Question

find the standard form of the equation of the hyperbola shown.
the standard form of the equation is \square

Explanation:

Step1: Identify hyperbola type and center

The hyperbola opens left/right, so standard form is $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$. Center $(h,k)$ is $(0,0)$ (midpoint of vertices).

Step2: Find $a$ value

Vertices are at $(\pm2,0)$, so $a=2$, $a^2=4$.

Step3: Find $b$ value

Asymptotes are $y=\pm3x$ (slope $\pm\frac{b}{a}$). $\frac{b}{a}=3\Rightarrow b=3a=6$, $b^2=36$.

Step4: Write standard equation

Substitute $h=0,k=0,a^2=4,b^2=36$ into the form.

Answer:

$\frac{x^2}{4}-\frac{y^2}{36}=1$