - x3)]
  + y3[(x - x1)(x - x2)] / [(x3 - x1)(x3 - x2)]

After expansion, the matching coefficients are:

a = y1/[(x1-x2)(x1-x3)] + y2/[(x2-x1)(x2-x3)] + y3/[(x3-x1)(x3-x2)]
b = -y1(x2+x3)/[(x1-x2)(x1-x3)] - y2(x1+x3)/[(x2-x1)(x2-x3)] - y3(x1+x2)/[(x3-x1)(x3-x2)]
c = y1x2x3/[(x1-x2)(x1-x3)] + y2x1x3/[(x2-x1)(x2-x3)] + y3x1x2/[(x3-x1)(x3-x2)]

The vertex is found with x = -b / 2a. The discriminant is b² - 4ac.

How to Use This Calculator

Enter the x and y values for three points.
Make sure each x value is different.
Enter an optional x value when you need a predicted y value.
Choose the decimal precision for rounded output.
Press the calculate button.
Review the equation, coefficients, vertex, roots, and residual table.
Use the CSV or PDF buttons to save your result.

Understanding the Three Point Quadratic Method

A quadratic equation can be built when three points are known. The points must have different x values. This calculator uses those points to form y = ax² + bx + c. It then solves the coefficients and reports useful curve features.

Why Three Points Matter

A parabola has three unknown coefficients. One point gives one equation. Three points give three equations. When the x values are unique, the system has one matching curve. This makes the method useful in algebra, physics modeling, projectile examples, trend fitting, and classroom graph work.

What the Calculator Checks

The tool first checks whether any x values repeat. Repeated x values can break the function rule or make the coefficient system invalid. It also reports whether the quadratic term is effectively zero. In that case, the three points may form a straight line instead of a curved parabola.

Result Details

After calculation, the result shows a clear equation. It also gives the vertex, axis of symmetry, discriminant, roots, y intercept, focus, directrix, and opening direction when possible. These values help you understand both the algebra and the shape of the graph.

Using Results Carefully

Exact decimals depend on rounding settings. More decimal places give more detail, but may make the equation harder to read. For homework, use enough precision to match your required answer format. For engineering or data work, keep extra precision and verify the model with residual checks.

Export and Review

CSV export is useful for spreadsheets and records. PDF export is useful for sharing a compact report. The example table shows how sample points become a known equation. You can change the values, calculate again, and compare the coefficients to your own manual solution.

Best Practice

Choose points from the same curve whenever possible. Avoid very close x values when measurements contain noise. Small measurement errors can strongly affect the coefficients. Review the predicted y value at each original point. A correct exact model should return the same y values within rounding tolerance.

Common Use Cases

Students use this method for graphing practice. Analysts use it for simple curved trends. Physics learners use it for motion paths. Builders can model arches, slopes, and other smooth project shapes quickly.

FAQs

1. What does this calculator find?

It finds the equation y = ax² + bx + c that passes through three given points, when the x values are different.

2. Why must the x values be different?

A normal function cannot have two different y values for the same x value. Repeated x values also make the coefficient formula invalid.

3. Can the answer become a straight line?

Yes. If the calculated a coefficient is zero or nearly zero, the points form a linear equation instead of a true parabola.

4. What is the vertex?

The vertex is the turning point of the parabola. It is the lowest point for an upward curve and the highest point for a downward curve.

5. What does the discriminant show?

The discriminant shows root type. A positive value gives two real roots. Zero gives one repeated root. A negative value gives complex roots.

6. What is a residual check?

A residual compares the original y value with the predicted y value. Exact input points should have residuals near zero after rounding.

7. Can I use decimal points?

Yes. You can enter whole numbers, decimals, negative values, and mixed signs for x and y values.

8. What can I export?

You can export the equation, coefficients, important features, prediction value, and residual check as a CSV file or PDF report.