Use this calculator to analyze the key characteristics of a trigonometric function in the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D.
Analysis Results:
function calculateTrigProperties() {
var amplitudeA = parseFloat(document.getElementById('amplitudeA').value);
var bValue = parseFloat(document.getElementById('bValue').value);
var cValue = parseFloat(document.getElementById('cValue').value);
var dValue = parseFloat(document.getElementById('dValue').value);
var functionType = document.querySelector('input[name="functionType"]:checked').value;
var resultEquation = document.getElementById('resultEquation');
var resultAmplitude = document.getElementById('resultAmplitude');
var resultPeriod = document.getElementById('resultPeriod');
var resultPhaseShift = document.getElementById('resultPhaseShift');
var resultVerticalShift = document.getElementById('resultVerticalShift');
var resultNote = document.getElementById('resultNote');
if (isNaN(amplitudeA) || isNaN(bValue) || isNaN(cValue) || isNaN(dValue)) {
resultEquation.innerHTML = "Please enter valid numbers for all fields.";
resultAmplitude.innerHTML = "";
resultPeriod.innerHTML = "";
resultPhaseShift.innerHTML = "";
resultVerticalShift.innerHTML = "";
resultNote.innerHTML = "";
return;
}
// Construct the equation string
var equationStr = "y = ";
if (amplitudeA !== 1 || functionType === 'cosine') { // Always show A for cosine, or if A is not 1 for sine
if (amplitudeA === -1) {
equationStr += "-";
} else if (amplitudeA !== 1) {
equationStr += amplitudeA + " ";
}
}
if (functionType === 'sine') {
equationStr += "sin(";
} else {
equationStr += "cos(";
}
if (bValue === 1) {
equationStr += "x";
} else if (bValue === -1) {
equationStr += "-x";
} else {
equationStr += bValue + "x";
}
if (cValue > 0) {
equationStr += " + " + cValue;
} else if (cValue 0) {
equationStr += " + " + dValue;
} else if (dValue < 0) {
equationStr += " – " + Math.abs(dValue);
}
resultEquation.innerHTML = "Equation: " + equationStr;
// Calculate properties
var calculatedAmplitude = Math.abs(amplitudeA);
var calculatedPeriod;
var calculatedPhaseShift;
var calculatedVerticalShift = dValue;
if (bValue === 0) {
calculatedAmplitude = 0; // For a constant function, amplitude is 0
calculatedPeriod = "Undefined (constant function)";
calculatedPhaseShift = "Undefined (constant function)";
// If B is 0, the function is y = A*sin(C) + D or y = A*cos(C) + D, which is a constant line.
// The effective vertical shift is this constant value.
if (functionType === 'sine') {
calculatedVerticalShift = amplitudeA * Math.sin(cValue) + dValue;
} else {
calculatedVerticalShift = amplitudeA * Math.cos(cValue) + dValue;
}
resultNote.innerHTML = "Note: When B=0, the function is a horizontal line.";
} else {
calculatedPeriod = (2 * Math.PI / Math.abs(bValue)).toFixed(4);
calculatedPhaseShift = (-cValue / bValue).toFixed(4);
resultNote.innerHTML = "";
}
resultAmplitude.innerHTML = "Amplitude: " + calculatedAmplitude;
resultPeriod.innerHTML = "Period: " + calculatedPeriod;
resultPhaseShift.innerHTML = "Phase Shift: " + calculatedPhaseShift;
resultVerticalShift.innerHTML = "Vertical Shift: " + calculatedVerticalShift;
}
// Initial calculation on page load
window.onload = calculateTrigProperties;
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Understanding Trigonometric Graphs: Amplitude, Period, Phase Shift, and Vertical Shift
Trigonometric functions like sine and cosine are fundamental in mathematics and physics, describing oscillatory phenomena from sound waves to alternating currents. Their graphs are characterized by four key properties: Amplitude, Period, Phase Shift, and Vertical Shift. Understanding these properties allows us to accurately model and interpret cyclical patterns.
The General Form of a Trigonometric Function
A standard way to represent a sinusoidal function (sine or cosine) is using the general form:
y = A sin(Bx + C) + D
or
y = A cos(Bx + C) + D
Let's break down what each parameter (A, B, C, D) signifies:
1. Amplitude (A)
The amplitude, denoted by |A|, is the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. In simpler terms, it's half the distance between the maximum and minimum values of the function. A larger amplitude means a "taller" wave.
Calculation: Amplitude = |A|
Example: If A = 3, the amplitude is 3. The wave oscillates between D - 3 and D + 3. If A = -2, the amplitude is 2, and the graph is also reflected across the x-axis.
2. Period (B)
The period is the length of one complete cycle of the wave. It tells us how often the function's pattern repeats. The standard period for sin(x) and cos(x) is 2π (or 360 degrees). The B value in our general equation affects this period.
Calculation: Period = 2π / |B|
Example: If B = 2, the period is 2π / 2 = π. This means the wave completes a full cycle in half the usual horizontal distance, making it "compressed" horizontally. If B = 0.5, the period is 2π / 0.5 = 4π, stretching the wave horizontally.
Special Case: If B = 0, the function becomes a constant value (e.g., y = A sin(C) + D), and its period is considered undefined as it does not oscillate.
3. Phase Shift (C)
The phase shift, often denoted as -C/B, represents the horizontal displacement of the graph. A positive phase shift means the graph shifts to the right, while a negative phase shift means it shifts to the left. It indicates where the cycle "starts" relative to the y-axis.
Calculation: Phase Shift = -C / B
Example: If B = 1 and C = π/2, the phase shift is -π/2, meaning the graph shifts π/2 units to the left. If B = 2 and C = -π, the phase shift is -(-π)/2 = π/2, shifting the graph π/2 units to the right.
Special Case: If B = 0, the concept of phase shift is undefined as there is no horizontal oscillation.
4. Vertical Shift (D)
The vertical shift, denoted by D, is the vertical displacement of the graph. It moves the entire graph up or down. It represents the midline of the oscillation.
Calculation: Vertical Shift = D
Example: If D = 5, the entire graph shifts 5 units upwards, and the new midline is y = 5. If D = -3, the graph shifts 3 units downwards.
Special Case: If B = 0, the function is a horizontal line, and its value (which is its vertical position) is A sin(C) + D or A cos(C) + D.
Practical Application
By manipulating these four parameters, you can model a vast array of periodic phenomena. For instance, you can adjust the amplitude to represent the intensity of a sound wave, the period to match the frequency of an electrical current, the phase shift to account for a time delay, and the vertical shift to represent an average temperature around which daily fluctuations occur.
Using the calculator above, you can input different values for A, B, C, and D to see how they affect the resulting equation and its key characteristics, helping you visualize and understand the behavior of trigonometric functions.