The image you uploaded contains trigonometric functions where you're asked to determine the amplitude, period, and sketch the graph for one period. Let's solve each part:

General form of Trigonometric Functions:

1. Sine Function: $y = A \sin(Bx)$
2. Cosine Function: $y = A \cos(Bx)$

• Amplitude: $|A|$ (the coefficient of sine or cosine)
• Period: $\frac{2\pi}{|B|}$ (where $B$ is the coefficient of $x$)

Let's analyze each function:

1. $y = 2 \sin x$

• Amplitude: $|2| = 2$
• Period: $\frac{2\pi}{1} = 2\pi$

2. $y = 3 \sin x$

• Amplitude: $|3| = 3$
• Period: $\frac{2\pi}{1} = 2\pi$

3. $y = 4 \sin x$

• Amplitude: $|4| = 4$
• Period: $\frac{2\pi}{1} = 2\pi$

4. $y = 6 \cos x$

• Amplitude: $|6| = 6$
• Period: $\frac{2\pi}{1} = 2\pi$

5. $y = 6 \cos \left( \frac{x}{4} \right)$

• Amplitude: $|6| = 6$
• Period: $\frac{2\pi}{\frac{1}{4}} = 8\pi$

6. $y = 6 \cos(4x)$

• Amplitude: $|6| = 6$
• Period: $\frac{2\pi}{4} = \frac{\pi}{2}$

Would you like details on how to sketch these graphs or how the calculations were made?

Here are 5 related questions to deepen your understanding:

1. How do you determine the phase shift of a sine or cosine function?
2. What is the relationship between period and frequency in trigonometric functions?
3. Can you explain how to determine the vertical shift of a sine function?
4. What would happen to the graph if the sine or cosine function had a negative amplitude?
5. How do you sketch multiple periods of a trigonometric function?

Tip: Always check the coefficient of $x$ to determine the period of a trigonometric function. It controls how "stretched" or "compressed" the graph is along the x-axis.