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The table you provided gives pairs of values for \( x \) and \( y \) as follows:

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

To determine whether this function is linear or nonlinear, we check the rate of change of \( y \) with respect to \( x \). A function is linear if the rate of change (slope) is constant between all points.

### Step 1: Calculate the rate of change (slope) between consecutive points.
- From \( (x_1 = 0, y_1 = 9) \) to \( (x_2 = 1, y_2 = 2) \):
  \[
  \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 9}{1 - 0} = \frac{-7}{1} = -7
  \]
  
- From \( (x_2 = 1, y_2 = 2) \) to \( (x_3 = 2, y_3 = 1) \):
  \[
  \text{slope} = \frac{y_3 - y_2}{x_3 - x_2} = \frac{1 - 2}{2 - 1} = \frac{-1}{1} = -1
  \]

### Step 2: Analyze the results.
The slope between the points changes from \(-7\) to \(-1\), which means the rate of change is not constant.

### Conclusion:
Since the rate of change is not constant, the function is **nonlinear**.

Would you like more details or have any questions?

Here are 5 follow-up questions to explore further:
1. What is a linear function, and how can you identify it from a table?
2. Can we determine if the function is quadratic based on the given data?
3. What other methods can be used to check if a function is nonlinear?
4. How does the graph of this function differ from a linear function graph?
5. Could this function have any special properties (e.g., exponential)?

**Tip:** To quickly spot if a function is linear, check if the difference in \(y\)-values divided by the difference in \(x\)-values remains constant across all points.

The table shows a function. Is the function linear or nonlinear?

Learn how to determine if a function is linear or nonlinear using a table of values. This problem involves calculating the rate of change between points to check for a constant slope. Suitable for students in Grades 6-8.

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