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

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Answer 1 · 2018-06-14T01:17:21

There are two such parabolas.

One has the form:

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

The other has the form:

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

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

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

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

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

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

$a = - \frac{1}{2}$ $a = -1/2$ and $a = - \frac{1}{2}$ $a = -1/2$

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

Substitute the value for $a$ $a$ into both forms:

$y = - \frac{1}{2} {(x - 3)}^{2} + 4$ $y = -1/2(x-3)^2+4$ and $x = - \frac{1}{2} {(y - 4)}^{2} + 3$ $x =-1/2(y-4)^2+3$

The following is a graph of both parabolas:

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

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

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)?