The hyperbola in the image has its branches opening upwards and downwards, and the center of the hyperbola appears to be at the origin $(0, 0)$.

Identifying the Form of the Equation:

For a hyperbola centered at the origin with a vertical transverse axis, the general equation is:

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

Where:

• $a$ is the distance from the center to the vertices along the $y$-axis.
• $b$ is related to the distance from the center to the points where the asymptotes intersect the rectangle formed by $a$ and $b$.

Analyzing the Graph:

• Vertices: The vertices seem to be at $(0, 3)$ and $(0, -3)$. This gives $a = 3$.
• Asymptotes: The asymptotes appear to intersect the grid points at $\pm 3$ along the $x$-axis and $\pm 3$ along the $y$-axis, suggesting that $b = 3$.

Equation:

Given $a = 3$ and $b = 3$, the equation of the hyperbola is:

$\frac{y^2}{3^2} - \frac{x^2}{3^2} = 1$

Simplifying this:

$\frac{y^2}{9} - \frac{x^2}{9} = 1$

This is the equation of the hyperbola shown in the image.

Would you like more details or have any questions?

Related Questions:

1. What would the equation be if the hyperbola were centered at a different point?
2. How do you derive the equations for the asymptotes of this hyperbola?
3. How does the orientation of the hyperbola change its standard equation?
4. What is the effect of changing the values of $a$ and $b$ on the shape of the hyperbola?
5. How can you determine the foci of the hyperbola from its equation?

Tip:

For hyperbolas, the relationship between $a$, $b$, and the distance to the foci $c$ is given by $c^2 = a^2 + b^2$.