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.

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loyamanny96 loyamanny96

05-11-2026
Mathematics

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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.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T13:02:28-05:00

Identify the cost functions: member cost $y = \frac{1}{10}x + 60$ and nonmember cost $y = \frac{1}{5}x$.
Compare slopes: the nonmember's slope ($0.2$) is steeper than the member's ($0.1$), indicating a higher variable cost per token.
Find the intersection: both lines cross at $(600, 120)$, representing the break-even point where costs are equal.
Visual description: The nonmember's graph starts at the origin $(0,0)$ and is steeper, while the member's starts at $(0,60)$, meaning nonmembers pay less for fewer than 600 tokens and more for more than 600 tokens.
Final answer: The nonmember's graph is steeper and starts at the origin, intersecting the member's graph at $\boxed{(600, 120)}$.

Explanation

Analyzing the cost functions

The arcade costs are given by two linear functions representing yearly expenses in terms of the number of tokens $x$ purchased:

Member's cost function: $y_m = \frac{1}{10}x + 60$
Nonmember's cost function: $y_n = \frac{1}{5}x$

The graph of each function is a line with a slope and a y-intercept. For the member, the slope is $m_m = \frac{1}{10} = 0.1$ and the y-intercept is $b_m = 60$. For the nonmember, the slope is $m_n = \frac{1}{5} = 0.2$ and the y-intercept is $b_n = 0$.

Comparing the graphs and finding the intersection

To compare the graphs, we look at their slopes and intercepts:

The nonmember's slope ($0.2$) is greater than the member's slope ($0.1$), which means the nonmember's graph is steeper, reflecting a higher cost per token.
The nonmember's y-intercept ($0$) is lower than the member's y-intercept ($60$), meaning that for a small number of tokens, the nonmember pays less initially.
The two lines intersect where $y_m = y_n$. Setting $\frac{1}{10}x + 60 = \frac{1}{5}x$, we solve for $x$:

$\frac{1}{5}x - \frac{1}{10}x = 60$
$\frac{1}{10}x = 60$
$x = 600$
At $x = 600$, both costs are $y = \frac{1}{5}(600) = 120$. This point is the break-even point.

Describing the visual difference

The graph of the nonmember's cost starts at the origin $(0,0)$, whereas the member's graph starts at $(0,60)$ on the y-axis. Because the nonmember's graph has a steeper slope, it starts lower but rises more quickly, intersecting the member's line at $(600, 120)$. For any number of tokens $x < 600$, the nonmember's cost is lower; for $x > 600$, the nonmember's cost is higher than the member's.

Examples

In everyday life, this comparison is common when choosing between subscription-based services and pay-as-you-go models. For example, a gym might offer a membership with a monthly fee but lower usage rates, compared to a pay-per-visit rate. By graphing both cost functions, you can easily determine at what level of usage (the "break-even point") it becomes more economical to commit to a membership. This approach helps in making informed financial decisions regarding recurring service costs.