Certainly! Let's break down the solution step-by-step for Karla's Gamestation monthly plans using the given values for each option.
### 1. Arithmetic Sequence and Linear Function for Each Option
We need to derive both the arithmetic sequence in explicit form and the linear function in slope-intercept form for each subscription option.
#### Option 1
Details:
- Monthly cost: [tex]$18
- No initiation fee
Explicit Form of Arithmetic Sequence:
\[ a_n = 18 \cdot n \]
where \( n \) is the number of months.
Slope-Intercept Form of Linear Function:
\[ y = 18x \]
where \( x \) is the number of months, and \( y \) is the total cost.
#### Option 2
Details:
- Initiation fee: $[/tex]10
- Monthly cost: [tex]$15
Explicit Form of Arithmetic Sequence:
\[ a_n = 10 + 15 \cdot n \]
where \( n \) is the number of months.
Slope-Intercept Form of Linear Function:
\[ y = 15x + 10 \]
where \( x \) is the number of months, and \( y \) is the total cost.
#### Option 3
Details:
- Initiation fee: $[/tex]75
- Monthly cost: [tex]$5
Explicit Form of Arithmetic Sequence:
\[ a_n = 75 + 5 \cdot n \]
where \( n \) is the number of months.
Slope-Intercept Form of Linear Function:
\[ y = 5x + 75 \]
where \( x \) is the number of months, and \( y \) is the total cost.
Here’s the summary in a table:
\begin{tabular}{|c|c|c|}
\hline
Option 1 & Explicit Form & Slope Intercept Form \\
\hline
& $[/tex]a_n = 18 \cdot n[tex]$ & $[/tex]y = 18x[tex]$ \\
\hline
Option 2 & Explicit Form & Slope Intercept Form \\
\hline
& $[/tex]a_n = 10 + 15 \cdot n[tex]$ & $[/tex]y = 15x + 10[tex]$ \\
\hline
Option 3 & Explicit Form & Slope Intercept Form \\
\hline
& $[/tex]a_n = 75 + 5 \cdot n[tex]$ & $[/tex]y = 5x + 75[tex]$ \\
\hline
\end{tabular}
### 2. Similarity between Explicit Form of Arithmetic Sequences and Slope-Intercept Form
The explicit form of arithmetic sequences and the slope-intercept form for linear functions both consist of a starting point (initial value) and a rate of change.
- In the arithmetic sequence \( a_n = a_1 + (n-1)d \):
- \( a_1 \) represents the initial term.
- \( d \) is the common difference between terms.
- In the linear function \( y = mx + b \):
- \( b \) represents the y-intercept (initial value when \( x \) is 0).
- \( m \) is the slope (rate of change).
Both forms identify an initial value and how much the value changes per unit (month).
### 3. Interpretation of \( y \)-Intercept and Slope for Each Option
Option 1:
- Y-intercept (0): No initiation fee.
- Slope (18): The total cost increases by $[/tex]18 per month.
Option 2:
- Y-intercept (10): The [tex]$10 initiation fee.
- Slope (15): The total cost increases by $[/tex]15 per month.
Option 3:
- Y-intercept (75): The [tex]$75 initiation fee.
- Slope (5): The total cost increases by $[/tex]5 per month.
For each option, the y-intercept reflects the initial cost Karla has to pay to start the plan, and the slope indicates the ongoing monthly charge based on the plan's rate.
