What are the vertex, focus and directrix of  y=x^2 + 3?

The vertex is at the point where the function is at its minimum (it would be the maximum if the ${x}^{2}$ factor were negative). Hence the vertex is at the point (0,3).
The focus is a distance $\frac{1}{4 a}$ above the vertex. It is therefore the point (0,$3 \cdot \frac{1}{4}$).
The directrix is the horizontal line an equal distance below the vertex and is therefore the line $y = 2 \cdot \frac{3}{4}$