https://file.aiquickdraw.com/mathbot/file/1726427047378yhjr1f59.jpg

The graph appears to depict a function of the form \( y = a \sec(bx) + c \) or \( y = a \csc(bx) + c \), based on the options provided in the problem. Let's break down the graph to determine the exact equation.

### Key observations:
1. **Type of function**: 
   The graph shows vertical asymptotes and a periodic behavior typical of secant or cosecant functions. The minimum point in the middle and the vertical asymptotes suggest that this is a **secant function**, \( y = a \sec(bx) + c \).
   
2. **Period**:
   The period of the secant function is \( \frac{2\pi}{b} \). From the graph, the vertical asymptotes occur at \( x = -\pi \) and \( x = \pi \), meaning the period is \( 2\pi \). Hence, \( b = 1 \).

3. **Amplitude and vertical shift**:
   - The minimum value occurs at \( y = -5 \) and the midline (horizontal axis around which the function oscillates) is at \( y = -5 \).
   - This indicates a vertical shift \( c = -5 \).
   - The maximum value is \( y = 5 \), meaning the amplitude is 10. Since secant functions typically range from \( (-\infty, -1] \cup [1, \infty) \), the coefficient \( a \) is 5 to scale the function appropriately.

### Final equation:
Thus, the equation of the function is:

\[
y = 5 \sec(x) - 5
\]

Would you like more details on any of these steps?

### Related questions:
1. How does the graph of \( \sec(x) \) differ from \( \csc(x) \)?
2. How is the period of a secant function calculated?
3. What is the significance of the vertical asymptotes in secant and cosecant graphs?
4. How do vertical shifts affect the equation of trigonometric functions?
5. How can you determine the amplitude and period just by looking at a graph?

### Tip:
To easily identify secant or cosecant functions from their graphs, look for their characteristic vertical asymptotes and the positioning of the curves relative to the midline.

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 secant or cosecant function based on its graph. This problem covers key trigonometric concepts, including the period, amplitude, and vertical shifts. Suitable for high school students (Grades 10-12).

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