Find The Equation of The Following Parabola Calculator

This calculator helps you find the equation of a parabola given three points. Whether you're a student studying algebra or a professional working with quadratic functions, this tool provides a quick and accurate way to determine the standard form of a parabola.

How to Use This Calculator

To find the equation of a parabola, you need to provide three points that lie on the parabola. The calculator will use these points to determine the quadratic equation in the standard form:

y = ax² + bx + c

Here's how to use the calculator:

Enter the coordinates of three points that lie on the parabola.
Click the "Calculate" button to find the equation.
Review the result and the visualization of the parabola.
If needed, adjust the points and recalculate.

Note: The three points must not be colinear, and they must not all lie on the same vertical line. If they do, the parabola cannot be uniquely determined.

Formula Used

The equation of a parabola in standard form is:

y = ax² + bx + c

To find the coefficients a, b, and c, we use the following system of equations derived from the three given points (x₁, y₁), (x₂, y₂), and (x₃, y₃):

ax₁² + bx₁ + c = y₁ ax₂² + bx₂ + c = y₂ ax₃² + bx₃ + c = y₃

Solving this system of equations gives us the values of a, b, and c.

Worked Example

Let's find the equation of a parabola that passes through the points (1, 2), (2, 5), and (3, 10).

Using the formula:

a(1)² + b(1) + c = 2 a(2)² + b(2) + c = 5 a(3)² + b(3) + c = 10

This simplifies to:

a + b + c = 2 4a + 2b + c = 5 9a + 3b + c = 10

Solving these equations gives a = 1, b = 0, and c = 1. Therefore, the equation of the parabola is:

y = x² + 1

Frequently Asked Questions

What is the standard form of a parabola?: The standard form of a parabola is y = ax² + bx + c, where a, b, and c are constants.
Can I find the equation of a parabola with fewer than three points?: No, you need at least three points to uniquely determine the equation of a parabola.
What if the points I enter are colinear?: If the points are colinear, they lie on a straight line rather than a parabola, and the calculator will not be able to find a valid equation.
How accurate is this calculator?: The calculator uses precise mathematical calculations to determine the equation of the parabola.
Can I use this calculator for vertical parabolas?: This calculator is designed for horizontal parabolas. For vertical parabolas, you would need to use a different approach or calculator.