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) = x^(2) +x-7`

Answer 1

Vertex form:
`f(x) =(x +(1)/(2))^2 -(29)/(4)`

The vertex is
`(-(1)/(2),-(29)/(4))`

Explanation:

For a general quadratic function the form is:

`ax ^ 2 + bx + c`

For the function

`f(x) = x ^ 2+ x -7`

The values of the coefficients for the function are the following:
`a = 1`,
`b =1`,
`c = -7`

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

`(b)/(2)= (1)/(2)`

`((b)/(2))^2=((1)/(2))^2=(1)/(4)`

Add and subtract
`(1)/(4)`

`f(x) = (x ^ 2 +x +(1)/(4)) -(1)/(4)- 7`

Write the expression of the form

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

`f(x) =(x +(1)/(2))^2 -(29)/(4)`

The vertex is (h, k)

The vertex is:
`(-(1)/(2),-(29)/(4))`