Question

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

Explanation:

Step1: Identify hyperbola type

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

Step2: Find $a$ from vertex

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

Step3: Find $b$ from asymptote

Asymptote $y=\frac{3}{4}x$, so $\frac{a}{b}=\frac{3}{4}$. Substitute $a=3$: $\frac{3}{b}=\frac{3}{4} \implies b=4$, $b^2=16$

Step4: Write final equation

Substitute $a^2=9$, $b^2=16$ into standard form: $\frac{y^2}{9} - \frac{x^2}{16} = 1$

Answer:

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