1. **State the problem:** We are given the function $g(x) = x^2 - 4x + 7$ and want to analyze its properties, including finding the vertex and understanding the shape of its graph. 2. **Formula and rules:** The function $g(x)$ is a quadratic function of the form $ax^2 + bx + c$ where $a=1$, $b=-4$, and $c=7$. The graph of a quadratic function is a parabola. 3. **Find the vertex:** The vertex of a parabola given by $ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. Here, $x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$. 4. **Calculate the y-coordinate of the vertex:** Substitute $x=2$ into $g(x)$: $$g(2) = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3$$ 5. **Interpretation:** The vertex is at $(2, 3)$, which is the minimum point since $a=1 > 0$ and the parabola opens upwards. 6. **Summary:** The parabola $g(x) = x^2 - 4x + 7$ has its vertex at $(2, 3)$ and opens upwards, matching the description and graph provided.