Question

What is the the vertex of `y = (x - 3)^2 + 7x-12`?

Answer 1

`color(blue)("Vertex" -> (x,y) -> (-1/2 ,-3 1/4)`

Explanation:

You have to combine these variables before you can do any thing else.

Given:`" "y=(x-3)^2+7x-12`

`y=x^2-6x+9+7x-12`

`y=x^2+x-3`...........................(1)

From this point you can 'complete the square' or do a part version of the process.

I am opting for the part process

Write as:

Consider the coefficient of `+x` in equation ( 1 ), which is 1
Apply `(-1/2)xx1 = -1/2`

`x_("vertex")=-1/2`....... Fast isn't it!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("A word of warning:")` Given the standard form of
`y=ax^2+bx+c` you need to convert this to

`y=a(x^2+b/ax)+c`

In your case `a=1`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute `x=-1/2` into equation (1)

`y_("vertex")=(-1/2)^2+(-1/2)-3`

`y_("vertex")=1/4-1/2-3 = -3 1/4`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Vertex" -> (x,y) -> (-1/2 ,-3 1/4)`