https://file.aiquickdraw.com/mathbot/file/1726093786900mq21a13k.jpg

Let's solve the math problems shown in the image step by step.

### Problem a:
**Write a function relating the value of the gift card, $ V $, to the number of visits, $ n $:**
- Andrea has a $40 gift card.
- Each swim visit costs $8.

The value of the gift card decreases by $8 for each visit. So, the function can be written as:

\[
V(n) = 40 - 8n
\]

Where:
- $ V(n) $ represents the value left on the gift card after $ n $ visits.
- $ 40 $ is the initial value of the gift card.
- $ 8n $ is the total cost after $ n $ visits, where each visit costs $8.

### Problem b:
**What is the appropriate domain and range of the function? Explain:**

- **Domain**: 
  The number of visits $ n $ must be non-negative and limited by how many visits are possible before the gift card runs out. To find the maximum number of visits, solve for $ n $ when $ V(n) = 0 $:

\[
40 - 8n = 0 \\
n = \frac{40}{8} = 5
\]

So, $ n $ can be between 0 and 5 (inclusive). The domain is $ n \in [0, 5] $.

- **Range**:
  The value of the gift card decreases from $40 to $0 as the number of visits increases from 0 to 5. Therefore, the range is $ V \in [0, 40] $.

### Problem c:
**How much is left on the gift card after 3 visits to the pool?**

To find out how much is left after 3 visits, substitute $ n = 3 $ into the function:

\[
V(3) = 40 - 8(3) = 40 - 24 = 16
\]

So, after 3 visits, there is $16 left on the gift card.

### Problem d:
**Make a table and graph the function.**

Here’s a table showing the number of visits $ n $ and the value left on the card $ V(n) $:

| $ n $ (Number of Visits) | $ V(n) $ (Value Left) |
|----------------------------|-------------------------|
| 0                          | 40                      |
| 1                          | 32                      |
| 2                          | 24                      |
| 3                          | 16                      |
| 4                          | 8                       |
| 5                          | 0                       |

To graph this, plot $ n $ on the horizontal axis and $ V(n) $ on the vertical axis. The graph will be a straight line starting at $ (0, 40) $ and decreasing to $ (5, 0) $.

---

Would you like help graphing the function or more details on any of the steps? Here are some related questions to further explore this topic:

1. How would the function change if the swim visits cost $10 instead of $8?
2. What would the function look like if the initial gift card amount were $50?
3. Can you express this situation using an inequality to represent how many visits can be made without exceeding the gift card value?
4. How would the domain and range change if the pool also had a one-time entrance fee of $5 in addition to the $8 per visit?
5. What happens to the graph if each visit cost $5 instead?

**Tip**: Always check if the function makes sense in the context of real-life situations, especially for the domain and range!

Andrea receives a $40 gift card to use at a town pool. It costs her $8 per visit to swim.

Write a function relating the value of the gift card, V, to the number of visits, n.
What is the appropriate domain and range of the function? Explain.
How much is left on the gift card after 3 visits to the pool?
Make a table and graph the function.

Learn how to write a linear function to represent the value of a $40 gift card after visits to a town pool that cost $8 per visit. Understand domain, range, and how to calculate the remaining value after 3 visits. Suitable for Grades 8-10.

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