https://file.aiquickdraw.com/mathbot/file/17270578152163cqz119q.jpg

Let's go over the math problems in the image:

### 1. **Rewrite the explicit formula in function form and identify the y-intercept**:
The given equation is $ a_n = -3 + 5(n-1) $. To convert this to function form:
\[
a_n = -3 + 5(n-1)
\]
First, simplify the equation:
\[
a_n = -3 + 5n - 5 = 5n - 8
\]
Now in function notation, we write it as:
\[
f(n) = 5n - 8
\]
The y-intercept is the constant term when $ n = 0 $, which is $-8$. So, the y-intercept is $ (0, -8) $.

---

### 2. **Determine if the tables represent a linear function**:

#### (a) 
The table is:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & 9 \\
-2 & 2 \\
-3 & -3 \\
-1 & -6 \\
\hline
\end{array}
\]
To determine if it's linear, check the differences between $ y $-values:
- From $ 9 \to 2 $, the change is $ -7 $
- From $ 2 \to -3 $, the change is $ -5 $
- From $ -3 \to -6 $, the change is $ -3 $

Since the differences are not constant, **this is not a linear function**.

#### (b)
The table is:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & 16 \\
-2 & 10 \\
0 & 4 \\
2 & -2 \\
\hline
\end{array}
\]
Check the differences:
- From $ 16 \to 10 $, the change is $ -6 $
- From $ 10 \to 4 $, the change is $ -6 $
- From $ 4 \to -2 $, the change is $ -6 $

Since the differences are constant, this **is a linear function**. The slope $ m = -3 $, and using the point $ (0, 4) $, the equation is:
\[
f(x) = -3x + 4
\]

---

### 3. **Elena's earnings:**
The table shows the amount of money Elena earned for various hours worked:
\[
\begin{array}{|c|c|}
\hline
\text{Time (hours)} & \text{Amount Earned (dollars)} \\
\hline
2.5 & 22.50 \\
3 & 27 \\
3.5 & 31.50 \\
4.5 & 40.50 \\
\hline
\end{array}
\]

#### (a) **Average rate of change**:
The average rate of change is calculated by finding the slope:
\[
\frac{\Delta \text{Amount Earned}}{\Delta \text{Time}} = \frac{40.50 - 22.50}{4.5 - 2.5} = \frac{18}{2} = 9 \text{ dollars per hour}.
\]
So, the rate of change is $ 9 $ dollars per hour.

#### (b) **Write the function**:
Since the rate of change (slope) is constant at $ 9 $, the function can be modeled as:
\[
f(t) = 9t
\]
Where $ t $ is the number of hours worked, and $ f(t) $ represents the amount earned.

#### (c) **Amount earned for 7.5 hours**:
Using the function $ f(t) = 9t $:
\[
f(7.5) = 9 \times 7.5 = 67.50
\]
So, Elena earns **$67.50** for working 7.5 hours.

---

Would you like more details or have any specific questions about these problems?

Here are 5 related questions to think about:
1. How do you identify a linear function from a table of values?
2. What is the significance of the slope in a linear equation?
3. Can you graph the function $ f(x) = -3x + 4 $ and find its intercepts?
4. What happens to the y-intercept if you change the slope of a linear function?
5. How would you explain the concept of average rate of change in real-life terms?

**Tip:** Always check the first differences in a table of values to quickly determine whether the relationship is linear.

Rewrite the explicit formula in function form. Identify the y-intercept. Determine if the tables represent a linear function and calculate the rate of change for the time/money scenario.

Explore how to rewrite an explicit formula in function form, identify the y-intercept, and determine if a table represents a linear function. Includes calculating average rate of change in a real-life context with time and earnings.

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