Parabola Property Analysis: Does a Downward Parabola with Vertex Below X-axis Stay Positive?

To solve this problem, we need to understand the behavior of a downward-opening parabola with its vertex below the x-axis.

A quadratic function of the form $f(x) = ax^2 + bx + c$ opens downward if $a < 0$. The vertex form is $f(x) = a(x-h)^2 + k$, where the vertex is $(h, k)$. In this problem, the vertex $(h, k)$ is below the x-axis, which means $k < 0$.

For a parabola opening downward with $a < 0$, the function will have values greater than at the vertex as $x$ moves away from $h$. However, since $k < 0$, at the vertex itself, $f(h) = k$ is negative. As $x$ increases significantly away from $h$, the value of $a(x-h)^2$ becomes large and negative, due to $a < 0$, and dominates the function, causing $f(x)$ to also be negative for sufficiently large or small $x$.

Therefore, despite the downward-bending parabola having a vertex below the x-axis, it is incorrect to say the entire function is positive. The parabola will take on negative values when $x$ is sufficiently far from the vertex.

The correct conclusion is that the statement, "If a parabola is bending downwards and its vertex is below the x-axis, then it is always positive," is incorrect.