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, do the points in the table below represent a linear or nonlinear rate of change.

To determine if the points in the table represent a linear or nonlinear rate of change, we need to examine how the values of \( y \) change as \( x \) changes. If the rate of change of \( y \) with respect to \( x \) is constant, the relationship is linear. If the rate of change is not constant, it is nonlinear.

Here’s the table you provided:

\[
\begin{array}{c|c|c|c|c|c}
x & -7 & -5 & -3 & -1 & 0 \\
\hline
y & 11 & 14 & 17 & 20 & 23 \\
\end{array}
\]

### Step 1: Calculate the rate of change (slope) between each pair of points:

\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

- Between \( x = -7 \) and \( x = -5 \):
  \[
  \frac{14 - 11}{-5 - (-7)} = \frac{3}{2} = 1.5
  \]
  
- Between \( x = -5 \) and \( x = -3 \):
  \[
  \frac{17 - 14}{-3 - (-5)} = \frac{3}{2} = 1.5
  \]

- Between \( x = -3 \) and \( x = -1 \):
  \[
  \frac{20 - 17}{-1 - (-3)} = \frac{3}{2} = 1.5
  \]

- Between \( x = -1 \) and \( x = 0 \):
  \[
  \frac{23 - 20}{0 - (-1)} = \frac{3}{1} = 3
  \]

### Step 2: Analyze the results

The rates of change between \( x = -7 \) and \( x = -1 \) are consistent (1.5). However, the rate of change between \( x = -1 \) and \( x = 0 \) differs (3). This indicates that the relationship is **nonlinear** because the rate of change is not constant across the interval.

Would you like further details or have any questions about this analysis? 

Here are five related questions you might consider:
1. What type of equation could model this nonlinear relationship?
2. How can you graphically represent this data to visualize the nonlinearity?
3. If the relationship is nonlinear, could it be quadratic or some other polynomial function?
4. How does the slope of a linear function differ from that of a nonlinear function?
5. What are some real-world examples of nonlinear relationships?

**Tip:** In a linear relationship, the slope between any two points remains constant. This makes linear functions easy to predict and model.

Explore how to determine if a table represents a linear or nonlinear rate of change. Analyze a given table and calculate the slope between points to identify the type of relationship. Suitable for Grades 7-9 students learning about rates of change and linear equations.

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