https://file.aiquickdraw.com/mathbot/file/1731513785764r0bqomd6.jpg

To analyze each relation, let’s go through the tables in the image and answer each question:

### A. Which table(s) do not represent a function?

A relation is a function if every \( x \)-value has a unique \( y \)-value.

1. **Relation 1:** Each \( x \)-value has a unique \( y \)-value.
2. **Relation 2:** Each \( x \)-value has a unique \( y \)-value.
3. **Relation 3:** Each \( x \)-value has a unique \( y \)-value.
4. **Relation 4:** Each \( x \)-value has a unique \( y \)-value.

**Answer:** All tables represent functions.

### B. Which table(s) represent a linear relationship?

To determine if a relation is linear, the change in \( y \) values should be consistent as \( x \) increases by a fixed amount.

1. **Relation 1:** The differences between consecutive \( y \)-values (3, 5, 7, 9, 11) are consistent, indicating a linear relationship.
2. **Relation 2:** The differences between \( y \)-values (31, 28, 25, 22, 19) are consistent as well, indicating a linear relationship.
3. **Relation 3:** The differences between \( y \)-values (3, 3, 3, 3) are consistent, indicating a linear relationship.
4. **Relation 4:** The differences between \( y \)-values (10, 20, 30, 40, 50) are also consistent, indicating a linear relationship.

**Answer:** All tables (Relations 1, 2, 3, and 4) represent linear relationships.

### C. Which table(s) show a decreasing relationship?

A relationship is decreasing if \( y \) values decrease as \( x \) values increase.

1. **Relation 1:** \( y \)-values are increasing.
2. **Relation 2:** \( y \)-values are decreasing.
3. **Relation 3:** \( y \)-values are constant.
4. **Relation 4:** \( y \)-values are increasing.

**Answer:** Only Relation 2 shows a decreasing relationship.

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If you need further details or have any questions, feel free to ask!

Here are some follow-up questions that expand on this topic:

1. What are the characteristics of a nonlinear relationship in a table format?
2. How can we determine if a function is increasing or decreasing just by looking at a graph?
3. Can a function have a mix of increasing and decreasing intervals? Give an example.
4. How would we identify if a table represents an exponential relationship?
5. What are the implications of having a constant function in real-world scenarios?

**Tip:** When analyzing tables for linear relationships, check if the difference between consecutive \( y \)-values is constant, as this indicates a linear pattern.

Looking at the 4 tables, determine which ones do not represent a function, which ones represent a linear relationship, and which show a decreasing relationship.

Learn how to analyze data tables to determine if they represent functions, identify linear relationships, and find decreasing patterns. This problem covers core concepts in functions and algebra, suitable for students in Grades 8-10.

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