write the equation of the hyperbola graphed below, whose vertices and foci are marked.

Explanation:

Step1: Identify the center and orientation

The vertices are at $(0, 8)$ and $(0, -8)$, so the center is $(0, 0)$ and the transverse axis is vertical. $$Center: (0, 0)$$

Step2: Determine the value of $a$

The distance from the center to a vertex is $a$. $$a = 8 \implies a^2 = 64$$

Step3: Determine the value of $c$

The foci are marked at $(0, 17)$ and $(0, -17)$, so the distance $c$ is 17. $$c = 17 \implies c^2 = 289$$

Step4: Calculate the value of $b^2$

Use the hyperbola relationship $c^2 = a^2 + b^2$. $$b^2 = c^2 - a^2 = 289 - 64 = 225$$

Step5: Write the standard equation

For a vertical hyperbola centered at the origin, the equation is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. $$\frac{y^2}{64} - \frac{x^2}{225} = 1$$

Answer:

$$\frac{y^2}{64} - \frac{x^2}{225} = 1$$