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

1 Answer
Douglas K.
Dec 13, 2017

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

Explanation:

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

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

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

15=a(7-5)^2+1215=a(7−5)2+12
7=a(15-12)^2+57=a(15−12)2+5

15=a(2)^2+1215=a(2)2+12
7=a(3)^2+57=a(3)2+5

3=a(2)^23=a(2)2
2=a(3)^22=a(3)2

a = 3/4a=34
a = 2/9a=29

Substitute the above values into its respective equation:

"[1.2] "y=3/4(x-5)^2+12[1.2] y=34(x−5)2+12
"[2.2] "x=2/9(y-12)^2+5[2.2] x=29(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):

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

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