Parabola Property: When Downward-Facing Vertex Above X-axis Implies Positive Values

To solve this problem, we need to carefully examine the nature of the parabolic function given its attributes:

• The parabola opens downward, which indicates the coefficient $a < 0$ in its standard form $y = ax^2 + bx + c$.
• The vertex is above the x-axis, suggesting that its y-coordinate, given by $c - \frac{b^2}{4a}$, is positive.

However, having the vertex above the x-axis alone does not guarantee that the entire parabola remains above the x-axis. A downward-opening parabola can have parts below the x-axis even if its vertex is above it.

Consider a simple example. Take $y = -x^2 + 6$, which is a downward-opening parabola with vertex $(0, 6)$. While the vertex is above the x-axis, the roots $x = -\sqrt{6}$ and $x = \sqrt{6}$ indicate that it crosses the x-axis, thus having negative values for $x$ in $(-\sqrt{6}, \sqrt{6})$.

Therefore, the statement that the parabola is always positive is Incorrect.