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.

Recognizing the perfect square trinomial:

The vertex is immediately identified at $(2, 0)$.

Step 2: Intercepts & Tangency to Axis

• $y$-intercept: $f(0) = (0 - 2)^2 = 4 \implies (0, 4)$.
• $x$-intercept: $(x - 2)^2 = 0 \implies x = 2$ (a repeated real root). The parabola is tangent to the $x$-axis at the vertex $(2, 0)$ and never crosses below it.
• Axis of Symmetry: $$x = 2$$

Step 3: Domain and Range

• Domain: Defined everywhere on $\mathbb{R}$: $$\text{Domain} = \mathbb{R} = (-\infty, \infty)$$
• Range: Because $(x - 2)^2 \ge 0$ for all real numbers $x$: $$\text{Range} = [0, \infty)$$