To determine the equation of the function graphed, we need to analyze the key features of the graph and match them to the general forms provided, which are either:

• $y = a \sec(bx) + c$
• $y = a \csc(bx) + c$

Let's break down the characteristics of the graph:

1. Vertical Asymptotes: The vertical asymptotes occur at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$, which suggests that the function has a secant form since secant functions have asymptotes where the cosine function is zero, i.e., $\sec(x)$ has vertical asymptotes where $\cos(x) = 0$.

2. Shape of the Graph: The graph has a U-like shape between the asymptotes, typical of a secant graph. This further supports the idea that the function is of the form $y = a \sec(bx) + c$.

3. Amplitude and Vertical Shift: The minimum value of the graph appears to be around $y = -2$, and the vertical shift seems to be centered around $y = 0$, which means there is no vertical shift, so $c = 0$.

4. Period of the Function: The distance between the asymptotes is $\pi$, which indicates that the period of the function is $\frac{2\pi}{b} = \pi$, giving $b = 2$.

5. Amplitude: The function reaches a maximum of about $y = 10$, and a minimum of about $y = -10$, suggesting $a = 10$.

Final Equation:

Given these observations, the equation of the function is: $y = 10 \sec(2x)$

Would you like a detailed explanation of any step? Here are some follow-up questions you could explore:

1. How do we determine the amplitude of secant and cosecant graphs?
2. Why do secant functions have vertical asymptotes at certain points?
3. How can we identify the period of a trigonometric function from its graph?
4. What changes in the graph would occur if there were a vertical shift (i.e., $c \neq 0$)?
5. How does the factor $b$ affect the frequency of the graph?

Tip: When analyzing the graph of trigonometric functions, identifying key points like vertical asymptotes, peaks, and troughs helps in determining the function's parameters.