Based on the description of the graph, it appears to represent a sinusoidal function, such as a sine or cosine function. Let’s analyze the details:
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Starting Point and Cycle Description: The curve starts at the origin (0, 0), which is typical for the sine function (y = A sin(Bx)) since it starts at the midline. The graph completes one-half cycle at a maximum of 3, which suggests that the amplitude (A) is 3. Therefore, we can express the amplitude as |A| = 3.
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Minimum Point: The graph reaches a minimum of -3 after another half cycle, reinforcing that the amplitude is indeed 3, as the sine function oscillates between -A and A.
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Cycle Count: The problem states that the curve completes 6 cycles within the interval from 0 to 3π. This suggests the period (T) can be calculated as:
- T = (3π) / 6 = (π/2).
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Frequency: The frequency (B) can be calculated using the relation T = 2π / B. Therefore,
- B = 2π / (π/2) = 4.
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Function Representation: With amplitude 3 and frequency 4, we can write the general sine function as: \[ y = 3 \sin(4x) \]
Alternatively, since the sine function starts from the origin, another possible representation is: \[ y = 3 \cos(4x - \pi/2) \] however, this function would only shift the phase. The sine function is more straightforward based on the initial point of the cycle.
To conclude, the function represented by the given graph is: \[ y = 3 \sin(4x) \]
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