Find Equation Of Curve From Points Calculator

This calculator finds the equation of a quadratic curve (a parabola of the form y = ax² + bx + c) that passes exactly through three given points. Enter the coordinates of the three points below.

Curve Equation Calculator

Data Table and Curve Plot

Point	Input X	Input Y	Y from Equation
1	–	–	–
2	–	–	–
3	–	–	–

Table showing input points and Y-values calculated from the derived equation.

Plot of the input points and the calculated quadratic curve.

What is a Find Equation of Curve from Points Calculator?

A "find equation of curve from points calculator" is a tool that determines the mathematical equation of a curve that best fits or passes exactly through a given set of data points. Specifically, our calculator focuses on finding the equation of a quadratic curve (a parabola) of the form y = ax² + bx + c that passes precisely through three distinct points provided by the user. If the three points are not collinear and have distinct x-values, a unique quadratic equation can be found.

This process is a form of polynomial interpolation, where we find a polynomial of a specific degree (in this case, degree 2) that goes through the specified points. Engineers, scientists, data analysts, and students often use such calculators to model relationships between variables based on observed data.

Common misconceptions include thinking that *any* set of points will yield a simple curve or that a quadratic is always the best fit. For more than three points, a quadratic might not pass through all of them, and a higher-degree polynomial or other curve fitting methods might be needed, which this specific find equation of curve from points calculator for quadratics doesn't address.

Find Equation of Curve from Points Formula and Mathematical Explanation

To find the equation of a quadratic curve y = ax² + bx + c that passes through three points (x₁, y₁), (x₂, y₂), and (x₃, y₃), we substitute these points into the equation, creating a system of three linear equations with three unknowns (a, b, c):

1. ax₁² + bx₁ + c = y₁
2. ax₂² + bx₂ + c = y₂
3. ax₃² + bx₃ + c = y₃

This system can be represented in matrix form and solved using methods like Cramer's Rule or Gaussian elimination. Using Cramer's Rule, we calculate the following determinants:

D = | x₁² x₁ 1 |
| x₂² x₂ 1 |
| x₃² x₃ 1 | = x₁²(x₂ – x₃) – x₁(x₂² – x₃²) + (x₂²x₃ – x₃²x₂)

Da = | y₁ x₁ 1 |
| y₂ x₂ 1 |
| y₃ x₃ 1 | = y₁(x₂ – x₃) – x₁(y₂ – y₃) + (y₂x₃ – y₃x₂)

Db = | x₁² y₁ 1 |
| x₂² y₂ 1 |
| x₃² y₃ 1 | = x₁²(y₂ – y₃) – y₁(x₂² – x₃²) + (x₂²y₃ – x₃²y₂)

Dc = | x₁² x₁ y₁ |
| x₂² x₂ y₂ |
| x₃² x₃ y₃ | = x₁²(x₂y₃ – x₃y₂) – x₁(x₂²y₃ – x₃²y₂) + y₁(x₂²x₃ – x₃²x₂)

The coefficients are then found by: a = Da / D, b = Db / D, c = Dc / D, provided D ≠ 0. If D = 0, the points are collinear, or the x-values are not distinct enough to define a unique quadratic, and a quadratic passing through them might not be unique or might not exist in the standard form (it could degenerate to a line if collinear).

Variable	Meaning	Unit	Typical Range
x₁, y₁	Coordinates of the first point	Varies	Any real numbers
x₂, y₂	Coordinates of the second point	Varies	Any real numbers
x₃, y₃	Coordinates of the third point	Varies	Any real numbers
a, b, c	Coefficients of the quadratic equation y = ax² + bx + c	Varies	Any real numbers
D	Determinant of the coefficient matrix	Varies	Any real number

Variables used in finding the equation of the curve.

Practical Examples (Real-World Use Cases)

Example 1: Projectile Motion

An object is thrown, and its height is measured at three different times: at 1 second, height is 30 meters; at 2 seconds, 40 meters; at 3 seconds, 30 meters. We want to find the quadratic equation modeling its height (y) over time (x).

• Point 1: (1, 30)
• Point 2: (2, 40)
• Point 3: (3, 30)

Using the find equation of curve from points calculator, we input these values. The calculator would solve for a, b, and c, giving an equation like y = -10x² + 40x + 0. This suggests the initial upward velocity and the effect of gravity.

Example 2: Cost Analysis

A company finds that producing 10 units costs $300, 20 units cost $400, and 30 units cost $700. They suspect the cost function might be quadratic over this range.

• Point 1: (10, 300)
• Point 2: (20, 400)
• Point 3: (30, 700)

Plugging these into the find equation of curve from points calculator, we get a quadratic cost function y = ax² + bx + c, which can help estimate costs for other production levels within this range.

How to Use This Find Equation of Curve from Points Calculator

1. Enter Point 1: Input the x and y coordinates (x1, y1) of the first point into the designated fields.
2. Enter Point 2: Input the x and y coordinates (x2, y2) of the second point.
3. Enter Point 3: Input the x and y coordinates (x3, y3) of the third point.
4. View Results: The calculator automatically updates and displays the equation y = ax² + bx + c, along with the values of a, b, c, and the determinant D.
5. Interpret the Equation: The primary result shows the quadratic equation that passes through your three points.
6. Check Determinant: If D is very close to 0, the points are nearly collinear, and the quadratic fit might be unstable or degenerate to a line.
7. Examine Table and Plot: The table confirms the input points and shows the y-values calculated by the equation. The chart visually represents the points and the curve.
8. Reset or Copy: Use the "Reset" button to clear inputs to default values or "Copy Results" to copy the equation and coefficients.

Key Factors That Affect Find Equation of Curve from Points Results

• Distinct X-values: The x-coordinates of the three points should ideally be distinct. If two x-values are the same but y-values differ, no function can pass through them. If x-values are identical, and y-values are too, the points are not distinct for finding a unique quadratic. Our calculator assumes distinct x-values for a stable quadratic.
• Collinearity of Points: If the three points lie on a straight line, the determinant D will be zero or very close to it. In this case, 'a' will be zero (or very small), and the equation degenerates to a linear one (y = bx + c), or the results might be unstable if D is very near zero due to precision.
• Magnitude of Coordinates: Very large or very small coordinate values can lead to large or small coefficients, potentially causing precision issues in calculations, though the calculator attempts to manage this.
• Measurement Errors: If the input points are from real-world measurements with errors, the calculated curve will pass exactly through these (possibly erroneous) points, which might not represent the true underlying relationship accurately. For noisy data, curve fitting (like least squares) is often preferred over exact interpolation through three points.
• Choice of Curve Degree: This find equation of curve from points calculator specifically finds a quadratic (degree 2). If the underlying relationship is linear, cubic, or something else, a quadratic might be a poor fit for the overall trend, even if it passes through the three chosen points.
• Extrapolation: The derived quadratic equation is most reliable *between* the given x-values (interpolation). Using it to predict y-values far outside the range of x1, x2, and x3 (extrapolation) can be highly inaccurate.

Frequently Asked Questions (FAQ)

What if my three points lie on a straight line?

The calculator will find a determinant D very close to zero, and the coefficient 'a' will be near zero, resulting in an equation very close to a linear one (y ≈ bx + c). The primary result might show 'a' as a very small number or indicate near-collinearity.

Can I use this calculator for more than three points?

No, this specific find equation of curve from points calculator is designed for exactly three points to find a unique quadratic equation. For more points, you'd typically look for a "best fit" curve using methods like least squares regression or higher-degree polynomial interpolation.

What if two of my points have the same x-coordinate?

If two points have the same x-coordinate but different y-coordinates, no single-valued function y=f(x) (including a quadratic) can pass through them. If they have the same x and y, they are the same point, and you effectively have only two distinct points, not enough for a unique quadratic.

What does it mean if the determinant D is zero?

If D=0, the three points are collinear (lie on a straight line) or the x-values are not distinct in a way that allows a unique quadratic solution (though our input structure encourages distinct x's if they are different points). A unique quadratic passing through three collinear points doesn't exist; it degenerates to a line.

Why a quadratic equation?

A quadratic equation (y = ax² + bx + c) is the simplest polynomial that can be made to pass through any three non-collinear points with distinct x-values. It represents a parabola.

Can I find a cubic equation with this calculator?

No, to find a unique cubic equation (y = ax³ + bx² + cx + d), you would need four distinct points. This tool is specifically for quadratics from three points.

How accurate is the result?

The calculator performs the mathematical operations with standard floating-point precision. The accuracy of the equation in representing a real-world phenomenon depends on how well a quadratic model fits your data and the accuracy of your input points.

What if my points form a very steep curve?

The coefficients 'a', 'b', or 'c' might become very large or small, but the calculator should handle it. The visual plot will adjust to show the curve, though extreme steepness might affect visual scaling.