https://r2.erweima.ai/mathbot/image/1725837453558lrfldrda.jpg

To determine the equation of the hyperbola based on the graph provided, let's analyze the features of the graph. 

1. **Center**: The hyperbola appears to have its center at the origin \((0,0)\), as it is symmetric around the axes.
   
2. **Vertices**: The vertices seem to be along the \(y\)-axis at points \((0, 9)\) and \((0, -9)\), meaning that the distance between the center and the vertices, \(a\), is 9. 

3. **Asymptotes**: From the shape of the hyperbola, it opens vertically, and the equations of the asymptotes are given by the slopes of the diagonals. Based on the graph, it appears that the slope of the asymptotes is approximately \( \pm 3/9 \) or \( \pm 1/3 \), meaning that \(b = 3\).

### Standard Form of a Vertically Oriented Hyperbola:
For a hyperbola that opens vertically, the equation takes the form:
\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]
Where:
- \(a = 9\) (distance to the vertices),
- \(b = 3\) (determined from the asymptotes).

### Substituting \(a\) and \(b\):
\[
\frac{y^2}{9^2} - \frac{x^2}{3^2} = 1
\]
\[
\frac{y^2}{81} - \frac{x^2}{9} = 1
\]

Now let's compare this to the options provided:
- **A:** \(\frac{x^2}{79} - \frac{y^2}{277} = 1\)
- **B:** \(\frac{x^2}{277} - \frac{y^2}{79} = 1\)
- **C:** \(\frac{y^2}{79} - \frac{x^2}{277} = 1\)
- **D:** \(\frac{y^2}{277} - \frac{x^2}{79} = 1\)

The correct equation corresponds to **D**.

Would you like more details or further explanation?

---

### Related Questions:
1. How do you calculate the eccentricity of a hyperbola?
2. What are the properties of the asymptotes of a hyperbola?
3. Can you explain the difference between a hyperbola that opens horizontally vs vertically?
4. How do the foci of a hyperbola relate to its equation?
5. How can we derive the equation of the hyperbola if the center is not at the origin?

**Tip**: When determining the equation of a hyperbola, always start by identifying its orientation (vertical or horizontal) based on how it opens!

Determine the equation of the hyperbola from its graph.

Learn how to find the equation of a hyperbola from its graph. Understand key concepts such as vertices, asymptotes, and the standard equation of a hyperbola. This problem is ideal for students in Grades 10-12 studying analytic geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation