https://file.aiquickdraw.com/mathbot/file/172650940636546mz6e7e.jpg

The graph you provided looks like it depicts a **tangent function** or some related function because of the periodic asymptotes and the general shape of the curves (similar to the behavior of tangent or cotangent functions).

### Key features in the graph:
1. **Vertical Asymptotes**: The vertical dashed lines appear at multiples of \(\frac{3\pi}{24} = \frac{\pi}{8}\), indicating periodic discontinuities, which is typical for tangent or cotangent functions.
2. **Increasing behavior**: The curve is increasing between the vertical asymptotes, which suggests a tangent function rather than cotangent (which would be decreasing).
3. **Period**: The period of the graph seems to be \(\frac{2\pi}{3}\) based on the spacing between the asymptotes.
4. **Horizontal Shift**: The asymptotes do not occur exactly at integer multiples of \(\pi\), but instead at \(\frac{3\pi}{24}\), indicating a phase shift.

Given these observations, a possible formula for the function could be:
\[
y = 5 \tan \left( \frac{8}{3} t \right)
\]
where:
- The coefficient \( \frac{8}{3} \) accounts for the period adjustment, as the tangent function normally has a period of \(\pi\), and here it is stretched to match the periodicity in the graph.
- The constant multiplier \(5\) scales the function vertically to match the height of 5 observed in the graph.

Would you like more details on how this formula was derived or have any specific questions?

Here are 5 related questions to deepen understanding:
1. How do vertical asymptotes relate to the period of a tangent function?
2. Why does the tangent function have vertical asymptotes?
3. What does the vertical stretch (the multiplier \(5\)) do to the graph of the tangent function?
4. How would a phase shift affect the position of the asymptotes?
5. Can you explain how the period of the tangent function is modified using the coefficient of \(t\) inside the function?

**Tip**: The period of the tangent function is \(\pi\), but it can be adjusted by multiplying \(t\) inside the function by a constant, which either stretches or compresses the period.

Give a possible formula for the function shown in the graph with periodic asymptotes and increasing behavior between vertical asymptotes at intervals of multiples of 3π/24, resembling a tangent function.

This problem involves deriving a tangent function formula based on the graph shown. The graph exhibits vertical asymptotes at multiples of 3π/24, a period of 2π/3, and increasing behavior. Explore key concepts like tangent function periodicity, asymptotes, and vertical scaling.

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