What is the equation of the parabola that has a vertex at # (10, 8) # and passes through point # (5,83) #?

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Answer 1 · 2017-05-28T17:38:51

Actually, there are two equations that satisfy the specified conditions:

#y = 3(x - 10)^2+8# and #x = -1/1125(y-8)^2+10#

A graph of both parabolas and the points is included in the explanation.

There are two general vertex forms:

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

where #(h,k)# is the vertex

This gives us two equations where "a" is unknown:

#y = a(x - 10)^2+8# and #x = a(y-8)^2+10#

To find "a" for both, substitute the point #(5,83)#

#83 = a(5 - 10)^2+8# and #5 = a(83-8)^2+10#

#75 = a(-5)^2# and #-5 = a(75)^2#

#a=3# and #a = -1/1125#

The two equations are: #y = 3(x - 10)^2+8# and #x = -1/1125(y-8)^2+10#

Here is a graph that proves that both parabolas have the same vertex and intersect the required point:

Demos.com