Question

find the standard form of the equation of the hyperbola shown.
the standard form of the equation is \boxed{}

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 $(±2,0)$, so $a=2$, $a^2=4$.

Step3: Find b using asymptote

Asymptote slope is $±\frac{b}{a}=±2$ (from graph: rise 4 over run 2). So $\frac{b}{2}=2 → b=4$, $b^2=16$.

Step4: Write equation

Substitute $h=0,k=0,a^2=4,b^2=16$ into standard form.

Answer:

$\frac{x^2}{4} - \frac{y^2}{16} = 1$