https://file.aiquickdraw.com/mathbot/file/1728288832791qgalth5o.jpg

Let's analyze the table shown in your image to determine if the function is linear or nonlinear. The table lists values for \(x\) and \(y\) as follows:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
15 & 2 \\
16 & 9 \\
17 & 15 \\
\hline
\end{array}
\]

To determine if a function is linear, the change in \(y\) must be constant for a constant change in \(x\). In other words, if the difference between successive \(x\) values is always the same, and the same is true for the differences in corresponding \(y\) values, the function is linear.

### Step 1: Calculate the change in \(x\)
- From 15 to 16, \( \Delta x = 16 - 15 = 1 \)
- From 16 to 17, \( \Delta x = 17 - 16 = 1 \)

### Step 2: Calculate the change in \(y\)
- From 2 to 9, \( \Delta y = 9 - 2 = 7 \)
- From 9 to 15, \( \Delta y = 15 - 9 = 6 \)

The changes in \(y\) are not constant (\(7 \neq 6\)), even though the changes in \(x\) are constant. This indicates that the function is **nonlinear**.

Would you like further details on why the function is nonlinear or have any other questions?

Here are 5 related questions to explore this topic further:
1. How do you find the equation of a linear function from a table?
2. What are the characteristics of a nonlinear function?
3. Can nonlinear functions ever be modeled with an equation?
4. How does the rate of change help identify a linear function?
5. What are examples of common nonlinear functions?

**Tip:** A function is linear if its graph forms a straight line, and the rate of change between any two points is constant.

The table shows a function. Is the function linear or nonlinear based on the values provided?

Learn how to determine whether a function is linear or nonlinear by analyzing a table of values. Explore the concept of constant rate of change, apply it to the given x and y values, and understand why this particular function is classified as nonlinear. Suitable for students in Grades 6-8.

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