Algebra 2 Solving Parabola and Line Equation Sets
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More Crossings
Solving Parabola and Line Equation Sets

 

 We have seen problems where we needed to find the point

 where two straight lines crossed.
 
 Now we're going to find the points
 where a straight line and a parabola cross.
 
 Say, like this one:

 

 
 We can see that in this problem there are 2 places where they cross.
 One where both X and Y are negative,
 and one where they are both positive.
 
 So how do we find these points???
 
 We start off the same way we did when we had 2 straight lines.
 
 Where the lines cross, the value of Y is the same for both,
 so our two equations can be joined together with the same Y.
 

 X2 - 1 = Y   Y = 2X + 1

 X2 - 1 = Y = 2X + 1

 
 Lose the Y ...

 X2 - 1 = 2X + 1

 
 Now we get all of the terms on one side ...

 

 
 Now we use our old buddy the quadratic formula ...
 
 Have you memorized it yet???

 

 A = 1   B = -2   C = -2

 So:

 

 

 X = 2.732     or     X = -.732

 
 These are the X values of the two points where our equation graph lines cross.
 
 We could put these X values back into either equation to find the two Y values.
 The straight line equation is easier to use ...
 
Y = 2X + 1 Y = 2X + 1
Y = 2(2.732) + 1 Y = 2(-.732) + 1
Y = 6.464 Y = -.464
 
 So the two equation lines cross at the point X = 2.732, Y = 6.464
 and the point X = -0.732, Y = -0.464.

 

 
 If we had a different straight line that was a bit lower
 there would only be one point where the two lines touch.
 We would get only one answer ...
 

 

 
 and if we had a straight line that was any lower than that,
 the two graph lines would not touch at all.
 We get NO answer ...
 

 

 
 The quick way to tell if we will get 2, 1, or 0 answers
 is to use the discriminant part of the quadratic formula.
 

 

 
IF B2 - 4AC > 0 we get 2 answers
IF B2 - 4AC = 0 we get 1 answer
IF B2 - 4AC < 0 we get 0 answers
 
 Since we only need to know if the discriminant is positive, negative, or zero,
 we don't even need to bother with the radical just the terms under it...
 
 One of the exciting things we do in calculus is figure out the equations 
 of straight lines that just touch curved lines
 
 That is just too exciting to do here ...
 
 (OK, well somebody might think it's exciting.)
 

   copyright 2005 Bruce Kirkpatrick

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