Generate an Input/Output table (x, y) based on a linear rate of change.
Input (x)
Output (y)
What is a Constant Rate of Change?
A constant rate of change describes a relationship between two variables where the change in the output variable (often denoted as y) is proportional to the change in the input variable (often denoted as x). In mathematics and physics, this creates a linear relationship, which means if you graph the data points, they will form a straight line.
The concept is mathematically expressed using the linear equation slope-intercept form:
y = mx + b
Where:
m = The Rate of Change (Slope). This tells you how much y changes for every 1 unit increase in x.
b = The Initial Value (y-intercept). This is the value of y when x is 0.
x = The Input (Independent Variable).
y = The Output (Dependent Variable).
Real-World Examples of Constant Rate of Change
We encounter constant rates of change daily. Here are a few practical examples:
Speed: If a car travels at a constant speed of 60 miles per hour, the rate of change is 60. The distance (y) increases by 60 for every hour (x).
Hourly Wages: If you earn 15 units of currency per hour, your total earnings increase by 15 for every hour worked. The initial value is 0 (assuming you earn nothing if you don't work).
Water Tank: If a tank contains 100 liters (Initial Value) and leaks 2 liters per minute (Rate = -2), the amount of water changes at a constant negative rate.
How to Identify Constant Rate of Change in a Table
If you are looking at a table of values, you can determine if it represents a linear function by checking the differences between consecutive rows. The ratio of the change in output to the change in input must be constant:
Rate of Change = (Change in y) / (Change in x)
If this calculation yields the same number for every pair of points in your table, the data has a constant rate of change.
function generateRocTable() {
// Get input values using var
var initialVal = document.getElementById('roc_initial_value').value;
var rateVal = document.getElementById('roc_rate').value;
var startX = document.getElementById('roc_start_x').value;
var stepVal = document.getElementById('roc_step').value;
var rowCount = document.getElementById('roc_rows').value;
// Validation
if (initialVal === "" || rateVal === "" || startX === "" || stepVal === "" || rowCount === "") {
alert("Please fill in all fields to generate the table.");
return;
}
// Parse to numbers
var b = parseFloat(initialVal);
var m = parseFloat(rateVal);
var xStart = parseFloat(startX);
var step = parseFloat(stepVal);
var rows = parseInt(rowCount);
if (isNaN(b) || isNaN(m) || isNaN(xStart) || isNaN(step) || isNaN(rows)) {
alert("Please enter valid numbers.");
return;
}
if (rows > 500) {
rows = 500; // Cap rows for performance
document.getElementById('roc_rows').value = 500;
}
var tableBody = document.getElementById('roc_table_body');
tableBody.innerHTML = ""; // Clear previous results
// Generate Formula String
var operator = (b >= 0) ? "+" : "-";
var absB = Math.abs(b);
var equationHtml = "Equation: y = " + m + "x " + operator + " " + absB;
document.getElementById('roc_equation_display').innerHTML = equationHtml;
// Loop to create table rows
for (var i = 0; i < rows; i++) {
var currentX = xStart + (i * step);
// Fix floating point errors (e.g. 0.1 + 0.2 = 0.30000000004)
currentX = Math.round(currentX * 10000) / 10000;
// Calculate Y = mx + b
var currentY = (m * currentX) + b;
currentY = Math.round(currentY * 10000) / 10000;
var tr = document.createElement('tr');
var tdX = document.createElement('td');
tdX.textContent = currentX;
var tdY = document.createElement('td');
tdY.textContent = currentY;
tr.appendChild(tdX);
tr.appendChild(tdY);
tableBody.appendChild(tr);
}
// Show result container
document.getElementById('roc_result_container').style.display = "block";
}