graph the following functions over one period:
y=sinx
y=cosx

identiry and label the amplitude, period and any key points
explain the significance of the amplitude and period in the context of waves or socillation 

Let's begin by graphing the functions \( y = \sin(x) \) and \( y = \cos(x) \) over one period. We'll also identify and label the amplitude, period, and key points on the graphs.

### Graphs of \( y = \sin(x) \) and \( y = \cos(x) \)

- **Function:** \( y = \sin(x) \)
- **Amplitude:** 1
- **Period:** \( 2\pi \)
- **Key Points:**
  - \( (0, 0) \)
  - \( \left(\frac{\pi}{2}, 1\right) \)
  - \( (\pi, 0) \)
  - \( \left(\frac{3\pi}{2}, -1\right) \)
  - \( (2\pi, 0) \)

- **Function:** \( y = \cos(x) \)
- **Amplitude:** 1
- **Period:** \( 2\pi \)
- **Key Points:**
  - \( (0, 1) \)
  - \( \left(\frac{\pi}{2}, 0\right) \)
  - \( (\pi, -1) \)
  - \( \left(\frac{3\pi}{2}, 0\right) \)
  - \( (2\pi, 1) \)

Let’s graph these functions.

### Explanation of Amplitude and Period

- **Amplitude**: The amplitude of a wave is the maximum displacement of the wave from its mean position (the central axis). For the functions \( y = \sin(x) \) and \( y = \cos(x) \), the amplitude is 1, meaning the wave oscillates between 1 and -1. In the context of waves or oscillations, the amplitude represents the energy or intensity of the wave. A larger amplitude would indicate a more powerful wave (e.g., louder sound or brighter light).

- **Period**: The period of a wave is the distance (along the x-axis) required for the wave to complete one full cycle. For both \( y = \sin(x) \) and \( y = \cos(x) \), the period is \( 2\pi \). In wave or oscillation contexts, the period is the time it takes for one complete cycle of the wave to pass a fixed point. This could represent, for example, the time between successive peaks of a sound wave.

Would you like to see the graph or need any further explanation? Here are some questions to expand on this topic:

1. How would the graphs of \( y = \sin(x) \) and \( y = \cos(x) \) change if the amplitude were doubled?
2. What would happen to the graphs if the period were halved?
3. Can you describe the graph of \( y = -\sin(x) \)? How does it compare to \( y = \sin(x) \)?
4. How do the sine and cosine functions relate to circular motion?
5. What is the significance of phase shift in wave functions?

**Tip:** When working with trigonometric functions, always pay attention to the amplitude and period as they significantly affect the shape and behavior of the graph.

graph the following functions over one period: y=sinx y=cosx identify and label the amplitude, period and any key points explain the significance of the amplitude and period in the context of waves or oscillation

Learn how to graph the sine and cosine functions over one period, identify and label the amplitude, period, and key points. Understand the significance of amplitude and period in the context of waves and oscillations. Suitable for students in Grades 9-12.

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