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Let's go through each part of the questions visible in the image:

### Question 12:
**Identify which graphs represent a sine function.**
- A sine function is continuous and smooth, oscillating between a maximum and minimum value in a repetitive wave pattern. The function will have no asymptotes and will not diverge to infinity.
  
**Answer for Question 12:**
- **Graph A** and **Graph C** resemble typical sine functions as they follow a smooth, wave-like oscillation without any vertical asymptotes. 

### Question 13:
**Identify which graphs represent a cosecant function.**
- The cosecant function, \( \csc(x) = \frac{1}{\sin(x)} \), has vertical asymptotes at points where the sine function is zero (where the function diverges to infinity or negative infinity). This leads to a characteristic set of curves that appear like "arches" or "inverse parabolas" with asymptotes at regular intervals.

**Answer for Question 13:**
- **Graph B** and **Graph D** illustrate cosecant functions because they show the characteristic vertical asymptotes and the upward and downward arches between these asymptotes.

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### II. Directions:
**Analyze the given functions and determine their properties, then sketch their graphs:**

#### For \( y = 3 \sin\left(x - \frac{\pi}{3}\right) \):
1. **Amplitude**: The coefficient of the sine function, which is \( 3 \).
2. **Period**: For a sine function \( y = a \sin(bx + c) \), the period is \( \frac{2\pi}{|b|} \). Here \( b = 1 \), so the period is \( 2\pi \).
3. **Phase Shift**: The function has \( -\frac{\pi}{3} \) inside the argument of sine, so the phase shift is to the **right by \( \frac{\pi}{3} \)**.
4. **Vertical Shift**: None, as there’s no constant term added or subtracted from the function.
5. **Graph Sketch**: The graph will resemble a sine wave with peaks at 3 and valleys at -3, shifted right by \( \frac{\pi}{3} \).

#### For \( y = 3 \cos\left(x + \frac{\pi}{3}\right) - 1 \):
1. **Amplitude**: The coefficient of the cosine function, which is \( 3 \).
2. **Period**: For \( y = a \cos(bx + c) \), the period is \( \frac{2\pi}{|b|} \). Here \( b = 1 \), so the period is \( 2\pi \).
3. **Phase Shift**: The function has \( +\frac{\pi}{3} \) inside the argument, which means a **left shift of \( \frac{\pi}{3} \)**.
4. **Vertical Shift**: There is a vertical shift down by 1 unit.
5. **Graph Sketch**: The graph will resemble a cosine wave with peaks at 2 and valleys at -4, shifted left by \( \frac{\pi}{3} \) and vertically shifted down by 1.

Would you like a detailed graph sketch for these functions or further explanation on any of the properties?

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### Related Questions:
1. What are the defining characteristics of a sine function compared to a cosine function?
2. How do amplitude and period changes affect the graph of trigonometric functions?
3. How is the graph of the cosecant function related to the sine function graph?
4. What is the impact of a vertical shift on the graph of a trigonometric function?
5. How can phase shifts be determined for functions with more complex expressions in the argument?

#### Tip:
When identifying trigonometric functions from graphs, remember that sine and cosine functions are continuous without asymptotes, while cosecant and secant functions will always exhibit asymptotes at specific intervals.

Which of the following graphs illustrate a sine function and a cosecant function? Additionally, analyze the given trigonometric functions to determine amplitude, period, phase shift, vertical shift, and sketch the graphs.

Learn to identify sine and cosecant functions from graphs and analyze trigonometric functions for amplitude, period, phase shift, and vertical shift. This problem includes determining graph properties and sketching trigonometric functions.

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