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.