Step-by-Step Solution

Step-by-Step Solution

Step 1: Identify Parameters and Vertex

For $f(x) = ax^2 + bx + c$:

Since $a = 1 > 0$, the parabola opens upwards, so the vertex is an absolute minimum point.

The $x$-coordinate of the vertex $h$:

The $y$-coordinate $k$:

The vertex point is $\left(\frac{5}{2}, -\frac{1}{4}\right)$.

Step 2: Intercepts & Axis of Symmetry

• $y$-intercept: $f(0) = 6 \implies (0, 6)$.
• $x$-intercepts: Factor $x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \implies x = 2$ and $x = 3$.
• Axis of Symmetry: $$x = \frac{5}{2} = 2.5$$

Step 3: Domain and Range

• Domain: All quadratic polynomials are defined everywhere on the real line: $$\text{Domain} = \mathbb{R} = (-\infty, \infty)$$
• Range: Since the vertex is the absolute minimum at $y = -0.25$: $$\text{Range} = \left[-\frac{1}{4}, \infty\right) = [-0.25, \infty)$$