Calculator Equations & Formulas
Interactive Equation Calculator
Enter the required variables to calculate the result of a chosen equation.
Simple Linear Equation (y = mx + b)
Quadratic Formula (x = [-b ± sqrt(b^2 – 4ac)] / 2a)
Distance Formula (d = sqrt[(x2 – x1)^2 + (y2 – y1)^2])
Choose the mathematical equation you wish to solve.
What are Calculator Equations?
Calculator equations, in the context of mathematics and science, refer to a set of defined mathematical relationships that can be solved using a calculator or computational tool. These equations represent underlying principles, laws, or formulas used across various disciplines, from basic arithmetic to advanced physics and engineering. They provide a structured way to model real-world phenomena, predict outcomes, or analyze data. Understanding calculator equations is fundamental for anyone looking to apply mathematical concepts practically, whether for academic purposes, problem-solving, or technological development. They are the building blocks for more complex mathematical models and computational processes.
Who Should Use This Calculator Equations Tool?
This calculator equations tool is designed for a broad audience, including:
- Students: High school and college students learning algebra, calculus, geometry, and physics will find this tool invaluable for verifying their manual calculations and gaining a deeper understanding of formulas.
- Educators: Teachers can use this as a demonstration tool in classrooms to explain complex equations and their applications.
- Engineers & Scientists: Professionals working in fields that require quantitative analysis can use it for quick calculations or as a reference for common formulas.
- Hobbyists & Enthusiasts: Anyone with an interest in mathematics, programming, or problem-solving can use it to explore different calculator equations.
- Researchers: Individuals who need to perform repetitive calculations or explore parameter variations in their models.
Common Misconceptions about Calculator Equations
Several misconceptions surround calculator equations:
- "Calculators replace understanding": While calculators automate computation, they don't replace the need to understand the underlying mathematical principles and the meaning of the variables involved.
- "All equations are simple": Many calculator equations are derived from complex theories and may require careful input of variables and interpretation of results.
- "A calculator is only for numbers": Modern calculators and computational tools can handle symbolic manipulation and complex data sets, not just basic arithmetic.
- "Equations are static": The application of calculator equations can vary greatly depending on the context and the specific values of the variables.
Calculator Equations Formula and Mathematical Explanation
1. Simple Linear Equation: y = mx + b
This is one of the most fundamental equations in mathematics, describing a straight line on a Cartesian plane. It's used to model proportional relationships where a constant rate of change is involved.
- y: The dependent variable (the output).
- m: The slope of the line, representing the rate of change of y with respect to x.
- x: The independent variable (the input).
- b: The y-intercept, representing the value of y when x is zero.
Explanation: For every unit increase in 'x', 'y' changes by 'm' units. 'b' is the starting point of 'y' when 'x' begins at 0.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| y | Dependent Variable | Unitless (often contextual) | Varies |
| m | Slope / Rate of Change | Unitless (often contextual) | Any real number |
| x | Independent Variable | Unitless (often contextual) | Varies |
| b | Y-intercept | Unitless (often contextual) | Any real number |
2. Quadratic Formula: x = [-b ± sqrt(b^2 – 4ac)] / 2a
This formula is used to find the roots (solutions) of a quadratic equation in the standard form ax^2 + bx + c = 0. These roots are the x-values where the parabola representing the equation crosses the x-axis.
- a, b, c: Coefficients of the quadratic equation ax^2 + bx + c = 0. 'a' cannot be zero.
- ±: Indicates that there are typically two possible solutions (one using '+' and one using '-').
- sqrt(): Square root.
- b^2 – 4ac: The discriminant. Its value determines the nature of the roots (two real, one real, or two complex).
Explanation: The formula provides the exact x-values where a quadratic function equals zero. It's derived from completing the square on the standard quadratic form.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Coefficient of x^2 | Unitless | Any real number except 0 |
| b | Coefficient of x | Unitless | Any real number |
| c | Constant term | Unitless | Any real number |
| x | Solutions / Roots | Unitless | Varies (can be real or complex) |
3. Distance Formula: d = sqrt[(x2 – x1)^2 + (y2 – y1)^2]
This formula calculates the straight-line distance between two points (x1, y1) and (x2, y2) in a Cartesian coordinate system. It is derived from the Pythagorean theorem.
- (x1, y1): Coordinates of the first point.
- (x2, y2): Coordinates of the second point.
- d: The distance between the two points.
Explanation: It essentially forms a right-angled triangle where the horizontal leg is the difference in x-coordinates (|x2 – x1|) and the vertical leg is the difference in y-coordinates (|y2 – y1|). The distance 'd' is the hypotenuse.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| x1, y1 | Coordinates of Point 1 | Unitless (or unit of measurement) | Varies |
| x2, y2 | Coordinates of Point 2 | Unitless (or unit of measurement) | Varies |
| d | Distance | Unitless (or unit of measurement) | Non-negative real number |
Practical Examples (Real-World Use Cases)
Example 1: Simple Linear Equation (Modeling Cost)
Imagine a small business that has fixed monthly costs of $500 (this is 'b') and variable costs of $10 per unit produced (this is 'm'). They want to know the total cost ('y') for producing a certain number of units ('x').
- Input:
- Equation Type: Simple Linear Equation
- Slope (m): 10
- Y-intercept (b): 500
- Independent Variable (x): 100 units
- Calculation:
- y = (10 * 100) + 500
- y = 1000 + 500
- y = 1500
- Output:
- Primary Result (Total Cost y): 1500
- Intermediate Value (Variable Costs m*x): 1000
- Intermediate Value (Fixed Costs b): 500
- Intermediate Value (Slope m): 10
- Interpretation: Producing 100 units will cost the business $1500 in total ($1000 in variable costs plus $500 in fixed costs). This is a direct application of a calculator equations concept.
Example 2: Quadratic Formula (Finding Project Timeline)
A project manager is analyzing the time it takes to complete a task. They have determined that the time ('t') in hours can be modeled by the equation 0.5t^2 – 5t + 30 = 0, where 't' represents potential time completion points. They need to find the actual completion times.
- Input:
- Equation Type: Quadratic Formula
- Coefficient a: 0.5
- Coefficient b: -5
- Coefficient c: 30
- Calculation:
- Discriminant (b^2 – 4ac) = (-5)^2 – 4*(0.5)*(30) = 25 – 60 = -35
- Since the discriminant is negative, there are no real solutions for 't'.
- Output:
- Primary Result (x1): No real solutions
- Primary Result (x2): No real solutions
- Intermediate Value (Discriminant b^2 – 4ac): -35
- Intermediate Value (2a): 1
- Interpretation: The negative discriminant indicates that the quadratic equation modeling this specific project timeline scenario has no real roots. This suggests that the model might be flawed, or perhaps the parameters used imply that the task, under these specific assumptions, cannot be completed within the modeled framework. This highlights the importance of interpreting results from calculator equations.
Example 3: Distance Formula (Calculating Travel Path)
Two hikers start at different points on a map. Hiker 1 is at coordinates (2, 3) and Hiker 2 is at (8, 10). We need to calculate the direct distance between them.
- Input:
- Equation Type: Distance Formula
- Point 1 (x1, y1): 2, 3
- Point 2 (x2, y2): 8, 10
- Calculation:
- d = sqrt[(8 – 2)^2 + (10 – 3)^2]
- d = sqrt[6^2 + 7^2]
- d = sqrt[36 + 49]
- d = sqrt[85]
- d ≈ 9.22
- Output:
- Primary Result (Distance d): 9.22
- Intermediate Value (Delta x squared (x2-x1)^2): 36
- Intermediate Value (Delta y squared (y2-y1)^2): 49
- Intermediate Value (Sum of Squares): 85
- Interpretation: The direct distance between the two hikers is approximately 9.22 units (e.g., miles, kilometers, or map units). This calculation, derived from fundamental calculator equations, is useful for navigation or resource allocation.
How to Use This Calculator Equations Tool
- Select Equation Type: From the dropdown menu, choose the specific mathematical equation you want to work with (e.g., Simple Linear Equation, Quadratic Formula, Distance Formula).
- Enter Variables: Based on your selection, specific input fields will appear. Carefully enter the required numerical values for each variable (e.g., slope 'm', coefficients 'a', 'b', 'c', or coordinates 'x1', 'y1', 'x2', 'y2'). Ensure you are using the correct units and formats as indicated by the helper text.
- Validate Inputs: The calculator will automatically check for common errors like empty fields, negative values where not applicable, or out-of-range values. Error messages will appear below the respective input fields if issues are detected.
- Calculate: Click the "Calculate" button to perform the computation.
- Read Results: The primary result will be displayed prominently. Key intermediate values, which show the steps of the calculation, will also be shown. The formula used will be explained in plain language.
- Interpret: Understand what the results mean in the context of your problem. Use the provided explanations and examples to guide your interpretation.
- Visualize (if applicable): For equations like the linear equation, a chart may be generated to visually represent the relationship.
- Copy Results: If you need to share or save the results, click the "Copy Results" button. This will copy the main result, intermediate values, and key assumptions to your clipboard.
- Reset: To start over with default values, click the "Reset" button.
Key Factors That Affect Calculator Equations Results
Several factors can influence the outcome of calculations using calculator equations:
- Accuracy of Input Data: The precision of the numbers you enter directly impacts the result. Inaccurate measurements or estimations will lead to inaccurate outputs. For example, in the distance formula, slightly off coordinates will yield a slightly off distance.
- Variable Definitions: Each variable in an equation has a specific meaning and unit. Misunderstanding or misapplying these definitions (e.g., confusing slope with intercept in a linear equation) will produce nonsensical results.
- Context of the Equation: An equation is a model. Its applicability depends on whether the real-world situation actually conforms to the assumptions of the model. For instance, the linear equation assumes a constant rate of change, which may not hold true for all scenarios over extended periods.
- Mathematical Constraints: Some equations have inherent limitations. For example, the quadratic formula requires the coefficient 'a' to be non-zero. Division by zero must be avoided. Complex numbers may arise from square roots of negative numbers (as seen in the quadratic formula example), which require specific interpretation.
- Rounding and Precision: Computational tools have limited precision. How rounding is handled during intermediate steps can affect the final result, especially in complex calculations. Using higher precision settings or symbolic calculators can mitigate this.
- Assumptions of the Model: Equations often simplify reality. A linear model assumes a constant rate, while a distance formula assumes Euclidean geometry. Deviations from these underlying assumptions in the real world can lead to discrepancies between calculated results and actual observations.
- Units of Measurement: Ensuring consistency in units across all input variables is crucial. Mixing units (e.g., using meters for one coordinate and kilometers for another in the distance formula without conversion) will lead to incorrect results.
Frequently Asked Questions (FAQ)
- What is the difference between an equation and a formula?
- Often used interchangeably, a formula is a specific type of equation that expresses a relationship between variables, often representing a physical law or rule (like the distance formula). An equation is a broader mathematical statement that asserts the equality of two expressions.
- Can these calculator equations handle negative numbers?
- Yes, most calculator equations can handle negative numbers. However, the interpretation and mathematical validity depend on the specific equation and context. For instance, in the quadratic formula, 'b' or 'c' can be negative, and the discriminant (b^2 – 4ac) can also be negative, leading to complex roots.
- What happens if I input zero for a variable where it's not allowed?
- The calculator should provide an error message. For example, in the quadratic formula, 'a' cannot be zero because it would turn the quadratic equation into a linear one. In the simple linear equation, 'm' can be zero (resulting in a horizontal line), but division by 'm' in other contexts would be invalid.
- How does the calculator handle non-integer inputs?
- This calculator is designed to handle decimal (floating-point) numbers for all inputs, allowing for precise calculations with non-integer values.
- Can I use this calculator for symbolic math (algebra)?
- No, this specific calculator is designed for numerical computation. It requires specific numerical values for variables to produce a numerical result. It does not perform symbolic manipulation (like simplifying algebraic expressions).
- What does it mean if the quadratic formula gives only one real solution?
- This occurs when the discriminant (b^2 – 4ac) is exactly zero. Graphically, it means the vertex of the parabola touches the x-axis at a single point. The formula simplifies to x = -b / 2a.
- Are calculator equations only used in math class?
- Absolutely not! Calculator equations are the foundation for countless applications in physics, engineering, economics, computer graphics, data science, and many other fields. They are used for modeling, simulation, prediction, and analysis in the real world.
- How can I be sure the results are correct?
- While this calculator is built with accurate logic, it's always good practice to double-check critical calculations manually or with another trusted tool, especially if the inputs or context are complex. Understanding the formula and its limitations, as provided here, also helps in verifying results.
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The rate of change (rise over run).
The value of y when x is 0.
The input value.
`;
} else if (currentEquationType === 'quadraticFormula') {
description = 'Quadratic Formula: x = [-b ± sqrt(b^2 – 4ac)] / 2aFind the roots (solutions) of a quadratic equation ax^2 + bx + c = 0.';
htmlContent = `
Must not be zero.
X-coordinate of the first point.
Y-coordinate of the first point.
X-coordinate of the second point.
Y-coordinate of the second point.
`;
}
inputArea.innerHTML = htmlContent;
formulaExplanationDiv.innerHTML = description;
addInputEventListeners();
}
function addInputEventListeners() {
var inputIds = [];
if (currentEquationType === 'simpleLinear') {
inputIds = ['slopeM', 'yInterceptB', 'independentX'];
} else if (currentEquationType === 'quadraticFormula') {
inputIds = ['coeffA', 'coeffB', 'coeffC'];
} else if (currentEquationType === 'distanceFormula') {
inputIds = ['point1X', 'point1Y', 'point2X', 'point2Y'];
}
for (var i = 0; i < inputIds.length; i++) {
var inputElement = document.getElementById(inputIds[i]);
if (inputElement) {
inputElement.addEventListener('input', function() {
calculate(); // Recalculate on input change
});
}
}
}
function calculate() {
var results = {};
var intermediateValues = {};
var allValid = true;
// Clear previous errors and results
var errorElements = document.querySelectorAll('.error-message');
for (var i = 0; i < errorElements.length; i++) {
errorElements[i].style.display = 'none';
}
document.getElementById('resultArea').style.display = 'none';
document.getElementById('chartArea').style.display = 'none';
document.getElementById('mainResult').textContent = '';
document.getElementById('intermediateResults').innerHTML = '';
if (currentEquationType === 'simpleLinear') {
var m = parseFloat(document.getElementById('slopeM').value);
var b = parseFloat(document.getElementById('yInterceptB').value);
var x = parseFloat(document.getElementById('independentX').value);
var mValid = validateInput(document.getElementById('slopeM').value, 'slopeM');
var bValid = validateInput(document.getElementById('yInterceptB').value, 'yInterceptB');
var xValid = validateInput(document.getElementById('independentX').value, 'independentX');
if (mValid && bValid && xValid) {
var variableCost = m * x;
var y = variableCost + b;
results.main = y.toFixed(4);
intermediateValues.variableCost = variableCost.toFixed(4);
intermediateValues.fixedCost = b.toFixed(4);
intermediateValues.slope = m.toFixed(4);
intermediateValues.inputX = x.toFixed(4);
renderResults("Total Cost (y)", results, intermediateValues, { m: m.toFixed(4), b: b.toFixed(4), x: x.toFixed(4) });
updateChartLinear(m, b, x);
} else {
allValid = false;
}
} else if (currentEquationType === 'quadraticFormula') {
var a = parseFloat(document.getElementById('coeffA').value);
var b = parseFloat(document.getElementById('coeffB').value);
var c = parseFloat(document.getElementById('coeffC').value);
var aValid = validateInput(document.getElementById('coeffA').value, 'coeffA');
var bValid = validateInput(document.getElementById('coeffB').value, 'coeffB');
var cValid = validateInput(document.getElementById('coeffC').value, 'coeffC');
if (aValid && bValid && cValid) {
if (a === 0) {
document.getElementById('coeffAError').textContent = 'Coefficient "a" cannot be zero for a quadratic equation.';
document.getElementById('coeffAError').style.display = 'block';
allValid = false;
} else {
var discriminant = (b * b) – (4 * a * c);
intermediateValues.discriminant = discriminant.toFixed(4);
if (discriminant = 0) {
updateChartQuadratic(a, b, c, x1, x2);
} else {
document.getElementById('chartArea').style.display = 'none';
}
}
} else {
allValid = false;
}
} else if (currentEquationType === 'distanceFormula') {
var x1 = parseFloat(document.getElementById('point1X').value);
var y1 = parseFloat(document.getElementById('point1Y').value);
var x2 = parseFloat(document.getElementById('point2X').value);
var y2 = parseFloat(document.getElementById('point2Y').value);
var x1Valid = validateInput(document.getElementById('point1X').value, 'point1X');
var y1Valid = validateInput(document.getElementById('point1Y').value, 'point1Y');
var x2Valid = validateInput(document.getElementById('point2X').value, 'point2X');
var y2Valid = validateInput(document.getElementById('point2Y').value, 'point2Y');
if (x1Valid && y1Valid && x2Valid && y2Valid) {
var deltaXSquared = Math.pow(x2 – x1, 2);
var deltaYSquared = Math.pow(y2 – y1, 2);
var sumOfSquares = deltaXSquared + deltaYSquared;
var distance = Math.sqrt(sumOfSquares);
results.main = distance.toFixed(4);
intermediateValues.deltaXSquared = deltaXSquared.toFixed(4);
intermediateValues.deltaYSquared = deltaYSquared.toFixed(4);
intermediateValues.sumOfSquares = sumOfSquares.toFixed(4);
renderResults("Distance (d)", results, intermediateValues, { x1: x1.toFixed(4), y1: y1.toFixed(4), x2: x2.toFixed(4), y2: y2.toFixed(4) });
updateChartDistance(x1, y1, x2, y2);
} else {
allValid = false;
}
}
if (allValid) {
document.getElementById('resultArea').style.display = 'block';
if (currentEquationType === 'quadraticFormula' && intermediateValues.discriminant = 0) {
document.getElementById('chartArea').style.display = 'block';
}
}
}
function renderResults(mainResultLabel, results, intermediateValues, assumptions) {
var mainResultDiv = document.getElementById('mainResult');
var intermediateResultsDiv = document.getElementById('intermediateResults');
intermediateResultsDiv.innerHTML = "; // Clear previous intermediate results
var mainResultHTML = ";
if (results.main) {
mainResultHTML = mainResultLabel + ": " + results.main;
} else if (results.x1 && results.x2) {
mainResultHTML = `
Root 1 (x1): ${results.x1}
Root 2 (x2): ${results.x2}
`;
}
mainResultDiv.innerHTML = mainResultHTML;
for (var key in intermediateValues) {
var label = key.replace(/([A-Z])/g, ' $1').trim(); // Add spaces to camelCase keys
label = label.charAt(0).toUpperCase() + label.slice(1); // Capitalize first letter
if (key === 'realPartX1' || key === 'imaginaryPartX1' || key === 'realPartX2' || key === 'imaginaryPartX2') {
// Skip these if they are not relevant for display or if they are 'N/A'
continue;
}
if (intermediateValues[key] !== "N/A") {
intermediateResultsDiv.innerHTML += `${label}: ${intermediateValues[key]}
`;
}
}
// Special handling for Quadratic Roots if needed
if (currentEquationType === 'quadraticFormula' && results.x1 !== "No real solutions") {
var realPartX1 = parseFloat(intermediateValues.realPartX1);
var imagPartX1 = parseFloat(intermediateValues.imaginaryPartX1);
var realPartX2 = parseFloat(intermediateValues.realPartX2);
var imagPartX2 = parseFloat(intermediateValues.imaginaryPartX2);
if (!isNaN(realPartX1)) {
intermediateResultsDiv.innerHTML += `Root 1 Real Part: ${realPartX1.toFixed(4)}
`;
}
if (!isNaN(imagPartX1) && imagPartX1 !== 0) {
intermediateResultsDiv.innerHTML += `Root 1 Imaginary Part: ${imagPartX1.toFixed(4)}
`;
}
if (!isNaN(realPartX2)) {
intermediateResultsDiv.innerHTML += `Root 2 Real Part: ${realPartX2.toFixed(4)}
`;
}
if (!isNaN(imagPartX2) && imagPartX2 !== 0) {
intermediateResultsDiv.innerHTML += `Root 2 Imaginary Part: ${imagPartX2.toFixed(4)}
`;
}
}
// Display assumptions
var assumptionsHTML = 'Key Assumptions:';
for (var key in assumptions) {
var label = key.replace(/([A-Z])/g, ' $1').trim();
label = label.charAt(0).toUpperCase() + label.slice(1);
assumptionsHTML += `${label}: ${assumptions[key]}`;
}
assumptionsHTML += '
';
intermediateResultsDiv.innerHTML += assumptionsHTML; // Append assumptions to intermediate results area
}
function copyResults() {
var mainResultText = document.getElementById('mainResult').innerText;
var intermediateResultsText = ";
var intermediateDivs = document.getElementById('intermediateResults').querySelectorAll('div');
for (var i = 0; i < intermediateDivs.length; i++) {
intermediateResultsText += intermediateDivs[i].innerText + '\n';
}
var fullText = "Calculator Equations Results:\n\n";
fullText += "— Primary Result —\n" + mainResultText + "\n\n";
fullText += "— Intermediate Values & Assumptions —\n" + intermediateResultsText;
var textArea = document.createElement("textarea");
textArea.value = fullText;
document.body.appendChild(textArea);
textArea.select();
try {
document.execCommand("copy");
alert("Results copied to clipboard!");
} catch (e) {
console.error("Failed to copy results: ", e);
alert("Failed to copy results. Please copy manually.");
}
textArea.remove();
}
function resetCalculator() {
// Reset inputs to default values
if (currentEquationType === 'simpleLinear') {
document.getElementById('slopeM').value = 2;
document.getElementById('yInterceptB').value = 5;
document.getElementById('independentX').value = 10;
} else if (currentEquationType === 'quadraticFormula') {
document.getElementById('coeffA').value = 1;
document.getElementById('coeffB').value = -3;
document.getElementById('coeffC').value = 2;
} else if (currentEquationType === 'distanceFormula') {
document.getElementById('point1X').value = 1;
document.getElementById('point1Y').value = 2;
document.getElementById('point2X').value = 4;
document.getElementById('point2Y').value = 6;
}
// Clear errors and results
var errorElements = document.querySelectorAll('.error-message');
for (var i = 0; i < errorElements.length; i++) {
errorElements[i].style.display = 'none';
}
document.getElementById('resultArea').style.display = 'none';
document.getElementById('chartArea').style.display = 'none';
document.getElementById('mainResult').textContent = '';
document.getElementById('intermediateResults').innerHTML = '';
// Optionally re-calculate after reset if needed, or just leave it clean
// calculate();
}
// Charting Functions
var myChart = null; // Global chart instance
function updateChartLinear(m, b, xInput) {
var canvas = document.getElementById('equationChart');
var ctx = canvas.getContext('2d');
if (myChart) {
myChart.destroy(); // Destroy previous chart instance
}
// Determine x-range for the chart
var xMin = Math.min(0, parseFloat(xInput) – 10);
var xMax = Math.max(10, parseFloat(xInput) + 10);
var yValues = [];
var xValues = [];
var step = (xMax – xMin) / 50; // Number of points for the line
for (var x = xMin; x <= xMax; x += step) {
xValues.push(x);
yValues.push(m * x + b);
}
// Add the specific input point if it's within range or slightly outside
if(parseFloat(xInput) xMax) {
xValues.push(parseFloat(xInput));
yValues.push(m * parseFloat(xInput) + b);
}
myChart = new Chart(ctx, {
type: 'line',
data: {
labels: xValues,
datasets: [{
label: 'y = mx + b (Line)',
data: yValues,
borderColor: 'var(–primary-color)',
fill: false,
tension: 0,
pointRadius: 0
}, {
label: 'Input Point (x, y)',
data: [{ x: parseFloat(xInput), y: m * parseFloat(xInput) + b }],
borderColor: 'var(–success-color)',
backgroundColor: 'var(–success-color)',
fill: false,
pointRadius: 5,
showLine: false // Do not draw a line for this dataset
}]
},
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
x: {
title: {
display: true,
text: 'Independent Variable (x)'
}
},
y: {
title: {
display: true,
text: 'Dependent Variable (y)'
}
}
},
plugins: {
title: {
display: true,
text: 'Linear Equation Visualization'
},
legend: {
position: 'top'
}
}
}
});
updateChartLegend(['y = mx + b', 'Input Point (x, y)']);
}
function updateChartQuadratic(a, b, c, root1, root2) {
var canvas = document.getElementById('equationChart');
var ctx = canvas.getContext('2d');
if (myChart) {
myChart.destroy();
}
var xValues = [];
var yValues = [];
var roots = [];
// Determine x-range for the parabola
var vertexX = -b / (2 * a);
var xMin = Math.min(vertexX – 10, root1 !== undefined ? root1 – 5 : vertexX – 10, root2 !== undefined ? root2 – 5 : vertexX – 10);
var xMax = Math.max(vertexX + 10, root1 !== undefined ? root1 + 5 : vertexX + 10, root2 !== undefined ? root2 + 5 : vertexX + 10);
var step = (xMax – xMin) / 100;
for (var x = xMin; x <= xMax; x += step) {
xValues.push(x);
yValues.push(a * x * x + b * x + c);
}
// Add roots if they exist
if (root1 !== undefined && !isNaN(root1)) roots.push({ x: root1, y: 0 });
if (root2 !== undefined && !isNaN(root2) && root1 !== root2) roots.push({ x: root2, y: 0 });
myChart = new Chart(ctx, {
type: 'line',
data: {
labels: xValues,
datasets: [{
label: 'Parabola (y = ax^2 + bx + c)',
data: yValues.map(function(y, index) { return { x: xValues[index], y: y }; }),
borderColor: 'var(–primary-color)',
fill: false,
tension: 0
}, {
label: 'Roots (x-intercepts)',
data: roots,
borderColor: 'var(–success-color)',
backgroundColor: 'var(–success-color)',
pointRadius: 5,
showLine: false
}]
},
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
x: {
title: { display: true, text: 'Variable (x)' }
},
y: {
title: { display: true, text: 'Value (y)' }
}
},
plugins: {
title: {
display: true,
text: 'Quadratic Equation Visualization'
},
legend: {
position: 'top'
}
}
}
});
updateChartLegend(['Parabola (y = ax^2 + bx + c)', 'Roots (x-intercepts)']);
}
function updateChartDistance(x1, y1, x2, y2) {
var canvas = document.getElementById('equationChart');
var ctx = canvas.getContext('2d');
if (myChart) {
myChart.destroy();
}
var dataPoints = [
{ x: parseFloat(x1), y: parseFloat(y1) },
{ x: parseFloat(x2), y: parseFloat(y2) }
];
myChart = new Chart(ctx, {
type: 'scatter', // Use scatter plot for two points
data: {
datasets: [{
label: 'Point 1',
data: [dataPoints[0]],
borderColor: 'var(–primary-color)',
backgroundColor: 'var(–primary-color)',
pointRadius: 6
}, {
label: 'Point 2',
data: [dataPoints[1]],
borderColor: 'var(–success-color)',
backgroundColor: 'var(–success-color)',
pointRadius: 6
}, {
label: 'Direct Distance Line',
data: dataPoints, // Use the same points to draw the line
borderColor: '#ffc107', // A distinct color for the distance line
backgroundColor: '#ffc107',
fill: false,
tension: 0,
pointRadius: 0 // Hide points for the line dataset
}]
},
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
x: {
title: { display: true, text: 'X-coordinate' },
type: 'linear',
position: 'bottom'
},
y: {
title: { display: true, text: 'Y-coordinate' },
type: 'linear'
}
},
plugins: {
title: {
display: true,
text: 'Distance Between Two Points'
},
legend: {
position: 'top'
}
}
}
});
updateChartLegend(['Point 1', 'Point 2', 'Direct Distance Line']);
}
function updateChartLegend(labels) {
var legendDiv = document.getElementById('chartLegend');
legendDiv.innerHTML = ''; // Clear previous legend
var legendHTML = 'Legend: ';
var colors = ['var(–primary-color)', 'var(–success-color)', '#ffc107']; // Default colors
if (currentEquationType === 'quadraticFormula') {
colors = ['var(–primary-color)', 'var(–success-color)'];
}
for (var i = 0; i < labels.length; i++) {
legendHTML += `
${labels[i]}
`;
}
legendDiv.innerHTML = legendHTML;
}
// Initial setup when the page loads
document.addEventListener('DOMContentLoaded', function() {
updateEquationDescriptionAndInputs();
// Add event listeners AFTER the inputs are dynamically created
addInputEventListeners();
});
// Chart.js library integration (must be included via CDN or locally for charts to work)
// For this self-contained HTML, we assume Chart.js is available.
// In a real-world scenario, you'd add:
// before this script.
// For this exercise, we'll include a placeholder definition that assumes Chart is globally available.
if (typeof Chart === 'undefined') {
console.warn("Chart.js library not found. Charts will not be displayed.");
// Define a dummy Chart object to prevent JS errors if Chart.js is missing
window.Chart = function() {
this.destroy = function() {};
};
}