Parabola Intersections: Analyzing X-Axis Crossing Points and Domain Properties

To analyze whether a quadratic function with two intersection points with the x-axis has sections where the function values are both positive and negative, consider the following:

• The general form of a quadratic function is $f(x) = ax^2 + bx + c$.
• The parabola intersects the x-axis at points where $f(x) = 0$, leading to the equation having real roots. This occurs when the discriminant $(b^2 - 4ac)$ is positive.
• When a parabola, which describes a quadratic function, has two distinct x-intercepts (roots), it must cross the x-axis twice.
• If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.
• The curve will be above the x-axis (positive values of $f(x)$) between or outside these intercepts, depending on the sign of $a$.
• Hence, for a parabola with two intersections, there are intervals on the x-axis where the function is both positive and negative, confirming dual domain signs.

Thus, it is correct to conclude that a parabola with two distinct x-axis intersections has both positive and negative function values, satisfying the problem's assertion regarding its range and confirming the correct answer choice is:

Correct