https://file.aiquickdraw.com/mathbot/file/1726454196441f2t9118p.jpg

The graph you have uploaded shows a rational function with two vertical asymptotes and a general shape that suggests a function of the form:

\[
y = \frac{a}{(x - p)(x - q)}
\]

### Key observations:
1. **Vertical asymptotes** occur at \(x = -2\) and \(x = 2\), which indicate the factors of the denominator are \((x + 2)\) and \((x - 2)\).
   
2. The graph suggests that the function is undefined at these vertical lines, and the curve approaches infinity or negative infinity near them.

3. The general shape of the graph between the vertical asymptotes shows the function crossing the y-axis at \(y = 0\). Therefore, the numerator must be \(x^2\) since the function is symmetric about the origin (in this case).

### Equation form:
The equation should look like:

\[
y = \frac{k \cdot x^2}{(x + 2)(x - 2)}
\]

where \(k\) is a constant that adjusts the steepness or scale of the graph.

### Simplifying further:
The denominator is simply the difference of squares:

\[
y = \frac{k \cdot x^2}{x^2 - 4}
\]

Based on the graph's symmetry and shape, we can assume that \(k = 1\) for now, but it can be adjusted based on further information or specific scale factors from the graph.

Thus, the final equation is:

\[
y = \frac{x^2}{x^2 - 4}
\]

Would you like more details on how I derived this equation, or do you have any further questions?

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Here are 5 related questions to explore this topic further:
1. How do vertical asymptotes affect the form of a rational function?
2. What would happen to the graph if the numerator had a higher power of \(x\)?
3. How do we determine the horizontal asymptotes for rational functions like this?
4. How does symmetry help identify the equation of a graph?
5. How does changing the constant \(k\) in the equation affect the steepness of the graph?

**Tip:** When analyzing rational functions, vertical asymptotes occur where the denominator equals zero, and horizontal asymptotes are determined by comparing the degrees of the numerator and denominator.

Write an equation for the function graphed below.

This problem involves determining the equation of a rational function based on a graph with vertical asymptotes at x = -2 and x = 2. Learn how to find the equation of such functions using key algebraic concepts like asymptotes and symmetry.

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