Rewriting f(x) = x² – 4x + 1 in Vertex Form
To rewrite the function f(x) = x² – 4x + 1 in vertex form, we will use the completing the square method. Vertex form of a quadratic function is expressed as:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola.
Step 1: Start with the original function
We begin with:
f(x) = x² – 4x + 1
Step 2: Group the x terms
Next, we focus on the x terms to complete the square:
f(x) = (x² – 4x) + 1
Step 3: Complete the square
To complete the square for x² – 4x, we take the coefficient of x (which is -4), halve it to get -2, and then square it:
- Halve: -4 / 2 = -2
- Square: (-2)² = 4
Now, we add and subtract this square inside the parentheses:
f(x) = (x² – 4x + 4 – 4) + 1
This simplifies to:
f(x) = ((x – 2)² – 4) + 1
f(x) = (x – 2)² – 3
Step 4: Write in vertex form
Thus, the function in vertex form is:
f(x) = (x – 2)² – 3
Conclusion
From this, we can observe that the vertex of the parabola represented by the function is at the point (2, -3). This illustrates how using the completing the square method allows us to easily convert a quadratic function into its vertex form.
Now, you have successfully rewritten the function and can better understand its properties!