function formatOutputNumber(num) {
// Use toFixed to limit decimal places, then parseFloat to remove trailing zeros
// This helps with floating point precision and cleaner output
return parseFloat(num.toFixed(6)).toString();
}
function calculatePointSlope() {
var x1 = parseFloat(document.getElementById('x1_coord').value);
var y1 = parseFloat(document.getElementById('y1_coord').value);
var m = parseFloat(document.getElementById('slope_m').value);
var errorDiv = document.getElementById('errorMessages');
errorDiv.innerHTML = "; // Clear previous errors
if (isNaN(x1) || isNaN(y1) || isNaN(m)) {
errorDiv.innerHTML = 'Please enter valid numbers for all fields.';
document.getElementById('pointSlopeResult').innerHTML = ";
document.getElementById('slopeInterceptResult').innerHTML = ";
document.getElementById('standardFormResult').innerHTML = ";
return;
}
// — Point-Slope Form: y – y1 = m(x – x1) —
var psLeftSide = "y";
if (y1 !== 0) {
psLeftSide += (y1 > 0 ? " – " : " + ") + formatOutputNumber(Math.abs(y1));
}
var psRightSide = formatOutputNumber(m) + "(x";
if (x1 !== 0) {
psRightSide += (x1 > 0 ? " – " : " + ") + formatOutputNumber(Math.abs(x1));
}
psRightSide += ")";
var pointSlopeForm = "
" + psLeftSide + " = " + psRightSide;
document.getElementById('pointSlopeResult').innerHTML = pointSlopeForm;
// — Slope-Intercept Form: y = mx + b —
var b = y1 – (m * x1);
var slopeInterceptForm = "
y = " + formatOutputNumber(m) + "x";
if (b !== 0) {
slopeInterceptForm += (b > 0 ? " + " : " – ") + formatOutputNumber(Math.abs(b));
}
document.getElementById('slopeInterceptResult').innerHTML = slopeInterceptForm;
// — Standard Form: Ax + By = C —
// Start with mx – y = mx1 – y1
var A_sf = m;
var B_sf = -1;
var C_sf = (m * x1) – y1;
// Normalize to make A positive, or B positive if A is zero
// This ensures a consistent standard form representation
if (A_sf < 0 || (A_sf === 0 && B_sf < 0)) {
A_sf = -A_sf;
B_sf = -B_sf;
C_sf = -C_sf;
}
var standardFormString = "
";
var terms = [];
if (A_sf !== 0) {
var aTerm = "";
if (A_sf === 1) aTerm = "x";
else if (A_sf === -1) aTerm = "-x"; // Should not happen after normalization if A_sf is first
else aTerm = formatOutputNumber(A_sf) + "x";
terms.push(aTerm);
}
if (B_sf !== 0) {
var bTerm = "";
var absB_sf = Math.abs(B_sf);
if (absB_sf === 1) bTerm = "y";
else bTerm = formatOutputNumber(absB_sf) + "y";
if (B_sf > 0) {
if (terms.length > 0) bTerm = " + " + bTerm;
} else { // B_sf < 0
bTerm = " – " + bTerm;
}
terms.push(bTerm);
}
if (terms.length === 0) { // This case should ideally not happen for a line with a defined slope
standardFormString += "0 = " + formatOutputNumber(C_sf);
} else {
standardFormString += terms.join('') + " = " + formatOutputNumber(C_sf);
}
document.getElementById('standardFormResult').innerHTML = standardFormString;
}
// Initialize with default values on page load
window.onload = function() {
calculatePointSlope();
};
.point-slope-calculator {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.point-slope-calculator h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.calculator-input-group {
margin-bottom: 15px;
}
.calculator-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.calculator-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
}
.point-slope-calculator button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
}
.point-slope-calculator button:hover {
background-color: #0056b3;
}
.calculator-results {
margin-top: 25px;
padding-top: 15px;
border-top: 1px solid #eee;
}
.calculator-results h3 {
color: #333;
margin-bottom: 15px;
text-align: center;
}
.calculator-results div {
background-color: #e9ecef;
border: 1px solid #dee2e6;
padding: 10px 15px;
margin-bottom: 10px;
border-radius: 4px;
font-size: 1.1em;
color: #333;
word-wrap: break-word; /* Ensures long equations wrap */
}
.calculator-results div strong {
color: #0056b3;
}
Understanding the Point-Slope Form of a Linear Equation
The point-slope form is a fundamental way to express the equation of a straight line in coordinate geometry. It's particularly useful when you know the slope of a line and at least one point that the line passes through.
What is Point-Slope Form?
The general formula for the point-slope form is:
y – y₁ = m(x – x₁)
Where:
(x₁, y₁) represents a specific point on the line.m represents the slope of the line.(x, y) represents any other point on the line.
This form directly reflects the definition of slope: the change in y divided by the change in x. If you rearrange the formula, you get m = (y - y₁) / (x - x₁).
How to Use the Point-Slope Form
To use the point-slope form, you need two pieces of information:
- The slope (m): This tells you how steep the line is and its direction.
- A point (x₁, y₁) on the line: This anchors the line in the coordinate plane.
Once you have these values, you simply substitute them into the formula. The calculator above automates this process for you, also providing conversions to other common forms.
Example Calculation
Let's say you have a line that passes through the point (2, 5) and has a slope of 3.
Here, x₁ = 2, y₁ = 5, and m = 3.
Substitute these values into the point-slope formula:
y - y₁ = m(x - x₁)
y - 5 = 3(x - 2)
This is the equation of the line in point-slope form.
Converting to Other Forms
1. Slope-Intercept Form (y = mx + b)
The slope-intercept form is another popular way to write linear equations, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). You can easily convert from point-slope to slope-intercept form by distributing the slope and isolating y.
Using our example y - 5 = 3(x - 2):
- Distribute the slope:
y - 5 = 3x - 6
- Add 5 to both sides to isolate
y: y = 3x - 6 + 5
- Simplify:
y = 3x - 1
Here, the slope m = 3 and the y-intercept b = -1.
2. Standard Form (Ax + By = C)
The standard form of a linear equation is typically written as Ax + By = C, where A, B, and C are integers, and A is usually positive. You can convert from slope-intercept form to standard form by moving the x term to the left side of the equation.
Using our slope-intercept example y = 3x - 1:
- Subtract
3x from both sides: -3x + y = -1
- (Optional) Multiply the entire equation by -1 to make the leading coefficient positive:
3x - y = 1
This is the equation in standard form, where A = 3, B = -1, and C = 1.
When to Use Point-Slope Form
The point-slope form is particularly useful in the following scenarios:
- When you are given a point and the slope of a line.
- When you are given two points on a line (you can first calculate the slope, then use one of the points).
- As an intermediate step to derive the slope-intercept or standard forms.
- In calculus, when finding the equation of a tangent line to a curve at a specific point.
By understanding and utilizing the point-slope form, you gain a powerful tool for working with linear equations in various mathematical and scientific contexts.