Question

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

Answer 1

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.

Explanation:

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):