I was originally given the value $(4,-2)$ as the vertex of a parabola and told that it also includes the value $(3,-5)$. From this point, I deduced that the next point would have the same y-value as the point whose x-value is equidistant from the vertex, so the next point would be $(5,-5)$. I also know that since the parabola's vertex is higher than the two values surrounding it, it is a negative parabola. I am now stuck and do not know how to continue about finding the equation for this limited input/output table.
[Math] How to find the equation for a parabola for which you are given two points and the vertex
quadratics
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Best Answer
Any parabola can be written as:
$ y = Ax^2 + Bx + C$
As you said, our parabola passes through:
$(4,-2)$, $(3,-5)$ and $(5,-5)$
Substituting these values into the general equation of the parabola we get:
$-2 = 16A +4B +C$
$-5 = 9A + 3B +C$
$-5 = 25A + 5B + C$
So we have to solve this 3 by 3 linear system. The solution is:
$A = -3 \quad \quad B = 24 \quad \quad C = -50$
The final answer is then:
$y = -3x^2 + 24x - 50$