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The image provides steps for graphing a hyperbola with the following details:

1. **Center**: \( (0, 0) \)
2. **Vertices**: \( (0, 4) \) and \( (0, -4) \)
3. **Foci**: \( (0, 5) \) and \( (0, -5) \)
4. **Asymptotes**: Slopes \( \frac{4}{3} \) and \( -\frac{4}{3} \)
5. **Shape of the Hyperbola**: Opens upwards and downwards, approaching the asymptotes.

This describes a **vertical hyperbola** centered at the origin. The formula for such a hyperbola typically takes the form:

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

Here:
- The distance from the center to the vertices is \( a = 4 \), so \( a^2 = 16 \).
- The asymptotes have slopes \( \pm \frac{a}{b} = \pm \frac{4}{3} \), giving \( b = 3 \), so \( b^2 = 9 \).

Thus, the equation of the hyperbola is:

\[
\frac{y^2}{16} - \frac{x^2}{9} = 1
\]

Would you like more details on this equation or graphing it? 

Here are five related questions to expand this topic:
1. How are the asymptotes of a hyperbola derived?
2. What is the relationship between the foci and the vertices of a hyperbola?
3. How does the orientation of the hyperbola change if the equation swaps \( x^2 \) and \( y^2 \)?
4. Can you explain how to find the equation of a hyperbola given its foci and vertices?
5. What is the significance of the transverse and conjugate axes in a hyperbola?

**Tip**: The slopes of asymptotes are determined by the ratio \( \frac{a}{b} \) for vertical hyperbolas and \( \frac{b}{a} \) for horizontal ones.

How do you graph a hyperbola with the center at (0, 0), vertices at (0, 4) and (0, -4), and foci at (0, 5) and (0, -5), along with asymptotes having slopes 4/3 and -4/3?

This tutorial explains how to graph a vertical hyperbola with the center at (0, 0), vertices at (0, 4) and (0, -4), and foci at (0, 5) and (0, -5). It includes details on drawing asymptotes with slopes 4/3 and -4/3, along with the equation of the hyperbola. Ideal for students in Grades 10-12.

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