Question

1. what equation corresponds to the graph of the hyperbola?

Explanation:

Step1: Identify hyperbola type

This is a vertical transverse axis hyperbola centered at $(0,0)$, standard form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Step2: Find $a$ from vertex

Vertex at $(0, 3)$, so $a=3$, $a^2=9$

Step3: Find $b$ from guide point

Point $(-2, 2)$ lies on the graph. Substitute $x=-2$, $y=2$, $a^2=9$ into the standard form:
$\frac{(-2)^2}{9} - \frac{2^2}{b^2} = 1$
$\frac{4}{9} - \frac{4}{b^2} = 1$
$-\frac{4}{b^2} = 1 - \frac{4}{9} = \frac{5}{9}$
Wait, correction: Use the asymptote. Asymptotes of this hyperbola are $y=\pm\frac{b}{a}x$. The asymptote passes through $(3,2)$, so $2=\frac{b}{3}\times3 \implies b=2$, $b^2=4$

Step4: Write final equation

Substitute $a^2=9$, $b^2=4$ into standard form:
$\frac{x^2}{9} - \frac{y^2}{4} = 1$

Answer:

$\frac{x^2}{9} - \frac{y^2}{4} = 1$