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Completing the square works
because you can change one
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parabola into another using
standard transformations.
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You can scale the parabola.
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And you can move the parabola or
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translate it. Horizontally.
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All vertically.
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You can also reflect the
parabola. That's like a
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scaling, but with the scale
factor of minus one.
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Now the standard quadratic
expression Y equals X squared
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gives this parabola.
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And any other quadratic will
always give a parabola, but it
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will be a different size and be
in a different place.
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So to get this new parabola,
we take the standard parabola
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and 1st change its eyes instead
of Y equals X squared. It's
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now Y equals 1/2 X squared.
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Next we move the parabola
sideways instead of Y equals 1/2
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X squared. The ex has become
X minus 2.
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And finally we move the parabola
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vertically. We add on one
to the formula and that
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gives us the quadratic in
completed square form.