The yearly cost in dollars, $y$, at a video game arcade based on total game tokens purchased, $x$, is $y=\frac{1}{10} x+$ 60 for a member and $y=\frac{1}{5} x$ for a nonmember. Explain how the graph of a nonmember's yearly cost will differ from the graph of a member's yearly cost.
Answer (2)
The non-member's cost graph is steeper than the member's because the non-member pays more per token and has no initial fixed cost, starting at zero dollars. In contrast, the member's graph starts at 60 dollars due to a fixed fee and increases at a slower rate. Thus, the member's graph is less steep and has a higher starting point on the y-axis.
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The member's cost is modeled by y = 10 1 x + 60 , while the non-member's cost is y = 5 1 x .
The non-member's cost graph has a steeper slope ( 5 1 ) compared to the member's ( 10 1 ), indicating a faster increase in cost per token.
The member's graph has a y-intercept of 60, while the non-member's has a y-intercept of 0, meaning the member has a fixed cost of $60.
Therefore, the non-member's graph is steeper and starts at a lower y-value than the member's graph, which can be summarized as: the nonmember's graph is steeper and starts at a lower y-value than the member's graph.
Explanation
Analyzing the Cost Functions Let's analyze the cost functions for members and non-members at the video game arcade. We have two linear equations:
Member's cost: y = 10 1 x + 60 Non-member's cost: y = 5 1 x
Here, y represents the yearly cost in dollars, and x represents the total game tokens purchased. Our goal is to explain how the graph of the non-member's cost differs from the member's cost.
Identifying Slopes and Y-intercepts The equations are in the slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. Let's identify the slope and y-intercept for each equation.
For the member's cost equation, y = 10 1 x + 60 :
Slope (m) = 10 1 Y-intercept (b) = 60
For the non-member's cost equation, y = 5 1 x :
Slope (m) = 5 1 Y-intercept (b) = 0
Comparing Slopes Now, let's compare the slopes. The slope represents the rate of change of the yearly cost with respect to the number of tokens purchased. A larger slope means the cost increases more rapidly as the number of tokens increases.
The slope for the non-member's cost ( 5 1 ) is greater than the slope for the member's cost ( 10 1 ). This means the non-member's cost increases faster with each token purchased compared to the member's cost. Therefore, the graph of the non-member's cost will be steeper than the graph of the member's cost.
Comparing Y-intercepts Next, let's compare the y-intercepts. The y-intercept represents the yearly cost when no tokens are purchased ( x = 0 ).
The y-intercept for the member's cost is 60, which means a member pays a fixed cost of $60 even if they don't buy any tokens. The y-intercept for the non-member's cost is 0, which means a non-member pays $0 if they don't buy any tokens.
Therefore, the graph of the member's cost starts at a higher y-value (60) than the graph of the non-member's cost (0).
Conclusion In conclusion, the graph of the non-member's yearly cost differs from the graph of the member's yearly cost in the following ways:
Steeper Slope: The non-member's graph is steeper because the cost increases more rapidly with each token purchased.
Lower Y-intercept: The non-member's graph starts at a lower y-value (0) compared to the member's graph (60).
Examples
Understanding linear cost functions is crucial in many real-life scenarios. For instance, businesses use these functions to model production costs, where 'x' could represent the number of units produced, and 'y' the total cost. Similarly, consumers can use them to compare different service plans, like mobile phone plans with a fixed monthly fee plus a per-use charge. By analyzing the slopes and y-intercepts, one can determine which plan is more economical based on their usage habits. This kind of analysis helps in making informed decisions, whether in business or personal finance.
