To check the behavior of the function \( y = -\sin(x) \), let's go over a few key points:

### 1. **Basic Shape and Symmetry:**
   - The function \( y = -\sin(x) \) is a reflection of the standard sine function \( y = \sin(x) \) across the x-axis.
   - The sine function typically oscillates between -1 and 1. The negative sign flips it, so now the function oscillates between 1 and -1.

### 2. **Periodicity:**
   - The sine function has a period of \( 2\pi \). This means \( y = -\sin(x) \) repeats itself every \( 2\pi \) units along the x-axis.

### 3. **Key Points:**
   - At \( x = 0 \), \( y = -\sin(0) = 0 \).
   - At \( x = \frac{\pi}{2} \), \( y = -\sin\left(\frac{\pi}{2}\right) = -1 \).
   - At \( x = \pi \), \( y = -\sin(\pi) = 0 \).
   - At \( x = \frac{3\pi}{2} \), \( y = -\sin\left(\frac{3\pi}{2}\right) = 1 \).
   - At \( x = 2\pi \), \( y = -\sin(2\pi) = 0 \).

### 4. **Graph:**
   - The graph of \( y = -\sin(x) \) will look similar to the sine wave but flipped upside down. It starts at 0, goes down to -1 at \( \frac{\pi}{2} \), returns to 0 at \( \pi \), rises to 1 at \( \frac{3\pi}{2} \), and returns to 0 at \( 2\pi \).

### 5. **Derivative and Critical Points:**
   - The derivative of \( y = -\sin(x) \) is \( y' = -\cos(x) \).
   - Critical points occur where the derivative equals zero, i.e., \( -\cos(x) = 0 \), which happens at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. These points correspond to the maxima and minima of the function.

Do you want further details on any specific aspect or have any questions?

Here are 5 related questions:
1. How does the graph of \( y = -\sin(x) \) compare to \( y = \cos(x) \)?
2. What is the amplitude and phase shift of \( y = -\sin(x) \)?
3. How does the transformation affect the sine wave in \( y = a \sin(bx + c) + d \)?
4. Can you find the integral of \( y = -\sin(x) \)?
5. How would \( y = -\sin(x) \) behave if you added a constant to the equation?

**Tip:** Reflecting a function across the x-axis changes the sign of all the y-values, effectively flipping the graph upside down.

Explore the behavior and properties of the function y = -sin(x) through key points, graph analysis, and derivative insights. Learn about its reflection across the x-axis, periodicity, and critical points. Suitable for students in Grades 10-12 studying trigonometry and function transformations.

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