A line in the xy -plane has a slope of - 1 2 and passes through

Choice B is correct. A line in the xy-plane with a slope of $m$ and a y-intercept of $\left(0,b\right)$ can be represented by the equation $y=mx+b$. It's given that the line has a slope of $-\frac{1}{2}$. Therefore, $m=-\frac{1}{2}$. It's also given that the line passes through the point $\left(0,3\right)$. Therefore, $b=3$. Substituting $-\frac{1}{2}$ for $m$ and $3$ for $b$ in the equation $y=mx+b$ yields $y=-\frac{1}{2}x+3$. Therefore, the equation $y=-\frac{1}{2}x+3$ represents this line.

Choice A is incorrect. This equation represents a line in the xy-plane that passes through the point $\left(0,-3\right)$, not $\left(0,3\right)$.

Choice C is incorrect. This equation represents a line in the xy-plane that has a slope of $\frac{1}{2}$, not $-\frac{1}{2}$, and passes through the point $\left(0,-3\right)$, not $\left(0,3\right)$.

Choice D is incorrect. This equation represents a line in the xy-plane that has a slope of $\frac{1}{2}$, not $-\frac{1}{2}$.