Calculate the slope from two points or a linear equation
Method 1: From Two Points (x, y)
Enter the coordinates for two points on the line.
Method 2: From Equation (Ax + By = C)
Enter the coefficients for a linear equation in standard form.
Calculation Result:
Understanding Rate of Change
The rate of change is a mathematical concept that describes how one quantity changes in relation to another. In the context of a linear equation, it is synonymous with the slope (m).
The Rate of Change Formula
When you have two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, the average rate of change is calculated using the formula:
m = (y₂ – y₁) / (x₂ – x₁)
Rate of Change from an Equation
Linear equations are often written in different forms. Identifying the rate of change depends on the format:
Slope-Intercept Form ($y = mx + b$): The rate of change is simply the coefficient m.
Standard Form ($Ax + By = C$): To find the rate of change, you must isolate y. This yields $y = (-A/B)x + C/B$. Therefore, the rate of change is -A/B.
Point-Slope Form ($y – y_1 = m(x – x_1)$): The rate of change is the value m outside the parentheses.
Real-World Example
Imagine a car traveling at a constant speed. If the car is at mile marker 50 ($y_1$) at 1:00 PM ($x_1$) and mile marker 170 ($y_2$) at 3:00 PM ($x_2$):
Change in Position (Rise): 170 – 50 = 120 miles
Change in Time (Run): 3 – 1 = 2 hours
Rate of Change: 120 / 2 = 60 miles per hour
function calculateTwoPoints() {
var x1 = parseFloat(document.getElementById('x1').value);
var y1 = parseFloat(document.getElementById('y1').value);
var x2 = parseFloat(document.getElementById('x2').value);
var y2 = parseFloat(document.getElementById('y2').value);
var resultBox = document.getElementById('result-box');
var resultText = document.getElementById('result-text');
var formulaText = document.getElementById('formula-text');
if (isNaN(x1) || isNaN(y1) || isNaN(x2) || isNaN(y2)) {
alert("Please enter valid numbers for all point coordinates.");
return;
}
if (x2 – x1 === 0) {
resultBox.style.display = 'block';
resultText.innerHTML = "Rate of Change: Undefined (Vertical Line)";
formulaText.innerHTML = "Reason: Division by zero (x2 – x1 = 0)";
return;
}
var slope = (y2 – y1) / (x2 – x1);
resultBox.style.display = 'block';
resultBox.style.borderLeftColor = '#3498db';
resultText.innerHTML = "Rate of Change (m) = " + slope.toFixed(4).replace(/\.?0+$/, "");
formulaText.innerHTML = "Formula used: (" + y2 + " – " + y1 + ") / (" + x2 + " – " + x1 + ")";
}
function calculateFromEquation() {
var a = parseFloat(document.getElementById('coeffA').value);
var b = parseFloat(document.getElementById('coeffB').value);
var c = parseFloat(document.getElementById('coeffC').value);
var resultBox = document.getElementById('result-box');
var resultText = document.getElementById('result-text');
var formulaText = document.getElementById('formula-text');
if (isNaN(a) || isNaN(b)) {
alert("Please enter valid numbers for coefficients A and B.");
return;
}
if (b === 0) {
resultBox.style.display = 'block';
resultText.innerHTML = "Rate of Change: Undefined";
formulaText.innerHTML = "Since B = 0, the equation describes a vertical line where x = C/A.";
return;
}
var slope = -a / b;
resultBox.style.display = 'block';
resultBox.style.borderLeftColor = '#27ae60';
resultText.innerHTML = "Rate of Change (m) = " + slope.toFixed(4).replace(/\.?0+$/, "");
formulaText.innerHTML = "Calculated from Standard Form Ax + By = C: m = -A/B = -" + a + " / " + b;
}