To determine which function in vertex form is equivalent to [tex]\( f(x) = x^2 + x + 1 \)[/tex], we need to complete the square and check each given option to see which matches the original function.
### Step-by-Step Solution:
1. Begin with the quadratic function [tex]\( f(x) = x^2 + x + 1 \)[/tex].
2. Complete the square for the quadratic part:
- Take the coefficient of [tex]\( x \)[/tex], which is 1.
- Divide it by 2: [tex]\( \frac{1}{2} = 0.5 \)[/tex].
- Square the result: [tex]\( (0.5)^2 = 0.25 \)[/tex].
3. Rewrite the quadratic expression by adding and subtracting 0.25 inside the equation:
[tex]\[
f(x) = x^2 + x + 1 = x^2 + x + 0.25 + 1 - 0.25 = (x + 0.5)^2 - 0.25 + 1
\][/tex]
4. Simplify the constant terms:
[tex]\[
f(x) = (x + 0.5)^2 + 0.75
\][/tex]
Thus, the vertex form of the quadratic function [tex]\( f(x) = x^2 + x + 1 \)[/tex] is:
[tex]\[
f(x) = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}
\][/tex]
### Verification of Options:
Let's compare this with the given options:
1. [tex]\( f(x) = \left(x + \frac{1}{4}\right)^2 + \frac{3}{4} \)[/tex]
2. [tex]\( f(x) = \left(x + \frac{1}{4}\right)^2 + \frac{5}{4} \)[/tex]
3. [tex]\( f(x) = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \)[/tex]
4. [tex]\( f(x) = \left(x + \frac{1}{2}\right)^2 + \frac{5}{4} \)[/tex]
Among these options, the one that matches our vertex form:
[tex]\[
\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}
\][/tex]
is option 3.
### Conclusion:
The function in vertex form that is equivalent to [tex]\( f(x) = x^2 + x + 1 \)[/tex] is:
[tex]\[
f(x) = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}
\][/tex]
