Let's analyze the yearly cost equations for both members and nonmembers and compare their graphs.
1. Member's Yearly Cost ([tex]$y$[/tex]) Equation:
[tex]\[
y = \frac{1}{10} x + 60
\][/tex]
This equation represents a linear relationship between the yearly cost [tex]\(y\)[/tex] and the number of tokens [tex]\(x\)[/tex]. There are two important parts to this equation:
- The slope ([tex]\(\frac{1}{10}\)[/tex]): This represents the rate at which the cost increases per token. For each additional token, the cost increases by [tex]$\frac{1}{10}$[/tex] or [tex]$0.10.
- The y-intercept (60): This represents the initial fixed cost for members, regardless of the number of tokens purchased. When \(x = 0\), the cost is $[/tex]60.
2. Nonmember's Yearly Cost ([tex]$y$[/tex]) Equation:
[tex]\[
y = \frac{1}{5} x
\][/tex]
Similarly, this is a linear equation but with a different slope and no fixed cost:
- The slope ([tex]\(\frac{1}{5}\)[/tex]): For nonmembers, the cost increases by [tex]$\frac{1}{5}$[/tex] or [tex]$0.20 for each additional token.
- The y-intercept (0): This shows that there is no initial fixed cost for nonmembers. When \(x = 0\), the cost is $[/tex]0.
Now, let's visually compare these two linear relationships on a graph:
- Y-intercepts:
- Member's graph starts at [tex]$60 when \(x = 0\), hence the y-intercept is 60.
- Nonmember's graph starts at $[/tex]0 when [tex]\(x = 0\)[/tex], hence the y-intercept is 0.
- Slopes:
- The slope of the member's graph is [tex]$\frac{1}{10}$[/tex], meaning the cost increases more slowly as tokens increase.
- The slope of the nonmember’s graph is [tex]$\frac{1}{5}$[/tex], meaning the cost increases faster as tokens increase.
When observing these characteristics on a graph:
- Initial Point Comparison:
- At [tex]\(x = 0\)[/tex], the cost for a member is [tex]$60, whereas the cost for a nonmember is $[/tex]0.
- Growth Rate Comparison:
- As [tex]\(x\)[/tex] increases from 0, the cost for a nonmember increases more quickly because [tex]$\frac{1}{5} x\) grows faster than $[/tex]\frac{1}{10} x\).
Let's summarize differences:
1. At [tex]\(x = 0\)[/tex]:
- Member's cost is [tex]$60.
- Nonmember's cost is $[/tex]0.
2. At [tex]\(x = 100\)[/tex]:
- Member's cost is \( \frac{1}{10} \times 100 + 60 = 10 + 60 = 70.
- Nonmember's cost is \( \frac{1}{5} \times 100 = 20
3. At \(x = 500:
- Member's cost can be extrapolated from the calculations shown, starting from 60 and increasing linearly but slower.
- Nonmember's cost starts from 0 and increases much faster.
From this detailed observation and the numerical data, we can conclude that the primary differences in the graphs are:
1. The member's graph starts higher on the y-axis (fixed cost of $60), whereas the nonmember's graph starts at the origin (no initial cost).
2. The nonmember's graph rises more steeply than the member's graph indicating a faster rate of cost increase per token.
3. These differences are reflected in the intercepts and slopes respectively, showing the difference in incremental costs and initial charges between members and nonmembers.
