Parabola With No X-Axis Intersection: Verify If Always Decreasing

To determine if a parabola is always decreasing when it does not intersect or touch the x-axis, we analyze the properties of quadratic functions.

• Step 1: For a parabola to not intersect the x-axis, its discriminant must be negative: $b^2 - 4ac < 0$. This condition ensures the quadratic equation has no real roots.
• Step 2: Depending on $a$, the parabola either opens upward ($a > 0$) or downward ($a < 0$).
• Step 3: A parabola is "always decreasing" only if it opens downward and does not have a minimum turning point in its domain, which is impossible for a standard quadratic function.
• Step 4: If $a > 0$, the parabola opens upwards, and cannot always decrease as it eventually increases after the vertex.
• Step 5: If $a < 0$, the parabola opens downwards; hence, it decreases after the vertex.

Therefore, regardless of whether the parabola opens upward or downward, it cannot "always be decreasing" because it either increases or decreases after the vertex. Thus, the statement is incorrect.

The correct answer is Incorrect.