1. **State the problem:** Find the vertex of the quadratic function $g(x) = x^2 - 4x + 7$ and determine its properties. 2. **Formula and rules:** For a quadratic function $ax^2 + bx + c$, the vertex $x$-coordinate is given by $x = -\frac{b}{2a}$. 3. **Calculate the vertex $x$-coordinate:** Here, $a=1$, $b=-4$, so $$x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$$ 4. **Calculate the vertex $y$-coordinate:** Substitute $x=2$ into $g(x)$: $$g(2) = 2^2 - 4 \times 2 + 7 = 4 - 8 + 7 = 3$$ 5. **Interpretation:** The vertex is at $(2, 3)$. Since $a=1 > 0$, the parabola opens upwards, so this vertex is a minimum point. 6. **Final answer:** The vertex of $g(x)$ is at $(2, 3)$, and the parabola opens upwards with a minimum at this point.