https://file.aiquickdraw.com/mathbot/file/1726528864918fmxmuwr8.jpg

The graph in the image shows a cosecant (csc) function. The cosecant function, \( \csc(x) \), is the reciprocal of the sine function, i.e.,

\[
\csc(x) = \frac{1}{\sin(x)}.
\]

### Analyzing the graph:
- The graph consists of two branches between asymptotes.
- The period (distance between consecutive asymptotes) is \( 2\pi \), as seen between \( -\pi \) and \( \pi \).
- The vertical asymptotes appear at \( x = -2\pi, -\pi, 0, \pi, 2\pi \), which is typical for a cosecant graph since \( \csc(x) \) is undefined wherever \( \sin(x) = 0 \).
- The amplitude appears to be 3 (the peak and trough values seem to be \( 3 \) and \( -3 \)).

### Equation of the graph:
The standard form of a cosecant function is:
\[
f(x) = A \csc(Bx)
\]
where:
- \( A \) controls the amplitude,
- \( B \) controls the period.

Since the period is \( 2\pi \), and the amplitude is 3, the equation for the given graph is:
\[
f(x) = 3 \csc(x).
\]

This is the equation for the graph.

Would you like more details, or do you have any further questions?

### Relative Questions:
1. What is the relationship between sine and cosecant graphs?
2. How do vertical asymptotes in cosecant graphs correspond to sine function zeros?
3. How does changing the amplitude \( A \) in \( A \csc(Bx) \) affect the graph?
4. What happens if the period of a cosecant function is altered?
5. Can we shift a cosecant graph horizontally or vertically? How?

**Tip:** Always remember that cosecant functions have vertical asymptotes where the sine function equals zero, and these asymptotes repeat at every multiple of \( \pi \).

Explore how to write the equation of a cosecant function graph with a period of 2π and an amplitude of 3. This guide covers the fundamental properties of the cosecant function, useful for high school students.

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