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September 9th, 2009, 07:38 PM  #1 
Newbie Joined: Sep 2009 Posts: 6 Thanks: 0  Parabola and Line intersections?
I have this problem that I just cannot figure out, and was wondering if I could get any help here. So basically we have to graph a generic parabola (y=x^2) and some lines (y=x+k) and figure out where it intersects once, twice, no times. That part is straightfoward enough, 1/4 is the single intersection, anything above is twice, anything below, none. What I'm having trouble with is to prove the conjecture that "When k>1/4, there will be 2 intersections. There will be one intersection at k=1/4, and no solutions when k<1/4." I know that we're supposed to play around with the discrimminant and such, and I've gotten to the point where I suppose I've proved that there is one intersection at 1/4, using b^24ac with the equation x^2x+1/4=k+1/4 (the previous two equations set equal to each other and modified by completing the square). What I'm really having a problem with is how to prove that numbers above this would have two intersections, and that numbers below would have none. Someone told me something about a "law of discriminants" but upon Googling this, I didn't really find anything. I get the feeling I'm missing something really obvious, but I don't know, and I can't figure this out at all. thanks to anybody who can help <3 
September 9th, 2009, 09:06 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,484 Thanks: 2041 
Any intersections have a real xcoordinate that is a root of x² = x + k. That's a quadratic equation with roots 1/2 ± ?(k + 1/4), so there are two real values of x when k > 1/4 and no real roots when x < 1/4. 

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intersections, line, parabola 
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