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To solve for \( y \) in the function \( f(x) = x^2 - 2x + 1 \) at the given values of \( x \), we need to substitute each \( x \) value into the equation.

Let's calculate each value:

1. **For \( x = -2 \):**
   \[
   f(-2) = (-2)^2 - 2(-2) + 1 = 4 + 4 + 1 = 9
   \]
   So, \( y = 9 \).

2. **For \( x = -1 \):**
   \[
   f(-1) = (-1)^2 - 2(-1) + 1 = 1 + 2 + 1 = 4
   \]
   So, \( y = 4 \).

3. **For \( x = 0 \):**
   \[
   f(0) = (0)^2 - 2(0) + 1 = 0 + 0 + 1 = 1
   \]
   So, \( y = 1 \).

4. **For \( x = 1 \):**
   \[
   f(1) = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0
   \]
   So, \( y = 0 \).

5. **For \( x = 2 \):**
   \[
   f(2) = (2)^2 - 2(2) + 1 = 4 - 4 + 1 = 1
   \]
   So, \( y = 1 \).

The completed table should be:

\[
\begin{array}{c|c}
x & y \\
\hline
-2 & 9 \\
-1 & 4 \\
0 & 1 \\
1 & 0 \\
2 & 1 \\
\end{array}
\]

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. What is the vertex of the parabola described by the equation \( f(x) = x^2 - 2x + 1 \)?
2. How can you determine the axis of symmetry for the function \( f(x) = x^2 - 2x + 1 \)?
3. What is the discriminant of the quadratic equation \( x^2 - 2x + 1 = 0 \) and what does it tell you about the roots?
4. How do you complete the square for the expression \( x^2 - 2x + 1 \)?
5. Can you find the roots of the equation \( x^2 - 2x + 1 = 0 \) using the quadratic formula?
6. What is the graphical representation of the function \( f(x) = x^2 - 2x + 1 \)?
7. How does changing the constant term in \( f(x) = x^2 - 2x + c \) affect the graph of the function?
8. How do you derive the general form of a quadratic function from its vertex form?

**Tip:** Quadratic functions are symmetrical, and the vertex can be found at \( x = \frac{-b}{2a} \) where the equation is \( ax^2 + bx + c \).

Learn how to solve the quadratic function f(x) = x² - 2x + 1 by evaluating it for various values of x. This detailed solution covers function evaluation, finding the vertex of a parabola, and the quadratic formula. Suitable for students in Grades 7-9.

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