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The graph you provided appears to represent a rational function. The curve seems to have a vertical asymptote near \( x = -2 \) and a horizontal asymptote along \( y = -2 \). Based on the shape and asymptotes, the function likely has the form:

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

where \( c \) corresponds to the vertical asymptote, and \( d \) to the horizontal asymptote. From the graph:
- The vertical asymptote is at \( x = -2 \), so \( c = -2 \).
- The horizontal asymptote is at \( y = -2 \), so \( d = -2 \).

This suggests the equation might look something like:

\[
f(x) = \frac{a}{x + 2} - 2
\]

To find \( a \), we can use a point from the graph. For example, the point \( (-1, 0) \) is on the curve. Plugging this into the equation:

\[
0 = \frac{a}{-1 + 2} - 2
\]

This simplifies to:

\[
0 = \frac{a}{1} - 2 \quad \Rightarrow \quad a = 2
\]

Thus, the equation of the function is:

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

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the domain of the function \( f(x) = \frac{2}{x+2} - 2 \)?
2. How do you determine the vertical and horizontal asymptotes of a rational function?
3. What are the key features of a graph of rational functions?
4. How would you find the x- and y-intercepts of the function?
5. Can you find the behavior of the graph as \( x \to \infty \) or \( x \to -\infty \)?

**Tip:** To identify asymptotes, observe where the function grows without bound (vertical) and where it levels off (horizontal).

What is the equation of the function represented in the graph?

Learn how to determine the equation of a rational function with vertical and horizontal asymptotes. The function in this example has a vertical asymptote at x = -2 and a horizontal asymptote at y = -2. Suitable for high school students in Grades 10-12.

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