Enter known points or properties of the graph to derive its equation. This calculator supports basic linear and quadratic equations. For more complex graphs, manual analysis may be required.
Equation will appear here.
Understanding the Graph to Equation Calculator
This calculator aims to help you find the mathematical equation that represents a given graph, focusing primarily on linear and quadratic functions. Understanding the relationship between a graph and its equation is fundamental in various fields, including mathematics, physics, engineering, economics, and data analysis.
How it Works (Linear Equations)
A linear equation represents a straight line on a graph. Its general form is y = mx + c, where:
m is the slope (gradient) of the line, indicating how steep the line is and its direction.
c is the y-intercept, the point where the line crosses the y-axis (when x=0).
To find the equation of a line using two points (x1, y1) and (x2, y2):
Calculate the slope (m):m = (y2 - y1) / (x2 - x1)
If x2 - x1 is zero, the line is vertical, and its equation is x = x1.
Calculate the y-intercept (c):
Once you have the slope m, you can use one of the points (x1, y1) and the slope-intercept form y = mx + c to solve for c:
c = y1 - m * x1
Form the equation:
Substitute the calculated values of m and c into y = mx + c.
How it Works (Quadratic Equations)
A quadratic equation represents a parabola on a graph. Its general form is y = ax^2 + bx + c. To determine this equation, you typically need three distinct points. The calculator uses these points to set up a system of three linear equations with three unknowns (a, b, and c).
Given three points (x1, y1), (x2, y2), and (x3, y3), we substitute them into the general quadratic form:
y1 = a*x1^2 + b*x1 + c
y2 = a*x2^2 + b*x2 + c
y3 = a*x3^2 + b*x3 + c
This system can be solved using various algebraic methods (like substitution or elimination) or matrix methods to find the values of a, b, and c.
Use Cases
Data Modeling: Fitting mathematical models to observed data points.
Engineering: Designing curves, trajectories, or control systems.
Physics: Analyzing projectile motion, forces, or energy levels.
Computer Graphics: Generating smooth curves and animations.
Education: Visualizing and understanding the relationship between algebraic expressions and geometric shapes.
Note: This calculator provides results for linear and quadratic equations. For graphs representing higher-order polynomials or other complex functions, advanced mathematical techniques or specialized software may be necessary.
function calculateEquation() {
var p1x = parseFloat(document.getElementById("point1_x").value);
var p1y = parseFloat(document.getElementById("point1_y").value);
var p2x = parseFloat(document.getElementById("point2_x").value);
var p2y = parseFloat(document.getElementById("point2_y").value);
var p3x = parseFloat(document.getElementById("point3_x").value);
var p3y = parseFloat(document.getElementById("point3_y").value);
var resultDiv = document.getElementById("result");
resultDiv.textContent = ""; // Clear previous result
var inputsValid = !isNaN(p1x) && !isNaN(p1y) && !isNaN(p2x) && !isNaN(p2y);
if (!inputsValid) {
resultDiv.textContent = "Please enter valid coordinates for at least two points.";
return;
}
// Check for vertical line first (x1 = x2)
if (p1x === p2x) {
if (p1y === p2y) {
resultDiv.textContent = "The two points are identical. Cannot determine a unique line or parabola.";
return;
}
resultDiv.textContent = "Vertical Line: x = " + p1x;
return;
}
// — Linear Equation Calculation (y = mx + c) —
var m = (p2y – p1y) / (p2x – p1x);
var c = p1y – m * p1x;
var equation = "y = " + formatNumber(m) + "x + " + formatNumber(c);
// — Quadratic Equation Calculation (y = ax^2 + bx + c) —
var inputsForQuadratic = inputsValid && !isNaN(p3x) && !isNaN(p3y);
if (inputsForQuadratic) {
// Check if the third point is collinear with the first two
var slope2 = (p3y – p2y) / (p3x – p2x);
if (!isNaN(slope2) && Math.abs(m – slope2) < 1e-9) { // Using tolerance for floating point comparison
resultDiv.textContent = "Points are collinear. The equation is linear: " + equation;
return;
}
// Solve for a, b, c using a system of equations
// y1 = a*x1^2 + b*x1 + c
// y2 = a*x2^2 + b*x2 + c
// y3 = a*x3^2 + b*x3 + c
// We can simplify by substituting c = y1 – a*x1^2 – b*x1
// and using the fact that m is slope between p1 and p2, and slope2 between p2 and p3.
// Let's use matrix or direct formula derivation for clarity.
// A common approach is solving determinant-based system.
// For simplicity here, we'll derive the system and solve:
var eq1_denom = (p1x – p2x) * (p1x – p3x) * (p2x – p3x);
if (eq1_denom === 0) {
resultDiv.textContent = "Error: Denominator is zero. Points may not form a unique parabola.";
return;
}
var a = p1x * (p2y – p3y) + p2x * (p3y – p1y) + p3x * (p1y – p2y);
a = a / eq1_denom;
var b_num = p1y * (p2x – p3x) + p2y * (p3x – p1x) + p3y * (p1x – p2x);
var b_den = (p1x – p2x) * (p2x – p3x) * (p3x – p1x);
if (b_den === 0) {
resultDiv.textContent = "Error: Denominator is zero. Points may not form a unique parabola.";
return;
}
b = b_num / b_den;
// Recalculate 'c' using one point and derived 'a' and 'b'
// c = y – ax^2 – bx
c = p1y – a * (p1x * p1x) – b * p1x;
// Adjust 'b' and 'c' calculation if needed based on common formulas
// A more stable method involves solving simultaneous equations or using matrix inversion.
// Let's re-verify 'a' and 'b' using a common formulaic approach for 3 points:
// y = ax^2 + bx + c
// Point 1: y1 = a*x1^2 + b*x1 + c
// Point 2: y2 = a*x2^2 + b*x2 + c
// Point 3: y3 = a*x3^2 + b*x3 + c
// System:
// (x1^2)a + (x1)b + c = y1
// (x2^2)a + (x2)b + c = y2
// (x3^2)a + (x3)b + c = y3
// Using Cramer's rule or equivalent:
var D = (p1x*p1x * (p2x – p3x) + p2x*p2x * (p3x – p1x) + p3x*p3x * (p1x – p2x));
if (D === 0) {
resultDiv.textContent = "Error: Determinant is zero. Points may be collinear or identical, not forming a unique parabola.";
return;
}
var Da = (p1y * (p2x – p3x) + p2y * (p3x – p1x) + p3y * (p1x – p2x));
var Db = (p1x*p1x * (p2y – p3y) + p2x*p2x * (p3y – p1y) + p3x*p3x * (p1y – p2y));
var Dc = (p1x*p1x * (p2x*p3y – p3x*p2y) + p2x*p2x * (p3x*p1y – p1x*p3y) + p3x*p3x * (p1x*p2y – p2x*p1y));
a = Da / D;
b = Db / D;
c = Dc / D;
equation = "y = " + formatNumber(a) + "x^2 + " + formatNumber(b) + "x + " + formatNumber(c);
resultDiv.textContent = "Quadratic Equation: " + equation;
} else {
// Only two points were provided, assume linear
resultDiv.textContent = "Linear Equation: " + equation;
}
}
function formatNumber(num) {
if (isNaN(num)) return "NaN";
// Format to a reasonable precision, avoiding trailing zeros unless necessary
if (Math.abs(num – Math.round(num)) < 1e-9) {
return Math.round(num).toString();
}
return parseFloat(num.toFixed(5)).toString(); // Limit to 5 decimal places, then convert to string to strip trailing zeros
}