which function in vertex form is equivalent to f(x) = x² + x + 1?\n○ f(x)=(x + 1/4)²+3/4\n○ f(x)=(x +…

Explanation:

Step1: Recall vertex - form formula

The vertex - form of a quadratic function is $f(x)=a(x - h)^2+k$, and for the quadratic function $y = ax^{2}+bx + c$, we complete the square. Given $f(x)=x^{2}+x + 1$, where $a = 1$, $b = 1$, $c = 1$.

Step2: Complete the square for the $x$ terms

For the expression $x^{2}+x$, we know that $(x + m)^{2}=x^{2}+2mx+m^{2}$. In $x^{2}+x$, if $2m = 1$, then $m=\frac{1}{2}$, and $x^{2}+x=(x+\frac{1}{2})^{2}-\frac{1}{4}$. So $f(x)=x^{2}+x + 1=(x+\frac{1}{2})^{2}-\frac{1}{4}+1$.

Step3: Simplify the expression

$f(x)=(x+\frac{1}{2})^{2}+\frac{-1 + 4}{4}=(x+\frac{1}{2})^{2}+\frac{3}{4}$.

Answer:

$f(x)=(x+\frac{1}{2})^{2}+\frac{3}{4}$ (the third option)