When the two equations are graphed on the coordinate plane, how are the graphs similar?
Understand the Problem
The question is asking how the graphs of two equations related to cable service costs for existing and new customers are similar when graphed on the coordinate plane. The existing customer pays a linear rate, while the new customer has an initial fee plus a linear rate.
Answer
The graphs are similar because they both have the same slope of 70, indicating the same rate of increase in cost over time, despite different starting points.
Answer for screen readers
The graphs of the two equations are similar because they have the same slope of 70, meaning both increase at the same rate. However, the graph for new customers starts higher on the y-axis due to an additional initial fee of $100.
Steps to Solve
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Identify the Equations
The equations given are:
For existing customers:
$$ y = 70x $$
For new customers:
$$ y = 70x + 100 $$ -
Determine the Slope
Both equations have the same slope of 70. This means that for every additional month, the cost increases by $70 for both existing and new customers. -
Identify the Y-Intercepts
The intercept for existing customers is at $(0, 0)$, indicating no charge before any months have passed. For new customers, the y-intercept is at $(0, 100)$, indicating a one-time additional fee of $100. -
Graph the Equations
When graphed, both lines will be parallel since they share the same slope. The line for new customers will be shifted upwards by 100 units due to the additional initial fee. -
Conclusion about Similarity
Despite the difference in starting points (the y-intercepts), both lines maintain the same rate of increase; thus, they are similar in terms of their linear relationship to monthly cost.
The graphs of the two equations are similar because they have the same slope of 70, meaning both increase at the same rate. However, the graph for new customers starts higher on the y-axis due to an additional initial fee of $100.
More Information
The existing customer's cost function goes directly from $0 as time progresses, while the new customer's cost starts at $100. The similarity lies in their linearity and consistent rate of change.
Tips
- Neglecting to identify the slope and intercept separately can lead to misunderstandings about how the graphs relate.
- Assuming the costs are the same without recognizing the initial fee for new customers.
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