Write the equation in vertex form equivalent to f(x) = x² + x +1?

Solution:

Vertex form of a quadratic equation refers to (x - h)² = 4a (y - k) form or (y - k)² = 4a (x - h) form depending on whether the square is on x-term or y-term respectively.

Given quadratic function is y = f(x) = x² + x +1

In the given equation square term is for x.

∴ Equation must be reduced to (x - h)² = 4a (y - k) form.

We have y = x² + x + 1

⇒ y = x² + x + 1 + 1/4 - 1/4 [by completing the square]

⇒ y = x² + x + 1/4 + 3/4

⇒ y - 3/4 = (x + 1/2)²

⇒ (x + 1/2)² = y - 3/4

The above equation is in the vertex form.

Write the equation in vertex form equivalent to f(x) = x² + x +1?

Summary:

The vertex form of the given quadratic function, f(x) = x² + x + 1, is (x + 1/2)² = y - 3/4.