1. The problem is to analyze the function $f(x) = x^2 - 7x + 12$. 2. This is a quadratic function of the form $f(x) = ax^2 + bx + c$ where $a=1$, $b=-7$, and $c=12$. 3. To find the roots (x-intercepts), we use the quadratic formula or factorization. Here, factorization is straightforward: $$x^2 - 7x + 12 = (x - 3)(x - 4)$$ 4. Setting each factor equal to zero gives the roots: $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 4 = 0 \Rightarrow x = 4$$ 5. The vertex of the parabola can be found using the formula for the x-coordinate of the vertex: $$x = -\frac{b}{2a} = -\frac{-7}{2 \times 1} = \frac{7}{2} = 3.5$$ 6. Substitute $x=3.5$ back into the function to find the y-coordinate of the vertex: $$f(3.5) = (3.5)^2 - 7(3.5) + 12 = 12.25 - 24.5 + 12 = -0.25$$ 7. The vertex is at $(3.5, -0.25)$, which is the minimum point since $a=1 > 0$ (parabola opens upwards). 8. Summary: - Roots: $x=3$ and $x=4$ - Vertex: $(3.5, -0.25)$ - Parabola opens upwards Final answer: The function $f(x) = x^2 - 7x + 12$ has roots at $x=3$ and $x=4$, and its vertex is at $(3.5, -0.25)$, which is the minimum point of the parabola.