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I am trying to find the function for a parabola. The things that are known are the points $(x_1,y_1)$ $(x_3,y_3)$ and in between somewhere the parabola has its vertex, which is its minimum. The $x$-coordinate of this point is not known only the $y_2$ is known. Now how am I supposed to find the function for the parabola? (I know that it is $ax^2+bx+c$ but how do I calculate the $a$,$b$ and $c$ from only the 2 and a half points?)

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2 Answers 2

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You have two equations:

$$y_1 = a{x_1}^2 + bx_1 + c$$ $$y_3 = a{x_3}^2 + bx_3 + c$$

and also know that the $x$-coordinate of the vertex is $x_2 = -\frac{b}{2a}$, so

$$y_2 = a{x_2}^2 + bx_2 + c$$

Substitute $-\frac{b}{2a}$ for $x_2$ in the above. You now have three equations whose unknowns are $a$, $b$ and $c$.

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Another approach: Let $y=a(x-x_2)^2+y_2\iff y-y_2=a(x-x_2)^2$. Plugging in the coordinates and dividing both equations gives $$\frac{y_3-y_2}{y_1-y_2}=\frac{(x_3-x_2)^2}{(x_1-x_2)^2}.$$ Now convince yourself that the left hand side is positive, hence $x_2$ is easily calculated from $$\sqrt{\frac{y_3-y_2}{y_1-y_2}}=\pm\frac{(x_3-x_2)}{(x_1-x_2)}.$$

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