Question

How do you write the equation for a graph that is a parabola with a vertex at `(5, 3)`?

Answer 1

You can't find any graph with just one point. If you just have a point it's just a point.

Explanation:

You can't find the equation for a parabola with just a vertex. You need another point.

The equation for a parabola is `y=a(x-h)^2+k`
You have `h = 5` and `k=3` but no `(x,y)` so then the equation becomes just `y=a(x-5)^2 + 3` which is not enough information to get the equation.

You can't find any graph with just one point. If you just have a point it's just a point.

Answer 2

`y = a(x^2 - 10x + 25) + 3`

Explanation:

`y = ax^2 + bx + c`

`-b/(2a) = 5 and - Delta/(4a) = 3`

`b = -10a and Delta = -12a = 100a^2 - 4ac => c = 25a + 3`

`y = ax^2 - 10ax + 25a + 3`

Example:

`x^2 - 10x + 28 = 0`
`Delta = 100 - 4 * 28 = -12`
`x = 5 ± i sqrt 3`