How do you write the equation of each parabola in vertex form given Vertex (5, 12); point (7,15)?

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Answer 1 · 2017-12-13T02:06:30

There are two vertex forms:
#"[1] "y=a(x-h)^2+k#
#"[2] "x=a(y-k)^2+h#
where #(h,k)# is the vertex and a is determined by the given point.

Given #(h,k) = (5,12)#:

#"[1.1] "y=a(x-5)^2+12#
#"[2.1] "x=a(y-12)^2+5#

Given #(x,y) = (7,15)#:

#15=a(7-5)^2+12#
#7=a(15-12)^2+5#

#15=a(2)^2+12#
#7=a(3)^2+5#

#3=a(2)^2#
#2=a(3)^2#

#a = 3/4#
#a = 2/9#

Substitute the above values into its respective equation:

#"[1.2] "y=3/4(x-5)^2+12#
#"[2.2] "x=2/9(y-12)^2+5#

Here is a graph with the vertex and the required point (in black), equation [1.2] (in blue), and equation [2.2] (in green):

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