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Let's go through each question to verify the answers and provide explanations for each section.

### Section 1: Determining if a Relationship is Proportional

1. **Question a.**
   - The table shows values of \( x \) and \( y \).
   - **Answer**: Non-Proportional (NP)  
     **Explanation**: For a relationship to be proportional, \( y/x \) should be constant for all pairs. Here, the values of \( y/x \) differ across the table, indicating it's non-proportional.

2. **Question b.**
   - The table shows values of \( x \) and \( y \).
   - **Answer**: Non-Proportional (NP)  
     **Explanation**: Similarly, the ratio \( y/x \) is not constant across all values, so it’s non-proportional.

3. **Question c.**
   - Given equation: \( y = 9x + \frac{1}{2} \)
   - **Answer**: Non-Proportional (NP)  
     **Explanation**: A proportional relationship in an equation form would be of the form \( y = kx \) with no constant added. Since there’s an added constant \(\frac{1}{2}\), this equation is non-proportional.

4. **Question d.**
   - Given equation: \( y = \frac{3}{4}x \)
   - **Answer**: Proportional (P)  
     **Explanation**: This is in the form \( y = kx \), with \( k = \frac{3}{4} \). Since there is no constant term added, it’s proportional.

5. **Question e.**
   - Graph interpretation.
   - **Answer**: Non-Proportional (NP)  
     **Explanation**: The graph shows a line that does not pass through the origin (0, 0), which is a requirement for proportional relationships in graphical form.

### Section 2: Finding the Constant of Proportionality (Unit Rate)

1. **Question a.**
   - Given equation: \( y = 8x \)
   - **Answer**: \( k = 8 \)  
     **Explanation**: In the form \( y = kx \), the constant \( k \) is directly given as 8.

2. **Question b.**
   - Table values are given.
   - **Answer**: \( k = 95 \)  
     **Explanation**: Calculate \( k \) by finding \( y/x \) for any pair (e.g., \( 190/2 = 95 \)).

3. **Question c.**
   - Scenario: \( 13.95 \) per person.
   - **Answer**: \( k = 13.95 \)  
     **Explanation**: The unit rate is simply \( 13.95 \) since it’s a direct price per person.

4. **Question d.**
   - Interpretation of the graph "Sara's English Essay".
   - **Answer**: Calculate \( k \) by finding the slope (words/minutes).  
     Using points (2, 40) and (8, 240):  
     \( k = \frac{240 - 40}{8 - 2} = \frac{200}{6} = 30 \)

Determine if relationships are proportional or non-proportional and find the constant of proportionality for given representations.

Understand how to determine if a relationship is proportional and find the constant of proportionality (unit rate). This review sheet includes examples of equations, tables, and graphs to help students in Grade 7 analyze proportional relationships effectively.

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