Evaluating the Statement: Parabolas Without X-Axis Intersection Are Always Increasing

To determine whether a parabola that does not intersect or touch the x-axis is always increasing, we need to analyze its general behavior:

• A parabola described by the quadratic function $y = ax^2 + bx + c$ will open upwards if $a > 0$ and downwards if $a < 0$.
• The point $(h, k)$, derived from the parabola's vertex form $y = a(x-h)^2 + k$, defines its vertex. The vertex is the point of symmetry in a parabola.
• The condition that it does not touch or intersect the x-axis implies its vertex is either completely above or below the x-axis.
• If the parabola opens upwards ($a > 0$), there are sections where the graph is both increasing and decreasing, divided by the vertex, hence it cannot be always increasing.
• Similarly, if the parabola opens downwards ($a < 0$), it is both increasing and decreasing around the vertex, and thus it cannot be always increasing.

In both scenarios, the understanding that a parabola does not always increase stems from the symmetry of its form about its vertex.

Therefore, the claim that the parabola is always increasing is incorrect.