https://file.aiquickdraw.com/mathbot/file/1728777336036cp1rt9k0.jpg

From the image provided, here is the information extracted and analyzed:

1. **Amplitude and Period** (Question a, part i):
   - The equation provided for the graph is \( y = 3 \cos(2x) - 1 \).
   - **Amplitude**: The amplitude of a cosine function \( y = A \cos(Bx) + C \) is given by the absolute value of \( A \). Here, \( A = 3 \), so the **amplitude is 3**.
   - **Period**: The period of a cosine function is calculated as \( \frac{2\pi}{B} \). Here, \( B = 2 \), so the period \( = \frac{2\pi}{2} = \pi \). In degrees, this is equivalent to \( 180^\circ \), so the **period is 2** (in terms of the problem, this likely represents a simplified unit).

2. **Graph** (Question ii):
   - The function \( y = 3 \cos(2x) - 1 \) is graphed on the \( y \)-axis against the \( x \)-axis. 
   - Key points at each quarter cycle (90°, 180°, 270°, 360°) are marked, showing the oscillation from a maximum of 2 to a minimum of -4.
   - The phase shift and vertical shift (downward by 1 unit due to "-1") are applied.

Would you like a step-by-step breakdown of the graph characteristics or other parts of the question?

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### Additional Questions to Explore:
1. How does changing the amplitude affect the shape of a cosine graph?
2. What impact does adjusting the period have on the cosine function's frequency?
3. How would a phase shift (e.g., \( y = 3 \cos(2x - \frac{\pi}{4}) \)) affect the graph?
4. What happens if a sine function is used instead of cosine, keeping amplitude and period the same?
5. How is the graph affected by altering the vertical shift (e.g., \( y = 3 \cos(2x) + 2 \))?

**Tip:** Remember that amplitude affects the "height" of peaks and troughs, while period determines the "width" or frequency of cycles.

Graph the function y = 3 cos(2x) - 1 and determine the amplitude and period.

Learn how to graph the cosine function y = 3 cos(2x) - 1, identifying key features such as the amplitude of 3 and the period of π. This problem covers trigonometric concepts and graphing techniques, ideal for students in Grades 10-12.

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