Transformations of Functions Calculator

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Function Transformations Calculator

Use this calculator to determine the new coordinates of a point on a function after applying various transformations. Enter the original (x, y) coordinates and the transformation parameters (A, B, C, D) to see the transformed point.

A > 1 stretches vertically, 0 < A < 1 compresses vertically. A < 0 reflects over x-axis.

B > 1 compresses horizontally, 0 < B < 1 stretches horizontally. B < 0 reflects over y-axis. Cannot be 0.

C > 0 shifts right, C < 0 shifts left.

D > 0 shifts up, D < 0 shifts down.

function calculateTransformation() { var originalX = parseFloat(document.getElementById('originalX').value); var originalY = parseFloat(document.getElementById('originalY').value); var factorA = parseFloat(document.getElementById('factorA').value); var factorB = parseFloat(document.getElementById('factorB').value); var shiftC = parseFloat(document.getElementById('shiftC').value); var shiftD = parseFloat(document.getElementById('shiftD').value); if (isNaN(originalX) || isNaN(originalY) || isNaN(factorA) || isNaN(factorB) || isNaN(shiftC) || isNaN(shiftD)) { document.getElementById('result').innerHTML = 'Please enter valid numbers for all fields.'; return; } if (factorB === 0) { document.getElementById('result').innerHTML = 'Horizontal Stretch/Reflection Factor (B) cannot be zero.'; return; } // Transformation rules for a point (x, y) on f(x) to (x', y') on g(x) = A * f(B * (x – C)) + D // x' = (x / B) + C // y' = A * y + D var transformedX = (originalX / factorB) + shiftC; var transformedY = (factorA * originalY) + shiftD; document.getElementById('result').innerHTML = 'Original Point: (' + originalX.toFixed(3) + ', ' + originalY.toFixed(3) + ')' + 'Transformed Point: (' + transformedX.toFixed(3) + ', ' + transformedY.toFixed(3) + ')'; } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 20px; border-radius: 8px; box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1); max-width: 600px; margin: 20px auto; border: 1px solid #ddd; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { color: #555; line-height: 1.6; margin-bottom: 15px; } .calc-input-group { margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 5px; color: #333; font-weight: bold; } .calc-input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 1em; } .calc-input-group .input-hint { font-size: 0.85em; color: #777; margin-top: 5px; margin-bottom: 0; } .calculate-button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 1.1em; cursor: pointer; transition: background-color 0.3s ease; margin-top: 20px; } .calculate-button:hover { background-color: #0056b3; } .calc-result { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 4px; color: #155724; font-size: 1.1em; text-align: center; } .calc-result p { margin: 5px 0; color: #155724; } .calc-result .error { color: #721c24; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 4px; }

Understanding Function Transformations

Function transformations involve altering the graph of a parent function by shifting, stretching, compressing, or reflecting it. These changes are typically represented by modifying the function's equation. Understanding these transformations is crucial in algebra and calculus for analyzing and sketching graphs of complex functions.

The General Transformation Form

A common way to express a transformed function, g(x), based on an original function, f(x), is through the form:

g(x) = A × f(B × (x - C)) + D

Each parameter (A, B, C, D) plays a specific role in transforming the graph:

  • A (Vertical Stretch/Reflection Factor):
    • If |A| > 1, the graph is stretched vertically.
    • If 0 < |A| < 1, the graph is compressed vertically.
    • If A < 0, the graph is reflected across the x-axis.
  • B (Horizontal Stretch/Reflection Factor):
    • If |B| > 1, the graph is compressed horizontally (by a factor of 1/|B|).
    • If 0 < |B| < 1, the graph is stretched horizontally (by a factor of 1/|B|).
    • If B < 0, the graph is reflected across the y-axis.
    • Note: B cannot be zero.
  • C (Horizontal Shift):
    • If C > 0, the graph shifts C units to the right.
    • If C < 0, the graph shifts |C| units to the left.
  • D (Vertical Shift):
    • If D > 0, the graph shifts D units upwards.
    • If D < 0, the graph shifts |D| units downwards.

How a Point Transforms

If an original point (x, y) lies on the graph of f(x), then after applying the transformations defined by A, B, C, D, the new point (x', y') on the graph of g(x) can be found using these rules:

  • New X-coordinate (x'): x' = (x / B) + C
  • New Y-coordinate (y'): y' = A × y + D

This calculator uses these exact formulas to determine the transformed coordinates.

Example Calculation

Let's say we have an original point (2, 4) on a function f(x). We want to apply the following transformations:

  • Vertical Stretch Factor (A) = 2
  • Horizontal Stretch Factor (B) = 0.5
  • Horizontal Shift (C) = 3
  • Vertical Shift (D) = -1

Using the transformation rules:

  • x' = (2 / 0.5) + 3 = 4 + 3 = 7
  • y' = (2 × 4) + (-1) = 8 - 1 = 7

So, the transformed point is (7, 7). This means the original point (2, 4) has been stretched vertically by a factor of 2, stretched horizontally by a factor of 2, shifted 3 units to the right, and 1 unit down.

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