Step-by-Step Solution

Step-by-Step Solution

Step 1: Identify Parameters and Vertex

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

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

The $x$-coordinate of the vertex $h$ is given by:

The $y$-coordinate of the vertex $k$ is:

Thus, the vertex is at $(2, -4)$.

Step 2: Intercepts & Axis of Symmetry

• $y$-intercept: Set $x = 0 \implies f(0) = 0$, giving $(0, 0)$.
• $x$-intercepts: Set $f(x) = 0 \implies x(x - 4) = 0 \implies x = 0$ or $x = 4$.
• Axis of Symmetry: The vertical line passing through the vertex: $$x = 2$$

Step 3: Domain and Range

• Domain: Any real number can be squared and multiplied by $4$, so the domain is all real numbers: $$\text{Domain} = \mathbb{R} = (-\infty, \infty)$$
• Range: Because the parabola opens upwards with minimum value $k = -4$: $$\text{Range} = [-4, \infty)$$