The yearly growth in the number of fast food restaurants by the function f(n)=5+8n. According to this model which statement is true?

Options

A. 8 is the initial number of restaurants, 5 is the number of restaurants added each year after the first year.
B. 5 is the starting number of restaurant: 8 is the number of restaurant added each year after the first year.
C. The result chain opened with 13 restaurant
D. The y-intercept of the graph of function f shows the year in which the retail chain made a zero profit.

Answer

To solve the problem, we need to analyze the function given for the yearly growth in the number of fast food restaurants, which is \( f(n) = 5 + 8n \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(n) \) represents the growth in the number of fast food restaurants after \( n \) years. Here, \( 5 \) is a constant term, and \( 8n \) represents the growth that occurs each year.

2. **Finding the Initial Number of Restaurants**:
To find the initial number of restaurants, we need to evaluate the function at \( n = 0 \):
\[ f(0) = 5 + 8 \times 0 = 5 \]
This means that the initial number of fast food restaurants is **5**.

3. **Understanding the Growth Rate**:
The coefficient of \( n \) in the function, which is \( 8 \), indicates the number of restaurants added each year. Therefore, every year, the number of restaurants increases by **8**.

4. **Verifying the Statements**:
- **Statement A**: "8 is the initial number of restaurants, 5 is the number of restaurants added each year after the first year." This is **false** because we found that the initial number is 5, not 8.
- **Statement B**: "5 is the starting number of restaurants, and 8 is the number of restaurants added each year." This is **true** based on our findings.
- **Statement C**: "The restaurant chain opened with 13 restaurants." This is **false** as we determined the initial number is 5.
- **Statement D**: "The y-intercept of the graph of the function shows the year in which the retail chain made a zero profit." This is **false** because the function does not provide information about profit.

5. **Conclusion**:
The only true statement according to the model is **Statement B**.