Find the equation of a parabola (in general form)

Question

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

Answer 1

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 : parabola with axis paraller to y axis

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

solution from wolfram alpha : wolfram alpha

Answer 2

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.