jmc

The graph of $y=\frac{1}{2}f(x)$ is produced by taking the graph of $y=f(x)$ and compressing it vertically by a factor of $\frac{1}{2}.$ We then get the graph of $y=\frac{1}{2}f(x)+3$ by shifting upward by three units. The correct graph is $\boxed{\text{C}}.$