Question

Transform the given quadratic function into vertex form f(x) = quadratic function into vertex form f(x) = quadratic function into vertex form
`f(x) = (x-h)^(2) + k` by completing the square.
`a(x-h)^(2) +k` by completing the square.


`f(x) = 3x^(2) -4x-6`

Answer 1

`f(x) = 3(x -(2)/(3))^2 -(22)/(3)`

The vertex is
`((2)/(3), -(22)/(3))`

Explanation:

For a general quadratic function the form is:

`ax ^ 2 + bx + c`

For the function

`f(x) = 3x ^ 2 -4x -6`

Take common factor 3.

`f(x) = 3(x ^ 2 -(4)/(3)x - 2)`

The values of the coefficients for the function within the parenthesis are the following:
`a = 1`,
`b = -(4)/(3)`,
`c = -2`

Take the value of b and divide it by 2. Then, the result obtained squares it.

`(b)/(2)= -(2)/(3)`

`((b)/(2))^2=(4)/(9)`

Add and subtract
`(4)/(9)`

`f(x) = 3([x ^ 2 -(4)/(3)x +(4)/(9)]- 2-(4)/(9))`

Write the expression of the form

`f(x) = (x-(b)/(2))^2 +k`

`f(x) = 3(x -(2)/(3))^2 -(22)/(3)`

The vertex is
`((2)/(3), -(22)/(3))`