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To find the equation of the hyperbola shown in the graph, we need to identify the key components of the hyperbola: the center, the transverse axis, the conjugate axis, and the slopes of the asymptotes.

From the graph:

1. **Center**: The center of the hyperbola appears to be at the point \((4, 2)\).

2. **Transverse Axis**: The transverse axis is horizontal. The distance between the vertices along the transverse axis is 6 units. This means that \(2a = 6\), so \(a = 3\).

3. **Conjugate Axis**: The conjugate axis is vertical. The distance between the co-vertices along the conjugate axis is 4 units. This means that \(2b = 4\), so \(b = 2\).

4. **Asymptotes**: The slopes of the asymptotes of a hyperbola centered at \((h, k)\) with a horizontal transverse axis are given by \(\pm \frac{b}{a}\). Here, the slopes are \(\pm \frac{2}{3}\).

Given these components, the standard form of the equation of a hyperbola centered at \((h, k)\) with a horizontal transverse axis is:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]

Plugging in the values \(h = 4\), \(k = 2\), \(a = 3\), and \(b = 2\), we get:
\[
\frac{(x - 4)^2}{3^2} - \frac{(y - 2)^2}{2^2} = 1
\]
\[
\frac{(x - 4)^2}{9} - \frac{(y - 2)^2}{4} = 1
\]

Therefore, the equation of the hyperbola is:
\[
\frac{(x - 4)^2}{9} - \frac{(y - 2)^2}{4} = 1
\]

If you have any questions or need further details, please let me know.

Here are 5 questions you can ask next:
1. How do you derive the standard form equation of a hyperbola?
2. What are the properties of hyperbolas that distinguish them from other conic sections?
3. Can you explain how to find the asymptotes of a hyperbola?
4. How do you determine the foci of a hyperbola?
5. What are the applications of hyperbolas in real-world problems?

**Tip:** When graphing hyperbolas, always start by identifying the center, vertices, and asymptotes to construct the graph accurately.

Learn how to find the equation of a hyperbola centered at (4, 2) with a horizontal transverse axis. Detailed step-by-step solution includes identifying center, transverse axis, conjugate axis, and asymptotes. Suitable for advanced high school students.

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