https://file.aiquickdraw.com/mathbot/file/1726606606673290e1ick.jpg

Let's analyze the function from the given table to determine whether it is linear or nonlinear.

### Given points:
\[
\begin{array}{c|c}
x & y = f(x) \\
\hline
0 & -25 \\
1 & -1 \\
2 & 7 \\
3 & 3 \\
4 & 16 \\
\end{array}
\]

### To check for linearity:
A linear function has a constant slope between all points. The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slopes between consecutive points:

1. From \((0, -25)\) to \((1, -1)\):
   \[
   \text{slope}_1 = \frac{-1 - (-25)}{1 - 0} = \frac{24}{1} = 24
   \]

2. From \((1, -1)\) to \((2, 7)\):
   \[
   \text{slope}_2 = \frac{7 - (-1)}{2 - 1} = \frac{8}{1} = 8
   \]

3. From \((2, 7)\) to \((3, 3)\):
   \[
   \text{slope}_3 = \frac{3 - 7}{3 - 2} = \frac{-4}{1} = -4
   \]

4. From \((3, 3)\) to \((4, 16)\):
   \[
   \text{slope}_4 = \frac{16 - 3}{4 - 3} = \frac{13}{1} = 13
   \]

Since the slopes between consecutive points are not the same (24, 8, -4, and 13), the function is **nonlinear**.

### Conclusion:
The given function is nonlinear.

Do you have any questions or want further clarification? 

Here are some related questions:
1. What is the definition of a linear function?
2. How can you graphically identify if a function is linear or nonlinear?
3. What are other types of functions besides linear and nonlinear?
4. Can a function have a varying slope and still be classified as linear?
5. How do you calculate the slope of a curve at a specific point?

**Tip**: Always check for a constant rate of change (slope) between points to determine if a function is linear.

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.

Analyze a function based on a table of values to determine if it is linear or nonlinear. Learn how to calculate the slope between points and identify non-constant rates of change. This example covers slope calculations between consecutive points and concludes that the function is nonlinear, suitable for Grades 8-10.

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