The equation $y = \sin(x) - 6$ represents a vertically shifted sine function. Here's a breakdown of its components:

Key Features of $y = \sin(x) - 6$:

1. Amplitude: The amplitude of the sine wave is 1, as the coefficient of $\sin(x)$ is 1.
2. Period: The period of the sine wave is $2\pi$, as the period of $\sin(x)$ is unchanged ($\frac{2\pi}{1} = 2\pi$).
3. Vertical Shift: The graph of $\sin(x)$ is shifted downward by 6 units. This means the midline of the sine wave is now at $y = -6$, instead of $y = 0$.
4. Range: The sine function $\sin(x)$ has a range of $[-1, 1]$. After the shift, the range becomes $[-7, -5]$.
5. Graph Behavior:
   • Peaks occur at $y = -5$ (when $\sin(x) = 1$).
   • Troughs occur at $y = -7$ (when $\sin(x) = -1$).

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Follow-up Questions:

1. What are the critical points of $y = \sin(x) - 6$ in a specific interval?
2. How does the derivative of $y = \sin(x) - 6$ describe the function's slope?
3. What are the solutions to $\sin(x) - 6 = 0$?
4. How would adding a coefficient, like $y = 2\sin(x) - 6$, affect the graph?
5. How does the phase shift affect $y = \sin(x) - 6$ if we modify it to $y = \sin(x - \pi/2) - 6$?

Tip: A vertical shift changes the baseline of the function, but does not affect its periodicity or amplitude.