Question

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

Answer 1

Answer:

See below.

Step-by-step explanation:

First, we are given the equation $f(x)=x^2+x+1$.

This is in standard form $ax^2+bx+c$. In other words, a=1, b=1, and c=1.

In order to convert this into vertex form, we need the vertex and the $a$ value. Recall the vertex form: $f(x)=a(x-h)^2+k$, where (h,k) is the vertex. The a is the same a as above.  

First, find the vertex. Find the x (or h) of the vertex using -b/2a. Thus:

$x=h=-b/2a=(-1)/2(1)=-1/2$

Plug this back into the equation for the y coordinate or k:

$(-1/2)^2+(-1/2)+1$

$1/4-2/4+4/4=3/4$.

Thus, the vertex is (-1/2, 3/4).

The a is 1 as we previously determined.

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