To determine the correct equation for the graph, analyze its properties:

Key observations:

1. Amplitude: The maximum and minimum values of the graph are $2$ and $-2$, so the amplitude ($A$) is $2$.
2. Period: The period of the graph is $2\pi$, matching the standard period of sine and cosine functions.
3. Phase shift:
   • At $x = 0$, the graph is at a minimum ($y = -2$).
   • Sine functions start at $y = 0$ or $y = \pm A$ with a phase shift, while cosine functions naturally start at a maximum or minimum at $x = 0$.
4. Reflection:
   • The curve starts at a minimum, meaning it is an inverted cosine function or a negative sine function.

Equation options:

• $y = A \sin(x)$: Starts at $y = 0$, so this is not correct without a phase shift.
• $y = A \cos(x)$: Starts at $y = A$ or $-A$. The graph starts at $y = -2$, suggesting $y = -2 \cos(x)$.

Correct equation:

The graph matches $y = -2 \cos(x)$, so the answer is:

D. $y = -2 \cos(x)$

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Would you like me to explain any part in more detail or graph the solution for visualization?

Related Questions:

1. How does the phase shift of sine and cosine functions affect their graphs?
2. What is the mathematical relationship between sine and cosine?
3. How do you determine the amplitude and period of a trigonometric function from its graph?
4. Why is the negative sign important in $y = -2 \cos(x)$?
5. What happens to the graph if the amplitude changes?

Tip:

When analyzing trigonometric graphs, always check the starting point and symmetry to distinguish between sine and cosine functions