Question

What is the range of this quadratic function?
y=x^2–4x+4

Answer 1

To find the range of the quadratic function \( y = x^2 - 4x + 4 \), we can start by rearranging it into vertex form by completing the square.

1. The given quadratic can be rewritten as:
\[
y = x^2 - 4x + 4
\]

2. We notice that the expression \( x^2 - 4x + 4 \) can be factored:
\[
y = (x - 2)^2
\]

3. The expression \( (x - 2)^2 \) reaches its minimum value when \( x - 2 = 0 \), or \( x = 2 \). At this point:
\[
y = (2 - 2)^2 = 0
\]

4. Since the squared term \( (x - 2)^2 \) is always non-negative (i.e., \( (x - 2)^2 \geq 0 \) for all \( x \)), the minimum value of \( y \) is \( 0 \).

5. Therefore, the range of the function is all values of \( y \) such that \( y \) is greater than or equal to \( 0 \).

Given this analysis, the range of the quadratic function \( y = x^2 - 4x + 4 \) is:
\[
\text{Range: } [0, \infty)
\]