# Find the equation of a parabola (in general form)

Find the equation of the parabola with axis parallel to the $y$-axis, passing through $(1/2,-5/2),(3/2,-9/4)$ and $(-7/2,3/2)$.

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possible duplicate of Find the equation of the parabola – Zia Aug 31 '13 at 10:13
This is not a duplicate . the idea is different although this also deals with finding equation of parabola – Harish Kayarohanam Aug 31 '13 at 10:16
What is the "axis" of a parabola? Is this a common term? – Stefan Smith Aug 31 '13 at 12:52


In order to determine $\ds{A, B , C}$ use $${\rm y}\pars{-\,{7 \over 2}} = {3 \over 2}\,,\qquad {\rm y}\pars{\half} = -\,{5 \over 2}\,,\qquad {\rm y}\pars{3 \over 2} = -\,{9 \over 4}$$

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The above answer is a good shortcut, but by convention it's as follows:

With axis parallel to the y axis, if the vertex of the parabola is on the origin, then the equation is $x^2=4ay$. But when you shift the parabola on a vertex with coordinates $(h,k)$, the equation becomes $(x-h)^2=4a(y-k)$. Substitute the values you gave to find h,k and a (three equations, three unknowns).

But I would also recommend going by $y=ax^2+bx+c$, since the convention might get messy.

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Using the standard form for the equation of the parabola doesn't create as much trouble as you might think, since the $\ 4ak \$ term will cancel between any pair of equations, reducing them to having two unknowns, $\ a \$ and $\ h \$ . You can then go back and pick up $\ k \$ . – RecklessReckoner Mar 28 '14 at 3:42

HINT:(Idea behind the problem) A parabola is a graph of a quadratic function

$$\displaystyle\boxed{y =ax^2 + bx + c}$$

substitute 3 points given , you will get 3 equations in a,b,c and from there find a , b , c solving the 3 equations .and then substitute this a,b,c in the equation above and that will be your answer .

refer an example : refer in case you have doubts and want example

solution from wolfram alpha : wolfram alpha

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Your parabola has axis parallel to the $x$-axis. – walcher Aug 31 '13 at 9:46
No . this is only for axis parallel to y axis . see en.wikipedia.org/wiki/File:Quadratic_function.png – Harish Kayarohanam Aug 31 '13 at 9:50
oh, is it different from x^2 +dx + ey + f = 0? – mona Aug 31 '13 at 9:51
My bad, I thought by axis you meant directrix. – walcher Aug 31 '13 at 9:56
@mona I didn't get what you are asking in comment 3. – Harish Kayarohanam Aug 31 '13 at 10:41

Standard form: $(x+20)^2 + (y+24)^2 = (\dfrac{185}{2})^2$

General form: $2x^2 + 2y^2 - 20x -24y - 63$

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