A linear function is a mathematical relationship that can be represented by a straight line on a Cartesian coordinate system. It is defined by a constant rate of change, known as the slope (m), and a starting point on the y-axis, known as the y-intercept (b).
The Slope-Intercept Form Formula
y = mx + b
y: The dependent variable (output).
x: The independent variable (input).
m: The slope, calculated as (y₂ – y₁) / (x₂ – x₁).
b: The y-intercept, where the line crosses the y-axis (when x = 0).
Real-World Example
Imagine you are taking a taxi. The taxi company charges a flat "entry fee" of 5.00 (the y-intercept) and then 2.00 per mile traveled (the slope). The function representing your total cost would be:
f(x) = 2x + 5
If you travel 10 miles (x=10), your cost would be 2(10) + 5 = 25.00.
Calculating the Slope Manually
To find the slope between two points (2, 4) and (5, 10):
Subtract the y-coordinates: 10 – 4 = 6
Subtract the x-coordinates: 5 – 2 = 3
Divide the difference in y by the difference in x: 6 / 3 = 2
The slope (m) is 2.
function calculateLinearFunction() {
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('linear_result');
if (isNaN(x1) || isNaN(y1) || isNaN(x2) || isNaN(y2)) {
resultDiv.style.display = 'block';
resultDiv.style.backgroundColor = '#f8d7da';
resultDiv.style.color = '#721c24';
resultDiv.innerHTML = 'Error: Please enter valid numerical values for all coordinates.';
return;
}
if (x1 === x2) {
resultDiv.style.display = 'block';
resultDiv.style.backgroundColor = '#fff3cd';
resultDiv.style.color = '#856404';
if (y1 === y2) {
resultDiv.innerHTML = 'Undefined: Both points are identical. An infinite number of lines can pass through a single point.';
} else {
resultDiv.innerHTML = 'Vertical Line: The slope is undefined (division by zero). The equation of this line is: x = ' + x1 + '';
}
return;
}
// Calculate Slope (m)
var m = (y2 – y1) / (x2 – x1);
// Calculate Y-intercept (b) -> y = mx + b => b = y – mx
var b = y1 – (m * x1);
// X-intercept -> 0 = mx + b => x = -b / m
var xIntercept = (m !== 0) ? (-b / m) : null;
// Build the equation string
var mDisp = Math.round(m * 1000) / 1000;
var bDisp = Math.round(b * 1000) / 1000;
var bSign = bDisp >= 0 ? "+ " : "- ";
var bAbs = Math.abs(bDisp);
var equation = "y = " + mDisp + "x " + (bDisp !== 0 ? bSign + bAbs : "");
if (mDisp === 0) equation = "y = " + bDisp;
resultDiv.style.display = 'block';
resultDiv.style.backgroundColor = '#e8f4fd';
resultDiv.style.color = '#0c5460';
resultDiv.style.border = '1px solid #bee5eb';
var html = '
Results:
';
html += 'Equation of the Line:' + equation + '';
html += '