Step-by-Step Solution

Step-by-Step Solution

Step 1: Identify Parameters and Vertex

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

Because $a = 1 > 0$, 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 is located at $(-1, -9.3)$.

Step 2: Intercepts & Axis of Symmetry

• $y$-intercept: $f(0) = -8.3 \implies (0, -8.3)$.
• $x$-intercepts: Set $f(x) = (x + 1)^2 - 9.3 = 0 \implies (x + 1)^2 = 9.3 \implies x = -1 \pm \sqrt{9.3} \approx -1 \pm 3.05$, giving $x \approx 2.05$ and $x \approx -4.05$.
• Axis of Symmetry: $$x = -1$$

Step 3: Domain and Range

• Domain: Defined for all real inputs: $$\text{Domain} = \mathbb{R} = (-\infty, \infty)$$
• Range: The lowest point on the curve is $y = -9.3$, and the graph opens upwards without bound: $$\text{Range} = [-9.3, \infty)$$