To graph the functions \( y = -8 \sin x \) and \( y = \sin x \), you can follow these steps to capture their key features, such as amplitude, period, and reflection. Below is a detailed explanation for each function:

### Graphing \( y = \sin x \)

1. **Amplitude:** The amplitude is the maximum distance from the midline of the graph. For \( y = \sin x \), the amplitude is 1.

2. **Period:** The period of a sine function is the length of one complete cycle. The period is given by \( 2\pi \).

3. **Key Points:** 
   - Start at \( (0, 0) \).
   - \( \left(\frac{\pi}{2}, 1\right) \) is the maximum point.
   - \( (\pi, 0) \) is the point where the graph crosses the x-axis again.
   - \( \left(\frac{3\pi}{2}, -1\right) \) is the minimum point.
   - \( (2\pi, 0) \) completes one full cycle.

4. **Shape:** The graph oscillates between -1 and 1, completing a wave every \( 2\pi \).

5. **Plot:** Use these points to plot the graph, connecting them smoothly in a sinusoidal shape.

### Graphing \( y = -8 \sin x \)

1. **Amplitude:** The amplitude is 8, which means the graph will oscillate between -8 and 8.

2. **Period:** The period is still \( 2\pi \) because the period of the sine function does not change with vertical scaling.

3. **Reflection:** The negative sign indicates a reflection over the x-axis.

4. **Key Points:**
   - Start at \( (0, 0) \).
   - \( \left(\frac{\pi}{2}, -8\right) \) is the minimum point due to the reflection.
   - \( (\pi, 0) \) is where the graph crosses the x-axis again.
   - \( \left(\frac{3\pi}{2}, 8\right) \) is the maximum point.
   - \( (2\pi, 0) \) completes one full cycle.

5. **Shape:** The graph follows a similar shape to \( y = \sin x \), but it is vertically stretched and flipped.

6. **Plot:** Plot these points and connect them smoothly, noting the increased amplitude and reflection.

### Visualizing the Graphs

- **\( y = \sin x \):** This function oscillates between -1 and 1, creating a smooth wave-like pattern across the x-axis.

- **\( y = -8 \sin x \):** This graph is a vertical stretch by a factor of 8 and is reflected across the x-axis compared to \( y = \sin x \). The wave now oscillates between -8 and 8.

### Summary

- **Amplitude:**
  - \( y = \sin x \): 1
  - \( y = -8 \sin x \): 8 (with a reflection)

- **Period:** Both functions have a period of \( 2\pi \).

- **Reflection:** The negative sign in \( y = -8 \sin x \) causes it to flip over the x-axis compared to \( y = \sin x \).

By identifying these features, you can accurately plot both graphs, clearly illustrating the differences between them.

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Would you like further assistance or have any questions about these graphs? Here are some related topics that might interest you:

1. How do you find the phase shift of a trigonometric function?
2. What is the effect of a vertical shift in sine and cosine functions?
3. How does the graph of \( y = \cos x \) compare to \( y = \sin x \)?
4. What transformations affect the period of a sine function?
5. How would you graph a sine function with a phase shift?
6. What is the effect of combining transformations on a trigonometric function?
7. How can you graph a sine function with a different frequency?
8. What are the steps to sketching the graph of a transformed cosine function?

**Tip:** When graphing trigonometric functions, carefully consider each transformation, such as amplitude changes, reflections, phase shifts, and vertical shifts, to accurately depict their behavior.

Learn how to graph y=-8 sinx and y=sin x with detailed steps on understanding amplitude, period, and reflection. Explore the differences between these sine functions and how to plot them accurately. Suitable for Grades 10-12 students studying trigonometry.

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