A lawn-mowing company is trying to grow its business. It had 26 clients when they started its business and wants to increase by 3 new clients each week. Use an arithmetic sequence to write a function to represent this real-world situation and determine the range of the function for the first four weeks of data. (1 point)

f(x) = 3x + 26; 0 ≤ y ≤ 4

f(x) = 3x + 23; 26 ≤ y ≤ 35

f(x) = 3x + 26; 26 ≤ y ≤ 35

f(x) = 3x + 23; 0 ≤ y ≤ 4

To solve this problem, we need to model the growth of the lawn-mowing company’s client base as an arithmetic sequence. Let's analyze the information provided and determine the correct function and range.

### Step-by-Step Analysis:

1. **Initial Information:**
   - The company starts with **26 clients**.
   - It adds **3 new clients each week**.

2. **Formulate the Arithmetic Sequence:**
   - In an arithmetic sequence, the number of clients after \( x \) weeks can be expressed using the formula:
   \[
   f(x) = a + d \cdot x
   \]
   where:
   - \( a \) is the initial number of clients,
   - \( d \) is the number of new clients added each week (common difference),
   - \( x \) is the number of weeks.

3. **Apply Given Values:**
   - Initial clients, \( a = 26 \)
   - Clients added per week, \( d = 3 \)

   Plugging in these values into the formula gives:
   \[
   f(x) = 26 + 3x
   \]

4. **Determine the Range for the First Four Weeks:**
   - We need to calculate \( f(x) \) for the first four weeks (\( x = 0, 1, 2, 3, 4 \)).

   Computing the values:
   - For \( x = 0 \): \( f(0) = 26 + 3 \cdot 0 = 26 \)
   - For \( x = 1 \): \( f(1) = 26 + 3 \cdot 1 = 29 \)
   - For \( x = 2 \): \( f(2) = 26 + 3 \cdot 2 = 32 \)
   - For \( x = 3 \): \( f(3) = 26 + 3 \cdot 3 = 35 \)
   - For \( x = 4 \): \( f(4) = 26 + 3 \cdot 4 = 38 \)

   Thus, the range of the function \( f(x) = 26 + 3x \) for the first four weeks (\( x = 0 \) to \( x = 4 \)) is from 26 to 38.

5. **Choose the Correct Option:**
   Comparing with the provided options:

   - The correct function that represents this scenario is \( f(x) = 3x + 26 \).

   - The corresponding range for \( y \) (representing \( f(x) \)) after evaluating for the first four weeks is from 26 to 38. 

   Since the range for \( y \) starts from 26 (when \( x = 0 \)) and goes up to 35 (for \( x = 3 \)), and 38 (for \( x = 4 \)), the correct range for four weeks should cover \( 26 \leq y \leq 38 \).

   **Thus, the closest correct option is:**

   \[
   \boxed{f(x) = 3x + 26; \, 26 \leq y \leq 35} \text{ (range should ideally be \(26 \leq y \leq 38\))}
   \]

However, none of the given options lists the complete range up to 38, but the correct function form is matched by this choice.

### Conclusion:

The correct function is \( f(x) = 3x + 26 \). For the first four weeks, the range for \( y \) is from 26 to 38.

Would you like more details or have any questions?

### Related Questions:
1. What is an arithmetic sequence, and how can it be identified?
2. How do you determine the common difference in an arithmetic sequence?
3. How would the function change if the company added 5 clients per week instead?
4. How can the function \( f(x) = 3x + 26 \) be used to predict the number of clients after 10 weeks?
5. What is the difference between arithmetic and geometric sequences?

### Tip:
When modeling real-world scenarios with arithmetic sequences, always identify the initial value and the consistent change (common difference) to form the correct function.

Explore how to model the growth of a lawn-mowing company's client base using arithmetic sequences. Learn the formula and steps involved in determining the number of clients each week, starting with 26 clients and increasing by 3 clients per week. Suitable for Grades 6-8 students.

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