1. **Problem:** Find the quadratic function passing through (0,4) with vertex at (2,3). 2. **Formula:** Vertex form of quadratic function is $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. 3. **Apply vertex:** Here, $h=2$, $k=3$, so $$y = a(x - 2)^2 + 3$$ 4. **Use point (0,4):** Substitute $x=0$, $y=4$: $$4 = a(0 - 2)^2 + 3$$ $$4 = a(4) + 3$$ $$4 - 3 = 4a$$ $$1 = 4a$$ $$a = \frac{1}{4}$$ 5. **Final function:** $$y = \frac{1}{4}(x - 2)^2 + 3$$ 6. **Expand:** $$y = \frac{1}{4}(x^2 - 4x + 4) + 3$$ $$y = \frac{1}{4}x^2 - x + 1 + 3$$ $$y = \frac{1}{4}x^2 - x + 4$$ 7. **Answer:** The function is $$y = \frac{1}{4}x^2 - x + 4$$ which corresponds to option D.