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The function provided in the table can be determined as linear or nonlinear by examining the relationship between the \( x \) and \( y \) values. A function is linear if the rate of change (slope) between the pairs of points is constant.

Let’s calculate the slope between the points.

1. For points \((-2, -9)\) and \((4, 1)\):
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-9)}{4 - (-2)} = \frac{1 + 9}{4 + 2} = \frac{10}{6} = \frac{5}{3}
   \]

2. For points \((4, 1)\) and \((10, 7)\):
   \[
   \text{slope} = \frac{7 - 1}{10 - 4} = \frac{6}{6} = 1
   \]

The slopes between the pairs of points are not equal (\( \frac{5}{3} \) and \( 1 \)), indicating that the rate of change is not constant. Therefore, **the function is nonlinear**.

Would you like further details on this, or have any other questions?

Here are 5 related questions to explore:
1. What is the general equation of a linear function?
2. How can we determine if a function is nonlinear from its equation?
3. What are some real-life examples of linear and nonlinear relationships?
4. How does the slope affect the graph of a linear function?
5. How would you graph the data points in the table?

**Tip:** For linear functions, always check if the slope between every pair of points is the same. If it isn’t, the function is nonlinear!

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

Learn how to determine if a function is linear or nonlinear based on data points provided in a table. This problem involves calculating the slope between points to check for consistency in the rate of change. Suitable for students in Grades 7-9.

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