https://file.aiquickdraw.com/mathbot/file/1728272392997ll2tehf1.jpg

The graph in the image shows a curve resembling a rational function with a vertical asymptote around \(x = 1\) and behavior typical of a function like \(y = \frac{1}{x-1}\).

From visual inspection:

1. **Vertical Asymptote**: The curve shoots upwards around \(x = 1\), indicating a vertical asymptote at \(x = 1\).
2. **End Behavior**: As \(x\) moves further to the left, the curve approaches the x-axis, suggesting that as \(x \to -\infty\), \(y \to 0\). For positive \(x\), the curve becomes negative as \(x\) increases from 1 to 2.
3. **Shape**: The curve behaves similarly to a rational function where the numerator and denominator have a specific relationship, typically something like \(y = \frac{A}{x - 1} + B\).

### Likely Equation
Based on the graph, the function is most likely of the form:

\[
y = \frac{-1}{x - 1}
\]

This rational function accounts for the vertical asymptote at \(x = 1\) and the behavior of the curve decreasing as \(x\) increases from 1. The negative sign ensures the correct orientation of the curve.

Would you like a detailed step-by-step explanation on how this form was derived? Let me know!

### Related Questions:
1. What is the effect of changing the numerator in a rational function?
2. How do vertical asymptotes affect the shape of a graph?
3. How would the graph change if the asymptote were at \(x = 2\)?
4. What happens when the degree of the polynomial in the numerator is greater than that of the denominator?
5. Can you describe how horizontal asymptotes influence the behavior of a function at extreme values of \(x\)?

**Tip**: For rational functions, the vertical asymptote occurs where the denominator equals zero, while the horizontal asymptote depends on the degree of the numerator and denominator.

Write an equation for the function graphed above.

Learn how to find the equation of a rational function with a vertical asymptote, as seen in the graph. This problem involves analyzing the asymptotic behavior and using the standard rational function formula y = A/(x - a). Suitable for students in Grades 10-12.

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