Answer
$\frac{x^2}{4}-y^2=1$
Work Step by Step
Step 1. Identify the given quantities: hyperbola vertices $V(0,\pm2)$, asymptotes $y=\pm\frac{1}{2}x$
Step 2. As the vertices are along the x-axis, we assume the equation as $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Step 3. With the vertices value, we have $a=2$
Step 4. The asymptotes is given by $y=\pm\frac{b}{a}x=\pm\frac{b}{2}x$ and with the quantity in step-1, we have $\frac{b}{2}=\frac{1}{2}$, thus $b=1$
Step 5. Conclusion: the equation for the hyperbola is $\frac{x^2}{4}-y^2=1$
