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.