 5 coordinates y equals one fourth sine left parenthesis negative 3 x right parenthesis
.

The equation you've given is:

\[
y = \frac{1}{4} \sin(-3x)
\]

This is a sine function with a few modifications:
- **Amplitude**: The amplitude is \(\frac{1}{4}\), which means the sine wave will be vertically compressed by a factor of 4.
- **Frequency**: The coefficient of \(x\) inside the sine function is \(-3\), which affects the period of the sine wave. The period is calculated as \(\frac{2\pi}{|3|} = \frac{2\pi}{3}\).
- **Phase Shift**: The negative sign indicates a reflection over the y-axis, but since sine is an odd function, this doesn't affect the overall shape of the graph (it just shifts the direction of the sine wave).

### To find 5 coordinates, we can plug in different \(x\)-values into the equation.

1. **At \(x = 0\)**:
   \[
   y = \frac{1}{4} \sin(0) = 0
   \]
   So, the coordinate is \((0, 0)\).

2. **At \(x = \frac{\pi}{6}\)**:
   \[
   y = \frac{1}{4} \sin\left(-3 \times \frac{\pi}{6}\right) = \frac{1}{4} \sin\left(-\frac{\pi}{2}\right) = \frac{1}{4} \times -1 = -\frac{1}{4}
   \]
   So, the coordinate is \(\left(\frac{\pi}{6}, -\frac{1}{4}\right)\).

3. **At \(x = \frac{\pi}{3}\)**:
   \[
   y = \frac{1}{4} \sin\left(-3 \times \frac{\pi}{3}\right) = \frac{1}{4} \sin(-\pi) = \frac{1}{4} \times 0 = 0
   \]
   So, the coordinate is \(\left(\frac{\pi}{3}, 0\right)\).

4. **At \(x = \frac{\pi}{2}\)**:
   \[
   y = \frac{1}{4} \sin\left(-3 \times \frac{\pi}{2}\right) = \frac{1}{4} \sin\left(-\frac{3\pi}{2}\right) = \frac{1}{4} \times 1 = \frac{1}{4}
   \]
   So, the coordinate is \(\left(\frac{\pi}{2}, \frac{1}{4}\right)\).

5. **At \(x = \frac{2\pi}{3}\)**:
   \[
   y = \frac{1}{4} \sin\left(-3 \times \frac{2\pi}{3}\right) = \frac{1}{4} \sin\left(-2\pi\right) = \frac{1}{4} \times 0 = 0
   \]
   So, the coordinate is \(\left(\frac{2\pi}{3}, 0\right)\).

### Summary of 5 Coordinates:
1. \((0, 0)\)
2. \(\left(\frac{\pi}{6}, -\frac{1}{4}\right)\)
3. \(\left(\frac{\pi}{3}, 0\right)\)
4. \(\left(\frac{\pi}{2}, \frac{1}{4}\right)\)
5. \(\left(\frac{2\pi}{3}, 0\right)\)

Would you like more details or have any questions on this? 

Here are 5 follow-up questions related to this problem:
1. How does the amplitude affect the sine wave?
2. What is the period of a sine function with a different coefficient for \(x\)?
3. How does the reflection over the y-axis affect the graph of sine functions?
4. How would the graph change if the coefficient of \(x\) was positive?
5. What happens if we change the sine function to cosine in this equation?

**Tip**: When graphing trigonometric functions, it's helpful to first identify the amplitude, period, and any phase shifts before plotting points.

5 coordinates y equals one fourth sine left parenthesis negative 3 x right parenthesis

Explore how to find five coordinates for the function y = (1/4)sin(-3x). Understand the impact of amplitude, frequency, and reflections on the graph of a sine wave. Suitable for high school students in Grades 10-12.

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