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We are given a table of ordered pairs and a quadratic equation \( y = (x - k)^2 \), where we are asked to find the value of \( k \).

From the table:

\[
\begin{array}{c|c}
x & y \\
\hline
-1 & 9 \\
1 & 1 \\
3 & 1 \\
5 & 9 \\
\end{array}
\]

Substituting one of these points into the equation \( y = (x - k)^2 \) will help us determine \( k \). Let's use the point \( (1, 1) \), which gives us \( x = 1 \) and \( y = 1 \):

\[
1 = (1 - k)^2
\]

Taking the square root of both sides:

\[
\pm 1 = 1 - k
\]

Now, solving for \( k \):

1. \( 1 = 1 - k \) gives \( k = 0 \) (this is not an option).
2. \( -1 = 1 - k \) gives \( k = 2 \).

Thus, the value of \( k \) is \( \boxed{2} \), which corresponds to option (D).

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would the graph of \( y = (x - k)^2 \) change as the value of \( k \) varies?
2. Why did we choose the point \( (1, 1) \) for substitution, and could we have used another point?
3. What is the significance of the vertex of a parabola in a quadratic equation like \( y = (x - k)^2 \)?
4. How would you verify the value of \( k \) using the other points from the table?
5. What is the difference between the general form of a quadratic equation and the vertex form used here?

**Tip:** Always check multiple points when solving for parameters in a function to verify the correctness of your solution.

The table above shows four ordered pairs on the graph of y = (x - k)^2. What is the value of k?

Determine the value of k for a quadratic function given four ordered pairs on the graph of y = (x - k)^2. The solution involves substitution of points into the function and solving using algebraic methods. Suitable for Grades 9-11.

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