https://file.aiquickdraw.com/mathbot/file/1729923369412v6cb2y18.jpg

The function \( f(x) \) in the image shows a vertical asymptote near \( x = -2 \). This indicates that the function might be in the form of a rational expression like:

\[
f(x) = \frac{a}{x + b} + c
\]

where:
- \( b \) would shift the vertical asymptote to \( x = -2 \),
- \( a \) affects the "steepness" or "stretch" of the curve, and
- \( c \) would indicate any horizontal shift.

For vertical asymptotes:
1. If the function has a vertical asymptote at \( x = -2 \), then we set \( b = 2 \) so that the denominator becomes zero when \( x = -2 \).
   
2. The general form could then be:
   \[
   f(x) = \frac{a}{x + 2} + c
   \]

Would you like to go further into finding the exact values for \( a \) and \( c \) based on other points on the curve?

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Here are 5 related questions for further exploration:
1. What is a vertical asymptote, and how is it determined for rational functions?
2. How does the value of \( a \) in \( \frac{a}{x + b} \) impact the graph's shape?
3. What role does the constant \( c \) play in shifting the graph vertically?
4. How can you use other points on the graph to find specific values of \( a \) and \( c \)?
5. How does the horizontal asymptote differ from the vertical asymptote in rational functions?

**Tip:** To find the vertical asymptote, identify the \( x \)-value where the function approaches infinity or negative infinity.

Write an expression of the form f(x) = for the given graph with a vertical asymptote.

Discover how to write an expression for a rational function based on its graph, focusing on identifying vertical asymptotes and determining the function's general form. This example includes concepts like asymptotes and rational functions, ideal for high school math students in Grades 10-12.

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