1. **State the problem:** We are given a semicircle with radius $2$ and want to understand its properties. 2. **Formula for a circle:** The equation of a circle centered at the origin with radius $r$ is $$x^2 + y^2 = r^2$$. 3. **Semicircle definition:** A semicircle is half of a circle. For the upper semicircle, the equation is $$y = \sqrt{r^2 - x^2}$$, and for the lower semicircle, $$y = -\sqrt{r^2 - x^2}$$. 4. **Apply radius:** Here, $r=2$, so the semicircle equation is $$y = \sqrt{4 - x^2}$$ for the upper semicircle. 5. **Domain:** Since the radius is $2$, $x$ ranges from $-2$ to $2$. 6. **Summary:** The semicircle with radius $2$ centered at the origin is described by $$y = \sqrt{4 - x^2}$$ for $-2 \leq x \leq 2$. This function represents the upper half of the circle of radius $2$.