How do you write the equation of the parabola that has a vertex of $(3, 4)$ and contains the point $(1, 2)$ ?

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Answer 1 · 2018-06-14T01:17:21

There are two such parabolas.

One has the form:

$y = a(x-h)^2+k$

The other has the form:

$x =a(y-k)^2+h$

Substitute the vertex $(h,k) = (3,4)$ into both forms:

$y = a(x-3)^2+4$ and $x =a(y-4)^2+3$

FInd the value of $a$ so that both parabolas contain the point (1,2):

$2 = a(1-3)^2+4$ and $1 =a(2-4)^2+3$

$2 = 4a+4$ and $1 =4a+3$

$a = -1/2$ and $a = -1/2$

NOTE: It is unusual that both forms have the same value for $a$ . Please do not assume that this will always be the case.

Substitute the value for $a$ into both forms:

$y = -1/2(x-3)^2+4$ and $x =-1/2(y-4)^2+3$

The following is a graph of both parabolas:

![www.desmos.com](useruploads.socratic.org)

Please observe that both parabolas have the vertex $(3,4)$ and both parabolas contain the point $(1,2)$

How do you write the equation of the parabola that has a vertex of (3, 4)(3, 4) and contains the point (1, 2)(1, 2)?