The average rate of change describes how much a function changes per unit on average over a specific interval. In geometry, this represents the slope of the secant line connecting two points on a graph.
The Formula
A = [f(x₂) – f(x₁)] / (x₂ – x₁)
Step-by-Step Calculation Example
Suppose you are tracking the height of a plant. On Day 2 (x₁), the plant is 10cm tall (y₁). On Day 5 (x₂), the plant is 25cm tall (y₂).
Change in Output (Δy): 25 – 10 = 15 cm
Change in Input (Δx): 5 – 2 = 3 days
Rate of Change: 15 / 3 = 5 cm per day
This means that, on average, the plant grew 5 centimeters every day between Day 2 and Day 5.
Common Applications
Physics: Calculating average velocity (change in position over time).
Economics: Determining the average growth of revenue over a fiscal quarter.
Biology: Measuring the growth rate of a bacterial population.
function calculateARC() {
var x1 = parseFloat(document.getElementById('x1_val').value);
var y1 = parseFloat(document.getElementById('y1_val').value);
var x2 = parseFloat(document.getElementById('x2_val').value);
var y2 = parseFloat(document.getElementById('y2_val').value);
var resultDiv = document.getElementById('arc-result-display');
var valueDiv = document.getElementById('arc-value');
var formulaDiv = document.getElementById('arc-formula-view');
if (isNaN(x1) || isNaN(y1) || isNaN(x2) || isNaN(y2)) {
alert("Please enter valid numbers in all fields.");
return;
}
if (x1 === x2) {
alert("The change in X cannot be zero (Division by zero error). x1 and x2 must be different values.");
return;
}
var deltaY = y2 – y1;
var deltaX = x2 – x1;
var rateOfChange = deltaY / deltaX;
// Format the result
var formattedResult = Number.isInteger(rateOfChange) ? rateOfChange : rateOfChange.toFixed(4);
valueDiv.innerHTML = formattedResult;
formulaDiv.innerHTML = "(" + y2 + " – " + y1 + ") / (" + x2 + " – " + x1 + ") = " + deltaY + " / " + deltaX;
resultDiv.style.display = 'block';
resultDiv.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}