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!