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

Let's re-evaluate whether the points in the table represent a linear or nonlinear rate of change by calculating the differences between the \( y \)-values corresponding to each step in \( x \).

### Table:

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

### 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 slope (rate of change) between \( x = -7 \) and \( x = -1 \) remains consistent at 1.5. However, the rate of change between \( x = -1 \) and \( x = 0 \) is different at 3. This change in the rate of change indicates that the relationship between \( x \) and \( y \) is **nonlinear**.

Would you like further details or have any questions?

Here are five related questions to consider:
1. How can the nonlinear relationship in the data be modeled mathematically?
2. What are the implications of a nonlinear relationship in a real-world context?
3. How would this table be plotted on a graph, and what shape would the graph take?
4. What would be the effect of extending the table further in both directions?
5. How can nonlinear relationships be approximated using linear segments?

**Tip:** To identify linearity, always check if the rate of change (slope) is constant across all intervals. Any variation suggests nonlinearity.

Explore whether the points in a given table represent a linear or nonlinear rate of change through detailed calculations of the slope between successive pairs of points. This problem covers key concepts such as rate of change, slope calculation, and identifying linear versus nonlinear relationships. Suitable for students in Grades 7-9.

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