https://r2.erweima.ai/mathbot/image/1726091308802d8n5pcah.jpg

Let's analyze the problem step by step. 

### Observing the image:

**Task Breakdown:**

1. **Table Completion**:
    - The table compares video rental costs for both non-members (no socios) and members (socios). 
    - Part (a) asks for the completion of the table based on given trends.

2. **Graph Completion**:
    - Part (b) requires plotting results on the graph: use red dots for non-members and green dots for members.

3. **Decision for Membership**:
    - Part (c) asks how many videos would justify becoming a member based on the costs.

4. **Equation Representation**:
    - Part (d) gives the cost equation for non-members \( y = 2.5x \), where \( x \) is the number of videos, and asks for a similar equation for members.

5. **Graph Type Analysis**:
    - Part (e) asks if the graph is made of discontinuous straight lines, and why.

---

### Part (a): Complete the Table

1. Non-Member Prices:
    - Price increases by 2.5 per video, which means the price for 4, 5, ..., 10 videos would be \( 10, 12.5, 15, 17.5, 20, 22.5, 25 \).

2. Member Prices:
    - Members' price decreases steadily by 1 per video, suggesting for videos 6–10, the prices are \( 16, 17, 18, 19, 20 \).

| # Videos    | Non-Members (Price No Socios) | Members (Price Socios) |
|-------------|-------------------------------|------------------------|
| 1           | 0                             | 12                     |
| 2           | 2.5                           | 13                     |
| 3           | 5                             | 14                     |
| 4           | 7.5                           | 15                     |
| 5           | 10                            | 16                     |
| 6           | 12.5                          | 17                     |
| 7           | 15                            | 18                     |
| 8           | 17.5                          | 19                     |
| 9           | 20                            | 20                     |
| 10          | 22.5                          | 21                     |

---

### Part (b): Complete the Graph

For this, you'd plot the points:
- **Non-Members (Red Dots)**: \( (1, 0), (2, 2.5), (3, 5), (4, 7.5), \dots \).
- **Members (Green Dots)**: \( (1, 12), (2, 13), (3, 14), (4, 15), \dots \).

---

### Part (c): At what point should one become a member?

- To find when it’s beneficial to become a member, compare the non-member and member prices. As you rent more videos, the non-member price increases faster than the member price.
    - For example, at 8 videos, non-member price = \( 17.5 \), and member price = \( 19 \). 
    - At 9 videos, non-member price = \( 20 \), and member price = \( 20 \).
    - **Conclusion**: It’s beneficial to become a member after renting more than 8 videos.

---

### Part (d): Equation for Members

For members, based on the table:
- The price starts at \( 12 \) for the first video and increases by \( 1 \) for each additional video.
- Thus, the equation for members can be expressed as:
    \[
    y = 11 + x
    \]
    where \( x \) is the number of videos.

---

### Part (e): Are the graphs discontinuous straight lines?

- **Answer**: Yes, the graphs are discontinuous.
- **Reason**: Since the number of videos rented must be a whole number (you cannot rent half a video), the graph will consist of distinct points rather than a continuous line.

---

Do you need further clarification or details on any of the points? Here are 5 related questions you might explore:

1. How would you calculate the total savings after renting a specific number of videos?
2. What does the slope represent in the non-member equation \( y = 2.5x \)?
3. How would the graph change if the video rental cost per video decreased for non-members?
4. What would the graph look like if the price for members was constant regardless of the number of videos?
5. Can you derive a break-even formula for becoming a member based on general prices?

**Tip**: Always consider both the slope and intercept when analyzing cost equations, as they can provide insights into initial fees and per-item charges.

Observing the following table, you can see that the prices of videos vary depending on whether or not you are a member (socio) of the video club. Complete the table and answer the related questions about video rental prices.

Explore how to compare video rental prices for members and non-members of a video club. This problem involves completing a table of costs, plotting graphs, and finding the point where it becomes beneficial to be a member. Suitable for students in Grades 6-8.

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