## Saturday, November 6, 2010

### Parabolas and Intersecting Lines

Suppose you have already plotted P1: y = x2 - 3x + 1, but the teacher asks you to find graphically the roots of P2: y = x2 - 4x + 3. What a bother – now you need to plot another parabola. Or do you?
Johnny can rearrange P2 to get an equation of the form: sloped line = parabola already plotted.

 0 = x2 - 4x + 3 Given P2 +x =( x2 - 4x + 3) +x Add 'x' to both sides x = x2 - 3x + 3 Simplify by collecting ‘x’ terms on the right x - 2 = (x2 - 4x + 3) - 2 Add '-2' to both sides, i.e., subtract x - 2 = x2 - 3x + 1 Simplify by collecting constant terms on the right line = given P1

Now Johnny has a sloped line = parabola that he has already graphed
The sloped line has m = 1 and c = -2, so it is easy to draw.
Reading from the graph, the solution to the teacher’s question is x = 1 or x = 3.

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As a check, Mary factorises y= x2 - 4x + 3
y = (x – 3) (x – 1)
When y = 0,
x – 3 = 0, so x = 3
or x – 1 = 0 so x = 1
Done!