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Find the equation which is an displacement of $x² - 3x + 4$ and passes though point $(-3, 3)$ and $(2, 8)$

I've already mounted an simple system of equations which looks obvious

$$\begin{align} 8 &= 4a + 2b + c \\ 3 &= 9a - 3b + c \end{align}$$

but I'm stuck here, does someone know where to go now ? thanks in advance.

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up vote 3 down vote accepted

We could think that since the new parabola is only a displacement of the first one, it must have the same leading coefficient, i.e., $a=1$ (otherwise we'd also be stretching, shrinking or reflecting it). Then we can solve the resulting system of two equations in two unknowns:

$\begin{cases} 8 =4 + 2b +c \\ 3 = 9 - 3b + c \end{cases} \Longleftrightarrow \begin{cases} 4 = 2b +c \\ -6 = - 3b + c \end{cases} $

We could try subtracting the last two equations to get $10 = 5b$, then $b=2$ and replacing anywhere we get $c=0$. So the equation of the desired parabola would be $y= x^2 +2x$.

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+1. To verify that the leading coefficients must match, simply observe that a displacement --say, by $(h,k)$-- of any parabola of the form $y=Px^2+Qx+R$ has the form $y-k=P(x-h)^2+Q(x-h)+R$. You don't even have to fully expand the new equation to see that the coefficient of $x^2$ will be $P$, just as in the original. – Blue Jun 20 '12 at 22:59
Thank you so much, it led me to the answer. – aajjbb Jun 20 '12 at 23:00

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