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9-66.

As you have seen, sometimes the graphs of two equations never intersect. However, when they do intersect, how many different intersection points can they have? How many intersection points can two lines have? How many intersections can two parabolas have? What about a parabola and a line? In this problem, you will investigate these questions. Sketch a graph that fits each description below. Not every graph is possible.

  1. A line and a parabola that intersect twice

    An upward parabola with vertex in third quadrant and an increasing line drawn so that the line is above the vertex of the parabola.

  2. A line and a parabola that only intersect once.

    An upward parabola with vertex in third quadrant and a decreasing line which is tangent to the parabola in the third quadrant.

  3. A line and a parabola that intersect more than twice.

    Can a straight line intersect a parabola more than twice?

  4. Two parabolas that intersect twice.

    An upward parabola with vertex in third quadrant and a downward parabola with vertex in first quadrant.

  5. Two parabolas that have an infinite number of intersections.

    Can different parabolas intersect more than twice? How must two parabolas relate to each other so that they intersect each other an infinite amount of times?

  6. Two parabolas that never intersect.

    An upward parabola with vertex in the third quadrant and a downward parabola with vertex in the third quadrant also but below the vertex of the upward parabola.

  7. Two parabolas that only intersect once.

    An upward parabola with vertex at the origin and a downward parabola whose vertex is only at the origin. is negative and points downwards.