https://file.aiquickdraw.com/mathbot/file/1729303243266v5r57hvr.jpg

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From the graph you uploaded, it looks like a rational function is being plotted with vertical asymptotes and a horizontal asymptote. Let’s break this down step by step:

### Key Observations:
1. **Vertical asymptotes**: 
   - There is a vertical asymptote at \( x = -2 \).
   - Another vertical asymptote is at \( x = 4 \).

2. **Horizontal asymptote**: 
   - There seems to be a horizontal asymptote along the \( x \)-axis at \( y = 0 \), meaning the function approaches \( 0 \) as \( x \) approaches very large positive or negative values.

3. **Behavior of the graph**:
   - As \( x \to -2^+ \), the function goes to \( \infty \), and as \( x \to -2^- \), it goes to \( -\infty \). This indicates the function has opposite behavior on either side of \( x = -2 \).
   - Similarly, around \( x = 4 \), it behaves similarly with opposite directions.
   - At large positive and negative \( x \)-values, the function approaches the horizontal asymptote \( y = 0 \).

4. **Zeros of the function**:
   - The graph crosses the x-axis at \( x = 1 \), indicating a zero there.

### Rational Function Form:
Since there are vertical asymptotes and a horizontal asymptote at \( y = 0 \), this suggests that the rational function can be written in the form:
\[
y = \frac{A(x - 1)}{(x + 2)(x - 4)}
\]
Where:
- \( A \) is a constant that might affect the steepness of the curve.
- The denominator corresponds to the vertical asymptotes at \( x = -2 \) and \( x = 4 \).
- The numerator includes \( (x - 1) \), accounting for the zero at \( x = 1 \).

Now, let's confirm that this form fits the overall behavior.

### Tentative Equation:
\[
y = \frac{(x - 1)}{(x + 2)(x - 4)}
\]
This rational function should have the same vertical asymptotes, horizontal asymptote, and zero as the graph you’ve provided.

Would you like more details or have any questions about the steps? Here are some related questions that could expand your understanding:

1. How do vertical and horizontal asymptotes influence the shape of a rational function?
2. Why does the graph have a horizontal asymptote at \( y = 0 \)?
3. How do we identify the location of zeros in a rational function?
4. What role does the numerator play in determining the zeros of a rational function?
5. How would changing the constant \( A \) in the function affect the graph?

**Tip**: Always check the sign and degree of both the numerator and denominator to predict the behavior at the asymptotes and far out along the \( x \)-axis.

Learn how to write the equation of a rational function with vertical asymptotes at x = -2 and x = 4, and a zero at x = 1. This problem focuses on key concepts like vertical and horizontal asymptotes and the behavior of rational functions.

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