Question

Which function in vertex form is equivalent to f(x) = x2 + x +1?

Answer 1

The function in vertex form is
`f(x)=(x-(-(1)/(2)))^2+(3)/(4)`

Explanation:

Given: Function
`f(x)=x^2+x+1`

We have to write the given function
`f(x)=x^2+x+1` in vertex form.

Consider the given function
`f(x)=x^2+x+1`

Using algebraic identity
`(a+b)^2=a^2+b^2+2ab`

Comparing a = x , 2ab = x

Also, 2b = 1 ⇒
`b=(1)/(2)`

Add and subtract
`b^2=(1)/(4)` , we get,

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

Simplify, we have,

`f(x)=(x+(1)/(2))^21-(1)/(4)`

Simplify, we have,

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

Thus, The function in vertex form is
`f(x)=(x-(-(1)/(2)))^2+(3)/(4)`

Answer 2

f(x) = (x +1/2)² +3/4