Bot GPT 4-o mini The function \( f(x) = \sin(x - \pi/6) \) is a transformation of the basic sine function, \( \sin(x) \). The differences between the two graphs come from the horizontal shift caused by the \(-\pi/6\) term.
Here's how the graph of \( f(x) = \sin(x - \pi/6) \) differs from the graph of \( \sin(x) \):
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Horizontal Shift: The term \(-\pi/6\) indicates a horizontal shift to the right by \(\pi/6\) units. This means that every point on the sine curve is moved \(\pi/6\) units to the right. As a result, the peaks, troughs, and zeros of \( f(x) \) will occur later along the x-axis compared to \( \sin(x) \).
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Same Amplitude and Period: Both functions have the same amplitude (1) and period \(2\pi\). This means that the height of the waves and the length of one complete cycle remain unchanged.
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Graph Characteristics: The general shape of the graph remains the same, but the starting point of the wave shifts. For example, while \( \sin(0) = 0\) for \( \sin(x)\), the function \(f(x) = \sin(x - \pi/6)\) will have \(f(0) = \sin(-\pi/6) = -\frac{1}{2}\).
In summary, the main difference between the graphs of \( f(x) = \sin(x - \pi/6) \) and \( \sin(x) \) is the rightward shift of \(\pi/6\) units. The overall appearance of the sine wave remains consistent, with just the starting position altered horizontally.
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