Cosine Graph Calculator
Use this calculator to define the parameters of a cosine function and understand its key characteristics. The standard form of a cosine function is y = A cos(Bx - C) + D.
Determines the height of the wave from its midline. A negative amplitude reflects the graph across the midline.
Affects the period of the wave. A larger B value means more cycles in a given interval.
Contributes to the horizontal shift of the graph. The actual phase shift is C/B.
Moves the entire graph up or down, defining the midline of the wave.
Cosine Function Properties:
"; resultsHTML += "Function Equation:" + functionString + "";
resultsHTML += "Amplitude (A): " + Math.abs(A).toFixed(2) + "";
resultsHTML += "Period (T = 2π/|B|): " + period.toFixed(4) + " (approx. " + (period / Math.PI).toFixed(2) + "π)";
resultsHTML += "Phase Shift (C/B): " + actualPhaseShift.toFixed(2) + " (Shift to the " + (actualPhaseShift > 0 ? "right" : (actualPhaseShift < 0 ? "left" : "none")) + ")";
resultsHTML += "Vertical Shift (D): " + D.toFixed(2) + "";
resultsHTML += "Midline: y = " + midline.toFixed(2) + "";
resultsHTML += "Maximum Value: " + maxVal.toFixed(2) + "";
resultsHTML += "Minimum Value: " + minVal.toFixed(2) + "";
resultDiv.innerHTML = resultsHTML;
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Understanding the Cosine Graph and its Properties
The cosine function is one of the fundamental trigonometric functions, widely used in mathematics, physics, engineering, and many other fields to model periodic phenomena like waves, oscillations, and cycles. Its graph is a smooth, continuous wave that oscillates between a maximum and minimum value.
The Standard Form: y = A cos(Bx – C) + D
To fully understand and graph a cosine function, it's crucial to recognize the role of its four main parameters:
1. Amplitude (A)
The amplitude, denoted by A, is the distance from the midline of the graph to its maximum or minimum value. It represents half the total height of the wave. A larger absolute value of A means a taller wave. If A is negative, the graph is reflected across the midline (it starts at a minimum instead of a maximum for a standard cosine wave).
- Example: If
A = 3, the wave goes 3 units above and 3 units below the midline. IfA = -2, the wave goes 2 units above and 2 units below the midline, but it starts by going down from the midline.
2. Frequency Multiplier (B)
The parameter B affects the frequency and period of the cosine wave. It determines how many cycles of the wave occur in a given interval. A larger B value means the wave completes more cycles in the same horizontal distance, making it appear "compressed" horizontally.
- Period (T): The period is the length of one complete cycle of the wave. It is calculated as
T = 2π / |B|. For example, ifB = 2, the period is2π / 2 = π, meaning the wave completes a full cycle inπunits. IfB = 0.5, the period is2π / 0.5 = 4π, stretching the wave horizontally.
3. Phase Shift Constant (C)
The constant C, in conjunction with B, determines the horizontal shift of the graph, also known as the phase shift. The actual phase shift is calculated as C / B.
- If
C / B > 0, the graph shifts to the right. - If
C / B < 0, the graph shifts to the left. - Example: In
y = cos(x - π/2),C = π/2andB = 1, so the phase shift isπ/2to the right. The graph ofy = cos(x)starts at its maximum atx=0; this shifted graph starts its maximum atx=π/2.
4. Vertical Shift (D)
The vertical shift, represented by D, moves the entire graph up or down. It defines the midline of the wave, which is the horizontal line about which the function oscillates. The equation of the midline is y = D.
- Example: If
D = 5, the entire graph shifts 5 units upwards, and the midline isy = 5. IfD = -3, the graph shifts 3 units downwards, and the midline isy = -3.
Key Characteristics Derived from Parameters:
- Maximum Value:
D + |A| - Minimum Value:
D - |A| - Midline:
y = D - Period:
2π / |B| - Phase Shift:
C / B
By adjusting these parameters, you can model a vast array of periodic behaviors, making the cosine function an indispensable tool in various scientific and mathematical applications.