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

1. f(x) = 4.5sin 3x 0<=^ x <= pi 2. f(x) = 5sin 32x 0 <= x <= 2pi

3. f(x) = sin 1/4 * x, 0 <= x <= 8pi

4. f(x) = - 3sin 2x, 0 <= x <= pi

5. f(x) = 4cos 1/2 * x, 0 <= x <= 4pi

6. f(x) = 1/2 * sin 1/2 * x, 0 <= x <= 2pi

7. f(x) = 7cos 1/3 * x, 0 <= x <= 6pi

8. f(x) = - 1/4 * sin 4x 0 <= x <= 2pi

Here are the sketches of the graphs for the given trigonometric functions:

1. \( f(x) = 4.5\sin(3x) \), for \( 0 \leq x \leq \pi \)
2. \( f(x) = 5\sin(32x) \), for \( 0 \leq x \leq 2\pi \)
3. \( f(x) = \sin\left(\frac{1}{4}x\right) \), for \( 0 \leq x \leq 8\pi \)
4. \( f(x) = -3\sin(2x) \), for \( 0 \leq x \leq \pi \)
5. \( f(x) = 4\cos\left(\frac{1}{2}x\right) \), for \( 0 \leq x \leq 4\pi \)
6. \( f(x) = \frac{1}{2}\sin\left(\frac{1}{2}x\right) \), for \( 0 \leq x \leq 2\pi \)
7. \( f(x) = 7\cos\left(\frac{1}{3}x\right) \), for \( 0 \leq x \leq 6\pi \)
8. \( f(x) = -\frac{1}{4}\sin(4x) \), for \( 0 \leq x \leq 2\pi \)

Would you like a detailed explanation of any specific graph or function?

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Here are 5 questions to explore further:
1. How does changing the amplitude affect the graph of a sine or cosine function?
2. How does the frequency (inside the sine or cosine) impact the period of the graph?
3. What is the effect of a negative sign in front of sine or cosine?
4. How do you determine the number of complete cycles within a given interval?
5. What is the phase shift, and how does it affect the graph?

**Tip:** The coefficient of \(x\) in the sine or cosine function controls the frequency, which determines how many cycles fit into \(2\pi\).

Sketch the graphs of the following equations on the specified intervals.
1. f(x) = 4.5sin 3x, 0 <= x <= pi
2. f(x) = 5sin 32x, 0 <= x <= 2pi
3. f(x) = sin (1/4) * x, 0 <= x <= 8pi
4. f(x) = -3sin 2x, 0 <= x <= pi
5. f(x) = 4cos (1/2) * x, 0 <= x <= 4pi
6. f(x) = (1/2) * sin (1/2) * x, 0 <= x <= 2pi
7. f(x) = 7cos (1/3) * x, 0 <= x <= 6pi
8. f(x) = -(1/4) * sin 4x, 0 <= x <= 2pi

Learn how to sketch the graphs of sine and cosine functions with different amplitudes and frequencies. This problem covers key trigonometric concepts, including how to calculate the amplitude, period, and cycles for functions like f(x) = 4.5sin(3x), f(x) = 5sin(32x), and more. Suitable for students in Grades 10-12.

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