1. ## Parabola question w/ answer

Consider all combinations of pairs of different parabolas in the standard (x,y) coordinate plane. Which of the following lists gives the number of points of intersections that are possible for 2 different parabolas?

Apparently they can intersect in 0,1,2,3 or 4 places?? I understand 0,1,2 but how do you get 3 and 4??

2. I don't see how 2 distinct parabolas can intersect in more than 2 points.

If you put both parabolas into general form (y=ax^2+bx+c), then you can find the x-values at which they intersect by setting the two right hand sides equal to each other. This is a quadratic equation in x which can have at most 2 solutions.

Edit: I was assuming that the two parabolas were graphs of functions. It is easy to make two parabolas intersect in 3 or 4 points if one has the form y=ax^2+bx+c, and the other has the form x=ay^2+by+c

3. Hi there--nope it's written correctly--maybe they mean a hyperbola is a parabola??? I have no idea though--this is an ACT practice question from years ago

4. Originally Posted by DrSteve
I don't see how 2 distinct parabolas can intersect in more than 2 points.
.

5. note the graph ...

red intersects blue at 3 points

red intersects green at 4 points

6. Not sure I understand guys--so it's not a hyperbola in red?? We used parametric equations?? Is a hyperbola technically a parabola then??

7. Maybe they are also referring to parabolas of the type:

ay = bx^2
cy^2 = dx + e

where a, b, c, d and e are constants...?

EDIT: A little too late.

But no, the red curve is still a parabola.

8. Oh ok..so it's still a parabola and NOT a hyperbola. But they never give parametric equations on the ACT--how are we supposed to know it without knowing that?

9. Originally Posted by donnagirl
Oh ok..so it's still a parabola and NOT a hyperbola. But they never give parametric equations on the ACT--how are we supposed to know it without knowing that?
It may surprise you but $x^2-2xy+y^2+2x-4y+3=0$ is in fact a parabola that has its axis of symmetry has rotated $45^o$.
Notice that it is not in parametric form.

10. Huh??? But this is the ACT, a minute a question--how can I deduce this problem in that time??

11. Originally Posted by donnagirl
Huh??? But this is the ACT, a minute a question--how can I deduce this problem in that time??
But the point is the question the question says that "all combinations of pairs of different parabolas in the standard (x,y) coordinate plane."
It assumes that you know that there many forms that a parabola can take on.
It is a one minute question.