Question

What is the vertex form of `y=-3x^2+4x -3`?

Answer 1

To complete the square of `-3x^2+4x-3`:
Take out the `-3`
`y=-3(x^2-4/3x)-3`
Within the brackets, divide the second term by 2 and write it like this without getting rid of the second term:
`y=-3(x^2-4/3x+(2/3)^2-(2/3)^2)-3`
These terms cancel each other out so adding them to the equation isn't a problem.

Then within the brackets take the first term, the third term, and the sign preceding the second term, and arrange it like this:
`y=-3((x-2/3)^2-(2/3)^2)-3`
Then simplify:
`y=-3((x-2/3)^2-4/9)-3`
`y=-3(x-2/3)^2+4/3-3`
`y=-3(x-2/3)^2-5/3`

You can conclude from this that the vertex is `(2/3, -5/3)`

Answer 2

`y=-3(x-2/3)^2-5/3`

Explanation:

`"the equation of a parabola in "color(blue)"vertex form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))`

`"where "(h,k)" are the coordinates of the vertex and a"`
`"is a multiplier"`

`"to obtain this form use the method of "color(blue)"completing the square"`

`• " the coefficient of the "x^2" term must be 1"`

`rArry=-3(x^2-4/3x+1)`

`• " add/subtract "(1/2"coefficient of x-term")^2" to"`
`x^2-4/3x`

`y=-3(x^2+2(-2/3)xcolor(red)(+4/9)color(red)(-4/9)+1)`

`color(white)(y)=-3(x-2/3)^2-3(-4/9+1)`

`color(white)(y)=-3(x-2/3)^2-5/3larrcolor(red)"in vertex form"`