https://file.aiquickdraw.com/mathbot/file/17270584872110xvhq1fx.jpg

Let's go through the questions in the image and work through each of them:

### 1. **Rewrite each explicit formula in function form:**

   **(a)** \( a_n = 19 - 7(n - 1) \)

   Expand the equation:
   \[
   a_n = 19 - 7n + 7 = 26 - 7n
   \]

   In function form, this becomes:
   \[
   f(n) = 26 - 7n
   \]

   **(b)** \( a_n = 1.5 + 4.2(n - 1) \)

   Expand the equation:
   \[
   a_n = 1.5 + 4.2n - 4.2 = 4.2n - 2.7
   \]

   In function form, this becomes:
   \[
   f(n) = 4.2n - 2.7
   \]

---

### 2. **Tree growth problem:**

   A tree is currently 8 feet tall and grows 3 feet per year.

   **(a)** Model the scenario with an arithmetic sequence:
   \[
   a_n = 8 + 3(n - 1)
   \]
   This simplifies to:
   \[
   a_n = 3n + 5
   \]

   **(b)** Rewrite in function notation:
   \[
   f(n) = 3n + 5
   \]

   **(c)** Find the height in 12 years:
   \[
   f(12) = 3(12) + 5 = 36 + 5 = 41 \, \text{feet}
   \]

---

### 3. **Recall on arithmetic sequences and linear functions:**

   **(a)** Can any arithmetic sequence be modeled with a linear function?
   
   Yes, any arithmetic sequence can be modeled with a linear function because both have constant rates of change (common difference for arithmetic sequences, slope for linear functions). 

   **(b)** Can any linear function be modeled with an arithmetic sequence?

   No, not every linear function can be modeled with an arithmetic sequence because arithmetic sequences only apply to discrete values (integers) of \(n\), whereas linear functions are continuous.

   **(c)** Compare the domain of arithmetic sequences and linear functions:

   - **Arithmetic sequences**: The domain consists of positive integers.
   - **Linear functions**: The domain consists of all real numbers unless otherwise restricted.

---

### 4. **Calculate the average rate of change for \(f(x)\):**

   From the graph, let's pick two points: \((-6, -6)\) and \((6, 6)\).

   The formula for the average rate of change is:
   \[
   \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
   \]

   Using the points:
   \[
   \text{Average rate of change} = \frac{6 - (-6)}{6 - (-6)} = \frac{12}{12} = 1
   \]

The average rate of change is \(1\).

---

Do you have any questions or need more details on any part? 

Here are five related questions for further exploration:

1. How can you derive the explicit formula for any given arithmetic sequence?
2. Can a sequence with a non-constant difference ever be linear?
3. What is the geometric interpretation of an arithmetic sequence?
4. What real-world scenarios can be modeled using arithmetic sequences or linear functions?
5. How does the rate of change affect the steepness of a line in linear functions?

**Tip**: Always double-check your expansion steps when converting from sequence notation to function notation to avoid errors.

Rewrite the explicit formulas in function form, model the tree growth scenario using an arithmetic sequence, and analyze the relationship between arithmetic sequences and linear functions.

Explore how to convert arithmetic sequences to function form and solve real-world problems, such as modeling tree growth. Learn how arithmetic sequences relate to linear functions, with examples and explanations for Grades 8-10.

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