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 downwards, meaning the vertex is an absolute maximum point.

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

The $y$-coordinate $k$:

The vertex is $(1, -7)$.

Step 2: Intercepts & Discriminant

• $y$-intercept: $f(0) = -8 \implies (0, -8)$.
• Discriminant: $\Delta = b^2 - 4ac = (2)^2 - 4(-1)(-8) = 4 - 32 = -28 < 0$. Because the discriminant is negative and $a < 0$, the parabola lies entirely below the $x$-axis and has no real $x$-intercepts.
• Axis of Symmetry: $$x = 1$$

Step 3: Domain and Range

• Domain: Defined for every real number: $$\text{Domain} = \mathbb{R} = (-\infty, \infty)$$
• Range: Since the maximum peak is at $y = -7$ and the parabola extends downwards to $-\infty$: $$\text{Range} = (-\infty, -7]$$