https://file.aiquickdraw.com/mathbot/file/1725587364343vnp0utqv.jpg

To identify the correct graph of the given function, let me first analyze the equation. The function provided in the image appears to be:

\[
f(x) = \frac{2x}{x^2 - 5}
\]

Now, let's discuss how this function behaves:

### Key Points to Consider:
1. **Vertical Asymptotes:**
   Vertical asymptotes occur where the denominator is zero. Set the denominator equal to zero:
   \[
   x^2 - 5 = 0 \implies x = \pm \sqrt{5}
   \]
   Therefore, vertical asymptotes will occur at \(x = \sqrt{5}\) and \(x = -\sqrt{5}\).

2. **Horizontal Asymptote:**
   The degree of the numerator and denominator are the same (both are degree 2 if simplified), so the horizontal asymptote is determined by the ratio of the leading coefficients. In this case, it is:
   \[
   y = 0
   \]
   This is the horizontal asymptote as \(x\) approaches infinity.

3. **Behavior Near Asymptotes:**
   The graph will approach these asymptotes but never touch them. The function could exhibit a reflection, depending on the values of \(x\).

### Based on these key features, we can now identify the graph.

From the graphs provided (Graph A through Graph F), the correct graph will have:
- Vertical asymptotes at \(x = \pm \sqrt{5}\),
- A horizontal asymptote at \(y = 0\),
- And the general behavior (shape) of a rational function.

After carefully inspecting the images of the graphs, **Graph E** appears to match this behavior.

Would you like further details on any part of the analysis? Here are five follow-up questions to consider:

1. How do vertical asymptotes affect the shape of a graph?
2. What is the relationship between the degrees of the numerator and denominator for determining horizontal asymptotes?
3. How does the function behave near zero?
4. Can you explain why the graph doesn't touch its asymptotes?
5. How would the graph change if we adjusted the function's coefficients?

**Tip:** Always start by analyzing asymptotes when graphing rational functions, as they provide critical boundaries for the graph's behavior.

Learn how to identify the correct graph of a rational function by analyzing its vertical and horizontal asymptotes. This problem involves key concepts like asymptote behavior and graph analysis, suitable for students in Grades 10-12.

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