The rate of change describes how one quantity changes in relation to another. When working with a table of values, the rate of change is the ratio of the change in the dependent variable (usually represented as y) to the change in the independent variable (usually x).
The Rate of Change Formula
To calculate the rate of change from a table, you select any two rows (points) from the table and use the slope formula:
Rate of Change = (y₂ – y₁) / (x₂ – x₁)
How to Find Rate of Change from a Table
Select Two Points: Pick any two sets of (x, y) values from your table. Let's call them (x₁, y₁) and (x₂, y₂).
Find the Change in Y: Subtract the first y-value from the second y-value (y₂ – y₁). This is often called the "rise."
Find the Change in X: Subtract the first x-value from the second x-value (x₂ – x₁). This is often called the "run."
Divide: Divide the change in y by the change in x.
Real-World Example
Imagine a table showing the distance a car travels over time:
Time (Hours)
Distance (Miles)
2
120
5
300
Using the points (2, 120) and (5, 300):
Δy: 300 – 120 = 180
Δx: 5 – 2 = 3
Rate of Change: 180 / 3 = 60 miles per hour.
function calculateRateOfChange() {
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 resultArea = document.getElementById('result-area');
var rateOutput = document.getElementById('rate-output');
var steps = document.getElementById('step-by-step');
if (isNaN(x1) || isNaN(y1) || isNaN(x2) || isNaN(y2)) {
alert("Please enter valid numbers for all fields.");
return;
}
if (x1 === x2) {
alert("The x-values cannot be the same (division by zero).");
return;
}
var deltaY = y2 – y1;
var deltaX = x2 – x1;
var rate = deltaY / deltaX;
resultArea.style.display = 'block';
rateOutput.innerHTML = "Rate of Change = " + rate.toLocaleString(undefined, {maximumFractionDigits: 4}) + "";
steps.innerHTML = "Step-by-step Calculation:" +
"1. Change in y (Δy) = " + y2 + " – " + y1 + " = " + deltaY + "" +
"2. Change in x (Δx) = " + x2 + " – " + x1 + " = " + deltaX + "" +
"3. Rate = Δy / Δx = " + deltaY + " / " + deltaX + " = " + rate.toLocaleString(undefined, {maximumFractionDigits: 4});
}