To determine the equation of a periodic function based on its graph, you need to consider the amplitude, the vertical shift, and whether it's a sine or cosine function.
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Amplitude: The amplitude is the maximum value of the function. If the function oscillates from -A to A, then the amplitude is A. For example, if you see that the graph peaks at 4 and troughs at -4, the amplitude is 4.
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Cosine vs. Sine: If the graph starts at its maximum point, it likely represents a cosine function (as cosine starts at the maximum). If it starts at the midline and goes up first, it is likely a sine function.
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Vertical Shift: If the graph is centered around a line that is not y=0, this means there is a vertical shift.
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Period: This is determined by how quickly the function repeats. For cosine and sine, this is typically influenced by the coefficient of x in the function.
Given the options provided, and without additional details on shifts or periodicity from the graph, if we assume maximum peaks of 4 and troughs of -4, and the function is cosine:
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The amplitude is 4.
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Assuming there is no vertical shift and it starts at the maximum, the equation would be:
y = 4 cos x (if starting at maximum).
If, however, it's reflected or if further details change it, you might choose one of the other options. But based on the peak-to-trough description, the most fitting choice is:
y = 4 cos x.
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