Downward-Opening Parabola Without X-Axis Intercepts: Analyzing Monotonicity Claims

Given any quadratic function given by $f(x) = ax^2 + bx + c$, it describes a parabola. Whether it opens upwards or downwards depends on the coefficient $a$.

A parabola opens upwards when $a > 0$, where it reaches a minimum at the vertex. It opens downwards when $a < 0$, where the vertex is a maximum point. Since the problem specifies a "decreasing parabola," we can infer that $a < 0$, indicating an opening downwards.

Importantly, if a parabola does not touch the $x$-axis, its discriminant $b^2 - 4ac$ is less than zero, indicating no real roots, and the parabola never crosses or touches the x-axis.

An opening downwards parabola has segments that are increasing on the left of the vertex and decreasing on the right. Therefore, it is incorrect to claim a downwards-opening parabola is always increasing, even if it doesn't touch the $x$-axis.

Therefore, the statement, "the parabola is always increasing," in this context is False.

Thus, the correct answer is:

False