Using the Completing the Square Method to Rewrite f(x) = x² – 4x + 1 in Vertex Form

To rewrite the quadratic function f(x) = x² – 4x + 1 in vertex form, we can use the completing the square method. The vertex form of a quadratic function is given by f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola.

Let’s follow these steps:

1. Start with the original function: f(x) = x² – 4x + 1.
2. To complete the square, we need to focus on the x² – 4x part. First, we take the coefficient of x, which is -4, divide it by 2 to get -2, and then square it to get 4.
3. Add and subtract this square inside the function: f(x) = (x² – 4x + 4) – 4 + 1.
4. This simplifies to: f(x) = (x – 2)² – 3.

Now, we can express it in vertex form:

f(x) = (x – 2)² – 3

Here, we see that the vertex of the parabola is at the point (2, -3). Completing the square has allowed us to easily identify the vertex and rewrite the function in a more useful form for graphing and analysis.