Question

What is the vertex form of `y= (3x+9)(x-2)`?

Answer 1

`y=3(x+0.5)^2 -18.75`

Explanation:

First let's expand the equation:

`(3x+9)(x−2)` `=` `3x^2 -6x+9x-18`

which simplifies to:
`3x^2 +3x-18`

Let's find our vertex using `x=-b/(2a)` where a and b are of `ax^2 +bx+c`

We find the x value of our vertex to be `-0.5`
(`-3/(2(3))`)

Plug it into our equation and find y to be `-18.75`
`3(-0.5)^2+3(-0.5)-18`

so our vertex is at `(-0.5, -18.75)`

We can also check this with a graph:
graph{(3x^2+3x-18) [-10.3, 15.15, -22.4, -9.68]}

Now that we have our vertex, we can plug it into the vertex form!

`f(x)=a(x-h)^2+k`

where `h` is our x value of the vertex, and `k` is the y value of the vertex.

so `h=-0.5` and `k=-18.75`

In the end we find:

`y=3(x+0.5)^2 -18.75`