https://file.aiquickdraw.com/mathbot/file/1728432503691nqn7yceu.jpg

The graph shows a rational function with vertical asymptotes at \( x = -2 \) and \( x = 4 \). The function has a hyperbola-like behavior with these asymptotes, indicating that the function likely involves terms of the form \( \frac{1}{x - a} \), where \( a \) corresponds to the location of the asymptotes.

The vertical asymptotes are at \( x = -2 \) and \( x = 4 \), suggesting that the denominator of the rational function should include the factors \( (x + 2) \) and \( (x - 4) \). The overall structure of the function seems to follow a reciprocal relationship, and the graph looks like the following function:

\[
y = \frac{1}{(x + 2)(x - 4)}
\]

This function will have vertical asymptotes where the denominator is zero, at \( x = -2 \) and \( x = 4 \), consistent with the graph.

Would you like further details or explanations on this? Here are some related questions to explore this concept further:

1. How do vertical asymptotes affect the shape of a graph?
2. What happens to the function as \( x \) approaches the vertical asymptotes?
3. How can you determine the horizontal asymptote of a rational function?
4. Can you manipulate the numerator to change the graph without changing the asymptotes?
5. How would adding constants to the function affect the graph?

**Tip:** Always check the locations of asymptotes and intercepts to identify the form of a rational function.

Write an equation for the function graphed below (from image).

This page helps you find the equation of a rational function with vertical asymptotes at x = -2 and x = 4, explaining the mathematical concepts and behavior of rational functions, including their asymptotes and structure.

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