https://file.aiquickdraw.com/mathbot/file/1728049797186z92unspr.jpg

Let's analyze the scatter plots in the image one by one and answer the questions.

1. **Figure 1**: The points form a clear upward trend, suggesting a **positive linear relationship**. As the \( x \)-values increase, the \( y \)-values also increase.
  
2. **Figure 2**: This plot shows no clear pattern; the points are scattered randomly, suggesting **no relationship** between the two variables.
  
3. **Figure 3**: The points follow a downward trend, indicating a **negative linear relationship**. As the \( x \)-values increase, the \( y \)-values decrease.

4. **Figure 4**: The points form a curved, parabolic shape, showing a **nonlinear relationship**.

### Answers to the questions:

- **(a)**: The data set that shows a **nonlinear relationship** is **Figure 4**.
- **(b)**: The data set that shows a **positive linear relationship** is **Figure 1**.
- **(c)**: The data set that shows a **negative linear relationship** is **Figure 3**.
- **(d)**: The data set that shows **no relationship** between the variables is **Figure 2**.

Would you like a detailed explanation of how these relationships are identified?

Here are 5 follow-up questions to explore this topic further:

1. What is the main difference between linear and nonlinear relationships in scatter plots?
2. Can a scatter plot display both linear and nonlinear characteristics simultaneously?
3. How can outliers in a scatter plot affect the perceived relationship between variables?
4. What are some real-world examples of data with positive and negative linear relationships?
5. How can we use the correlation coefficient to quantify the relationship between two variables?

**Tip**: To determine if a relationship is linear or nonlinear in a scatter plot, check if the data points can be approximated by a straight line. If not, it may indicate a nonlinear relationship.

Classify linear and nonlinear relationships from scatter plots for four data sets. Answer the following questions:

(a) Which data set appears to show a nonlinear relationship between its two variables?
(b) Which data set appears to show a positive linear relationship between its two variables?
(c) Which data set appears to show a negative linear relationship between its two variables?
(d) Which data set appears to show no relationship between its two variables?

Learn to classify scatter plots by identifying linear, nonlinear, positive, negative, and no-relationship patterns. Suitable for Grades 8-10, this problem focuses on analyzing four scatter plots and recognizing trends between variables.

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