Final answer:
The graphs of a linear function and a quadratic function can have no points of intersection, exactly one point of intersection if the line is a tangent to the parabola, or exactly two points of intersection if it intersects at two places. Three points or infinitely many points are not possible.
Step-by-step explanation:
The graphs of a linear function and a quadratic function can have the following points of intersection:
- B. No points of intersection - This situation occurs when the straight line and the parabola do not cross each other, typically when the line is above the vertex of an upward-opening parabola or below the vertex of a downward-opening parabola and the parabola opens in such a way that they will never intersect.
- C. Exactly one point of intersection - If the line is a tangent to the parabola, they will intersect at exactly one point. This happens when the straight line just grazes the curve of the parabola.
- E. Exactly two points of intersection - It is common for a straight line to intersect a parabola at two points, one on each 'side' of the parabola, provided the line is not parallel to the axis of symmetry.
Options A (exactly three points of intersection) and D (infinitely many points of intersection) are not possible between a straight line and a parabola because a straight line can cross a parabola at most twice and must be a line of symmetry to touch at an infinite number of points, which cannot occur with a parabola.