Using the vertex form, find a formula for the parabola with vertex (2,14) that passes through the point (1,7)?

There can be two parabolas with vertex at #(2,14)# and passing through #(1,7)#.

One of the form #(y-14)=a(x-2)^2# and other #(x-2)=a(y-14)^2#

Case 1 - If #(y-14)=a(x-2)^2# passes through #(1,7)# then

#7-14=a(1-2)^2# i.e. #a=-7#

and equation is #(y-14)=-7(x-2)^2# i.e. #y=-7(x-2)^2+14#, a vertical parabola

Case 2 - If #(x-2)=a(y-14)^2# passes through #(1,7)# then

#1-2=a(7-14)^2# i.e. #49a=-1# i.e. #a=-1/49#

and equation is #x-2=-1/49(y-14)^2# i.e. #x=-1/49(y-14)^2+2#, a horizontal parabola

graph{(y+7x^2-28x+14)(49x+y^2-28y+98)((x-1)^2+(y-7)^2-0.05)((x-2)^2+(y-14)^2-0.05)=0 [-10.21, 9.79, 4.52, 14.52]}

Given -

Vertex #(2, 14)#
Passes through point #(1, 7)#

This parabola may be either open down or open to left.

The equation of the parabola that opens down.

#y=a(x-h)+k#

Where #(h, k) # is the vertex

#y=a(x-2)^2+14#

The parabola is passing through point #(1,7)#

Substitute this to find the value of #a#

#a(1-2)^2+14=7#
#a+14 = 7#

#a=7-14=-7#

Then the required equation is -

#y=-7(x-2)^2+14#

Parabola May open to left. Then the equation is -

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

#x=a(y-14)^2+2#

#a(y-14)^2+2=x#

#a(7-14)^2+2=1#

#49a+2=1#

#49a=1-2=-1#

#a=-1/49#

Then the equation is -

#x=-1/49(x-14)^2+2#