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

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