https://file.aiquickdraw.com/mathbot/file/1726235678888h172wflr.jpg

The image you have uploaded contains a set of trigonometric functions, particularly in part B, where you are asked to **sketch the graphs** of these equations over specific intervals. I'll break down the general approach for sketching each function based on its format and interval. Here's how to analyze and sketch each one:

### 1. \( f(x) = 4.5 \sin(3x) \), \( 0 \leq x \leq \pi \)
- **Amplitude**: 4.5 (Vertical stretch)
- **Frequency/Period**: \( \frac{2\pi}{3} \), since \( f(x) = \sin(3x) \) means the period is \( \frac{2\pi}{3} \).
- **Sketch Tips**: Within the interval \( 0 \leq x \leq \pi \), plot one full cycle (since \( \frac{2\pi}{3} \) is less than \( \pi \)) and adjust the amplitude accordingly.

### 2. \( f(x) = 5 \sin(32x) \), \( 0 \leq x \leq 2\pi \)
- **Amplitude**: 5
- **Frequency/Period**: \( \frac{2\pi}{32} = \frac{\pi}{16} \), so it completes many oscillations (32 full cycles) over \( 0 \leq x \leq 2\pi \).
- **Sketch Tips**: Show multiple cycles within the interval, adjusting for amplitude.

### 3. \( f(x) = \sin\left( \frac{1}{4}x \right) \), \( 0 \leq x \leq 8\pi \)
- **Amplitude**: 1
- **Frequency/Period**: \( \frac{2\pi}{\frac{1}{4}} = 8\pi \), meaning one full cycle from \( 0 \) to \( 8\pi \).
- **Sketch Tips**: Show one complete sine wave over the entire interval.

### 4. \( f(x) = -3 \sin(2x) \), \( 0 \leq x \leq \pi \)
- **Amplitude**: 3 (negative means reflection across the x-axis)
- **Frequency/Period**: \( \frac{2\pi}{2} = \pi \), so one full cycle.
- **Sketch Tips**: Plot one full negative sine wave.

### 5. \( f(x) = 4 \cos(2x) \), \( 0 \leq x \leq 4\pi \)
- **Amplitude**: 4
- **Frequency/Period**: \( \frac{2\pi}{2} = \pi \), completing four full cycles.
- **Sketch Tips**: Draw four cycles with maximum amplitude 4 and minima at -4.

### 6. \( f(x) = \frac{1}{2} \sin\left( \frac{1}{2}x \right) \), \( 0 \leq x \leq 2\pi \)
- **Amplitude**: \( \frac{1}{2} \)
- **Frequency/Period**: \( \frac{2\pi}{\frac{1}{2}} = 4\pi \), completing half a cycle from \( 0 \) to \( 2\pi \).
- **Sketch Tips**: Plot half of a sine wave (because of the 4π period) with small amplitude.

### 7. \( f(x) = 7 \cos\left( \frac{1}{3}x \right) \), \( 0 \leq x \leq 6\pi \)
- **Amplitude**: 7
- **Frequency/Period**: \( \frac{2\pi}{\frac{1}{3}} = 6\pi \), completing one full cycle.
- **Sketch Tips**: Show one full cosine wave with large amplitude.

### 8. \( f(x) = -\frac{1}{4} \sin(4x) \), \( 0 \leq x \leq 2\pi \)
- **Amplitude**: \( \frac{1}{4} \)
- **Frequency/Period**: \( \frac{2\pi}{4} = \frac{\pi}{2} \), completing four full cycles.
- **Sketch Tips**: Show four full negative sine waves (reflected) with small amplitude.

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If you'd like more details on how to sketch these functions or need assistance with another part of the question, feel free to ask!

### 5 Related Questions:
1. How does changing the amplitude affect the graph of sine and cosine functions?
2. What is the relationship between period and frequency in trigonometric functions?
3. How would you modify the sketch if a phase shift was introduced in any of these functions?
4. How can you determine the key points (intercepts, maximums, and minimums) for these functions?
5. What differences would you expect when comparing sine and cosine graphs with the same amplitude and period?

**Tip**: For trigonometric functions, the period is calculated as \( \frac{2\pi}{|B|} \) for \( f(x) = A\sin(Bx) \) or \( f(x) = A\cos(Bx) \), where \( B \) is the coefficient of \( x \).

Sketch the graphs of the following equations on the specified intervals.

Learn how to sketch sine and cosine graphs, including transformations such as amplitude and period changes. This detailed guide covers various trigonometric functions and their corresponding graphs, suitable for high school students.

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