Question

What is the vertex form of `y= x^2 - 10x - 9`?

Answer 1

`y=x^2 + 10x -9`

First, we need to complete the square

`y=color (green)((x^2 + 10x)) -9`

What would make `color(green)(t h i s)` `(x^2+10x)` a perfect square? Well, `5+5` equals `10` and `5 xx 5` equals `25` so let's try adding that to the equation:

`x^2+10x+25`

As a perfect square:

`(x+5)^2`

Now let's look at our original equation.

`y= (x+5)^2 -9 color(red)(-25)`

NOTE that we subtracted `25` after we added it. That's because we added `25`, but as long as we later subtract it, we haven't changed the value of the expression

`y = (x+5)^2 -34`

To check our work, let's graph our original function and what we have. If we did it right, they should be the same

graph{y=x^2+10x-9}

graph{y=(x+5)^2-34}

Looks like we were right!