To convert the equation f(x) = x² + x + 1 into vertex form, we will complete the square.
1. Start with the original function:
f(x) = x² + x + 1
2. Take the coefficient of x, which is 1, divide it by 2, and square it. This gives us:
(1/2)² = 1/4.
3. Now, we need to add and subtract this value inside the function:
f(x) = (x² + x + 1/4) + 1 – 1/4
4. This simplifies to:
f(x) = (x + 1/2)² + 3/4
Now the equation is in vertex form, which is:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola. In this case, the vertex is at (-1/2, 3/4). So, the vertex form of the given quadratic equation is:
f(x) = (x + 1/2)² + 3/4