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We are given a function of the form \( y = a \sec(bx + c) \) or \( y = a \csc(bx + c) \), and the goal is to determine the equation of the function from the graph shown.

### Analysis of the Graph:

1. **Vertical Asymptotes**: 
   From the graph, we can observe vertical asymptotes at \( x = -\pi \), \( x = \pi \), and other multiples of \( \pi \). This indicates that the function likely involves a trigonometric function with periodic asymptotes, and \( \sec \) is a good candidate because it has vertical asymptotes at multiples of \( \frac{\pi}{b} \).

2. **Shape of the Graph**: 
   The function resembles the graph of a secant function, as secant functions have a characteristic U-shape between their asymptotes and extend infinitely.

3. **Period**:
   The distance between two vertical asymptotes is \( 2\pi \). This suggests that the period of the secant function is \( 2\pi \), which corresponds to a value of \( b = 1 \) in the equation \( y = a \sec(bx + c) \).

4. **Amplitude and Vertical Shift**:
   - The maximum and minimum values of the function appear to be \( y = 5 \) and \( y = -5 \), suggesting that \( a = 5 \).
   - There doesn't appear to be any vertical shift, so \( c = 0 \).

### Conclusion:
The equation of the function is \( y = 5 \sec(x) \).

Let me know if you need further clarification or additional details.

Here are some related questions:
1. What is the period of the secant function?
2. How do we determine vertical asymptotes of trigonometric functions?
3. What role does the amplitude \( a \) play in the secant function?
4. How does the vertical shift affect trigonometric graphs?
5. Can you graph a similar function with a phase shift?

**Tip:** In secant and cosecant functions, the vertical asymptotes occur where the corresponding cosine or sine function is zero.

The function graphed to the right is of the form y = a sec bx + c or y = a csc bx + c for some a ≠ 0, b > 0. Determine the equation of the function.

Learn how to determine the equation of a trigonometric function, secant or cosecant, based on graph features such as vertical asymptotes, amplitude, and period. The problem provides a step-by-step guide to solving functions of the form y = a sec(bx + c) or y = a csc(bx + c).

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