Question

assignment 7.5: hyperbolas
score: 10/15 answered: 9/15
question 10
write the standard equation for the hyperbola graphed above.
question help: video message instructor

Explanation:

Step1: Identify hyperbola type & center

The hyperbola opens up/down, center at $(0,0)$. Standard form: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.

Step2: Find $a$ from vertices

Vertices at $(0,2)$ and $(0,-2)$, so $a=2$. $a^2=4$.

Step3: Find slope of asymptote

Asymptote has slope $\frac{a}{b}=\frac{2}{3}$. Solve for $b$: $\frac{2}{b}=\frac{2}{3} \implies b=3$, $b^2=9$.

Step4: Plug values into standard form

Substitute $a^2=4$, $b^2=9$ into the equation.

Answer:

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