https://file.aiquickdraw.com/mathbot/file/1726668404788zdfp6qed.jpg

The equation given in the image is:

\[
\frac{x^2}{100} - \frac{y^2}{81} = 1
\]

This is the standard form of a hyperbola. The general form of a hyperbola centered at the origin is:

\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\]

From comparing the two equations, we have:
- \(a^2 = 100\), so \(a = 10\)
- \(b^2 = 81\), so \(b = 9\)

### Important Features of the Hyperbola:
- **Center**: \((0, 0)\) because the equation is in its standard form and there is no translation.
- **Vertices**: The vertices are located along the x-axis at \((\pm a, 0)\), which in this case gives the vertices at \((\pm 10, 0)\).
- **Co-vertices**: The co-vertices are along the y-axis at \((0, \pm b)\), which gives \((0, \pm 9)\). These help in defining the rectangle that guides the asymptotes.
- **Asymptotes**: The asymptotes of the hyperbola are lines that pass through the center, with slopes given by \(\pm \frac{b}{a}\). The asymptotes are:
  \[
  y = \pm \frac{9}{10}x
  \]
  
### Graphing the Hyperbola:
1. Start by plotting the center at \((0, 0)\).
2. Plot the vertices at \((10, 0)\) and \((-10, 0)\).
3. Plot the co-vertices at \((0, 9)\) and \((0, -9)\).
4. Draw a rectangle using the vertices and co-vertices as guides. The diagonals of this rectangle are the asymptotes of the hyperbola.
5. Sketch the hyperbola, which opens horizontally since the positive term is with \(x^2\), and approaches the asymptotes as it moves away from the center.

Would you like any further details or explanations?

### Follow-up questions:
1. How do we determine if a hyperbola opens horizontally or vertically?
2. Can we apply the same method if the hyperbola is not centered at the origin?
3. How do asymptotes help in sketching a hyperbola?
4. What is the significance of the foci of a hyperbola?
5. How would the equation change if the hyperbola opened vertically?

**Tip**: When graphing conic sections like hyperbolas, identifying the center, vertices, and asymptotes is crucial to sketching an accurate graph.

Find the coordinates and graph the hyperbola defined by the equation \(\frac{x^2}{100} - \frac{y^2}{81} = 1\).

Learn how to find the coordinates and graph the hyperbola defined by the equation \(\frac{x^2}{100} - \frac{y^2}{81} = 1\). This solution includes identifying vertices, asymptotes, and important features of the hyperbola, suitable for students in Grades 10-12.

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