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Certainly! Let's analyze and compare the two given cost functions for members and non-members at the video game arcade.
### Member's Cost Function
For members, the yearly cost [tex]\( y \)[/tex] is given by the equation:
[tex]\[ y = \frac{1}{10} x + 60 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of game tokens purchased.
- Slope (Rate of Change): The slope of this linear function is [tex]\(\frac{1}{10}\)[/tex], which means that for every additional game token purchased, the cost increases by [tex]\( \frac{1}{10} \)[/tex] dollars.
- Intercept (Fixed Cost): The y-intercept of this equation is 60, which means that there is a fixed yearly cost of [tex]$60 regardless of the number of tokens purchased. ### Non-Member's Cost Function For non-members, the yearly cost \( y \) is given by the equation: \[ y = \frac{1}{5} x \] - Slope (Rate of Change): The slope here is \(\frac{1}{5}\), meaning that for every additional game token purchased, the cost increases by \( \frac{1}{5} \) dollars. - Intercept (Fixed Cost): The y-intercept of this equation is 0, meaning there is no fixed yearly cost; the cost entirely depends on the number of tokens purchased. ### Comparison of the Graphs #### Slope: - The slope of the non-member's cost function (\(\frac{1}{5}\)) is greater than that of the member's cost function (\(\frac{1}{10}\)). This means the non-member's cost increases more steeply with each additional token purchased compared to the member's cost. #### Intercept: - The intercept for the member's cost function is 60, meaning members start with a higher base cost ($[/tex]60) even if they purchase zero tokens.
- The intercept for the non-member's cost function is 0, meaning non-members do not have an initial cost if they do not purchase any tokens.
### Graphical Representation:
1. Member's Graph:
- Starts at the point (0, 60) on the y-axis because the initial cost is $60.
- Rises slowly with a slope of [tex]\(\frac{1}{10}\)[/tex].
2. Non-Member's Graph:
- Starts at the origin (0, 0) because there’s no initial cost when no tokens are purchased.
- Rises more steeply with a slope of [tex]\(\frac{1}{5}\)[/tex].
### Conclusion:
The non-member's yearly cost graph will start at the origin and rise more steeply compared to the member's cost graph, which starts at a higher point (60 on the y-axis) but rises more slowly. This difference reflects how the costs accumulate differently for members and non-members based on the number of game tokens purchased.
Thus, the key differences are:
- The initial cost (intercept) is higher for members.
- The rate at which costs increase per token (slope) is higher for non-members.
### Member's Cost Function
For members, the yearly cost [tex]\( y \)[/tex] is given by the equation:
[tex]\[ y = \frac{1}{10} x + 60 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of game tokens purchased.
- Slope (Rate of Change): The slope of this linear function is [tex]\(\frac{1}{10}\)[/tex], which means that for every additional game token purchased, the cost increases by [tex]\( \frac{1}{10} \)[/tex] dollars.
- Intercept (Fixed Cost): The y-intercept of this equation is 60, which means that there is a fixed yearly cost of [tex]$60 regardless of the number of tokens purchased. ### Non-Member's Cost Function For non-members, the yearly cost \( y \) is given by the equation: \[ y = \frac{1}{5} x \] - Slope (Rate of Change): The slope here is \(\frac{1}{5}\), meaning that for every additional game token purchased, the cost increases by \( \frac{1}{5} \) dollars. - Intercept (Fixed Cost): The y-intercept of this equation is 0, meaning there is no fixed yearly cost; the cost entirely depends on the number of tokens purchased. ### Comparison of the Graphs #### Slope: - The slope of the non-member's cost function (\(\frac{1}{5}\)) is greater than that of the member's cost function (\(\frac{1}{10}\)). This means the non-member's cost increases more steeply with each additional token purchased compared to the member's cost. #### Intercept: - The intercept for the member's cost function is 60, meaning members start with a higher base cost ($[/tex]60) even if they purchase zero tokens.
- The intercept for the non-member's cost function is 0, meaning non-members do not have an initial cost if they do not purchase any tokens.
### Graphical Representation:
1. Member's Graph:
- Starts at the point (0, 60) on the y-axis because the initial cost is $60.
- Rises slowly with a slope of [tex]\(\frac{1}{10}\)[/tex].
2. Non-Member's Graph:
- Starts at the origin (0, 0) because there’s no initial cost when no tokens are purchased.
- Rises more steeply with a slope of [tex]\(\frac{1}{5}\)[/tex].
### Conclusion:
The non-member's yearly cost graph will start at the origin and rise more steeply compared to the member's cost graph, which starts at a higher point (60 on the y-axis) but rises more slowly. This difference reflects how the costs accumulate differently for members and non-members based on the number of game tokens purchased.
Thus, the key differences are:
- The initial cost (intercept) is higher for members.
- The rate at which costs increase per token (slope) is higher for non-members.
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