MHB Finding the equation of the parabola

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To find the equation of the parabola passing through points A(0,1), B(-1,-2), and C(-2,7), two types of parabolas can be derived: a vertical parabola and a horizontal parabola. The vertical parabola is given by the equation y = 6x^2 + 9x + 1, while the horizontal parabola is represented by x = -2/27y^2 + 7/27y - 5/27. A system of equations is created by substituting the points into the general forms of the parabolas, leading to the solutions for coefficients a, b, and c. The discussion also notes that rotating the axes can yield an infinite number of parabolas. The solutions provided clarify the equations for both orientations of the parabola.
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Hello guys, please help me, knowing that the parabola passes through the points A(0,1), B(-1,-2) e C(-2,7). How can i find the equation?
 
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This question has also been posted on MHF for which responses have been given.

I don't want to see the folks here take the time to post help when it has already been given elsewhere. ;)
 
Hello, Chipset3600!

Find the equation of the parabola passing through: A(0,1), B(-1,-2), C(-2,7).
There are two such parabolas: one "vertical" \cup, the other "horizontal" \supset.Vertical: .y \:=\:ax^2 + bx + c

Substitute the points and create a system of three equations.
The system has the solution: .a = 6,\:b = 9,\:c = 1

The equation is: .y \;=\;6x^2 + 9x + 1Horizontal: .x \;=\;ay^2 + by + c

Substitute the points and create a system of three equations.
The system has the solution: .a = \text{-}\tfrac{2}{27},\:b = \tfrac{7}{27},\:c = \text{-}\tfrac{5}{27}

The equation is: .x \;=\;\text{-}\tfrac{2}{27}y^2 + \tfrac{7}{27}y - \tfrac{5}{27}
 
I didn't consider anything but the parabola with vertical axis of symmetry...I suppose we could find an infinite number of parabolas by rotating the axes by any angle we choose. (Cool)
 
Thanks guys :)
 
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