Find Equation from 3 Points Calculator
Instantly find the quadratic or linear equation that passes through three given points with our find equation from 3 points calculator.
Equation Calculator
Graph of the equation passing through the three points.
| Point | X | Y (Input) | Y (Calculated) |
|---|---|---|---|
| 1 | 1 | 1 | – |
| 2 | 2 | 4 | – |
| 3 | 3 | 9 | – |
Input points and calculated y-values from the equation.
What is a Find Equation from 3 Points Calculator?
A "find equation from 3 points calculator" is a tool used to determine the equation of a curve, typically a quadratic (parabola) of the form y = ax² + bx + c, that passes exactly through three given points (x1, y1), (x2, y2), and (x3, y3) in a Cartesian coordinate system. If the three points happen to be collinear (lie on a straight line), the calculator will instead find the equation of the line y = mx + c passing through them.
This calculator is useful for students learning algebra, engineers, scientists, and anyone needing to model a relationship between two variables based on three data points. It essentially solves a system of three linear equations (derived by substituting the coordinates of the points into the general form of the equation) to find the coefficients a, b, and c (or m and c for a line).
Common misconceptions include believing that any three points will always define a unique parabola or that the calculator can find more complex curves. The standard find equation from 3 points calculator focuses on quadratic or linear relationships.
Find Equation from 3 Points Formula and Mathematical Explanation
Given three points (x1, y1), (x2, y2), and (x3, y3), we assume they lie on a parabola y = ax² + bx + c. Substituting each point into the equation gives us a system of three linear equations with three unknowns (a, b, c):
y1 = ax1² + bx1 + cy2 = ax2² + bx2 + cy3 = ax3² + bx3 + c
To solve for a, b, and c, we can use methods like substitution, elimination, or matrix methods (like Cramer's rule).
The determinant of the coefficient matrix for x², x, and 1 columns is D = x1²(x2-x3) + x2²(x3-x1) + x3²(x1-x2) = (x1-x2)(x1-x3)(x2-x3). If D is not zero, a unique quadratic solution exists.
The coefficients are found as:
a = (x1(y3-y2) + x2(y1-y3) + x3(y2-y1)) / ((x1-x2)(x1-x3)(x2-x3))b = (y1(x2²-x3²) + y2(x3²-x1²) + y3(x1²-x2²)) / ((x1-x2)(x1-x3)(x2-x3))c = (y1*x2*x3(x3-x2) + y2*x1*x3(x1-x3) + y3*x1*x2(x2-x1)) / ((x1-x2)(x1-x3)(x2-x3))
If the denominator (x1-x2)(x1-x3)(x2-x3) is zero, it means at least two x-values are the same, or the points are collinear. If the points are collinear (and not on a vertical line), a=0, and the equation becomes linear: y = mx + c, where m = (y2-y1)/(x2-x1) (if x1≠x2) and c = y1 - m*x1.
The calculator first checks for collinearity or identical x-values to determine whether to find a quadratic or linear equation, or if no unique function y=f(x) passes through them.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| x1, y1 | Coordinates of the first point | (as input) | Real numbers |
| x2, y2 | Coordinates of the second point | (as input) | Real numbers |
| x3, y3 | Coordinates of the third point | (as input) | Real numbers |
| a, b, c | Coefficients of the quadratic equation y = ax² + bx + c | (derived) | Real numbers |
| m, c | Slope and y-intercept of the linear equation y = mx + c | (derived) | Real numbers |
Explanation of variables in the equation finding process.
Practical Examples (Real-World Use Cases)
Example 1: Projectile Motion
Suppose an object is thrown, and its height (y) is recorded at three different times (x): (1 second, 5 meters), (2 seconds, 8 meters), (3 seconds, 9 meters). We use the find equation from 3 points calculator with points (1, 5), (2, 8), (3, 9).
Input: x1=1, y1=5; x2=2, y2=8; x3=3, y3=9
The calculator finds a=-1, b=6, c=0. The equation is y = -x² + 6x, modeling the height over time (ignoring air resistance, this is a simplified model).
Example 2: Simple Growth Curve
A plant's height (y) is measured over three weeks (x): (0 weeks, 2 cm), (1 week, 5 cm), (2 weeks, 10 cm). We input points (0, 2), (1, 5), (2, 10).
Input: x1=0, y1=2; x2=1, y2=5; x3=2, y3=10
The find equation from 3 points calculator determines a=1, b=2, c=2. The equation is y = x² + 2x + 2, representing the growth over these initial weeks.
Example 3: Collinear Points
If we have points (1, 2), (2, 4), (3, 6).
Input: x1=1, y1=2; x2=2, y2=4; x3=3, y3=6
The calculator will find a=0 and give the linear equation y = 2x + 0, or simply `y = 2x`.
How to Use This Find Equation from 3 Points Calculator
- Enter Point Coordinates: Input the x and y coordinates for each of the three points (x1, y1), (x2, y2), and (x3, y3) into the respective fields.
- Calculate: Click the "Calculate Equation" button (or the calculator will update automatically as you type if real-time updates are enabled).
- View Results: The calculator will display the primary result, which is the equation (either quadratic
y = ax² + bx + cor lineary = mx + c) that passes through the points. - Intermediate Values: It will also show the calculated coefficients (a, b, c or m, c).
- Graph and Table: A graph visualizing the points and the resulting equation, along with a table of input and calculated y-values, will be shown.
- Interpret: If the points are collinear, a linear equation is given. If two x-values are the same but y-values differ, no function y=f(x) passes through them, and an error or message will be shown.
Use the "Reset" button to clear inputs and "Copy Results" to copy the equation and coefficients.
Key Factors That Affect Find Equation from 3 Points Results
- Coordinates of the Points: The most direct factor. The specific x and y values determine the coefficients a, b, c or m, c.
- Collinearity of Points: If the three points lie on a straight line, the coefficient 'a' will be zero, resulting in a linear equation instead of a quadratic one. Our find equation from 3 points calculator detects this.
- Distinctness of X-values: If two or more points have the same x-coordinate but different y-coordinates, a function y=f(x) (and thus a standard parabola y=ax²+bx+c) cannot pass through them. The calculator should handle this. If all x are the same and y are different, it's a vertical line x=constant, not y=f(x).
- Precision of Input: Small changes in the input coordinates, especially if the points are close together or nearly collinear, can lead to significant changes in the coefficients of the equation, particularly 'a'.
- Scale of Coordinates: Very large or very small coordinate values might lead to very large or small coefficients, which could be sensitive to rounding in calculations.
- Whether a Quadratic is Expected: If you know the underlying relationship is linear, but the three points are not perfectly collinear due to measurement error, fitting a quadratic might not be the best model, though the find equation from 3 points calculator will provide the exact quadratic through those points.
Frequently Asked Questions (FAQ) about the Find Equation from 3 Points Calculator
The find equation from 3 points calculator will recognize this (the coefficient 'a' will be 0) and provide the equation of the straight line y = mx + c passing through them.
If two points have the same x-coordinate but different y-coordinates (e.g., (2,3) and (2,5)), no function of the form y=f(x) can pass through both, as it would violate the definition of a function. The calculator will indicate this or might default to a line if the third point also has the same x or other conditions are met.
If all three points are identical, there are infinitely many parabolas (and lines) that can pass through that single point. The calculator might give an error or indicate non-uniqueness.
No, this specific find equation from 3 points calculator is designed to find either a quadratic (parabola) or a linear equation, as three points uniquely define a parabola or a line (unless x-values are repeated for different y's, or points are identical).
The 'a' coefficient in y = ax² + bx + c determines the "width" and direction of the parabola. If a=0, it's not a parabola but a line.
The calculator uses standard mathematical formulas and is accurate for the given inputs. Accuracy depends on the precision of the input coordinates and the computational precision used.
The denominator (x1-x2)(x1-x3)(x2-x3) being zero indicates that at least two x-values are the same. If the corresponding y-values are different, no function y=f(x) passes through. If the y-values are also the same, the points are not distinct, or if all three x are the same, it's a vertical line case (if y's differ).
Yes, if you have exactly three data points and believe a quadratic or linear relationship is appropriate. For more than three points, you'd typically use regression methods (like least squares) to find the best-fit curve, which might not pass exactly through all points. This tool finds the curve *exactly* through three points.