The table represents a bicycle rental cost in dollars as a function of time in hours.
Bicycle Rental
| Time (hours) | Cost ($) |
|---|---|
| 0 | 0 |
| 2 | 10 |
| 4 | 20 |
| 6 | 30 |
| 8 | 40 |
Which explains whether or not the function represents a direct variation?
A. This function represents a direct variation because passes through the origin and has a constant rate of change of $5 per hour.
B. This function represents a direct variation because it has a positive, constant rate of change of $10 per hour.
C. This function does not represent a direct variation because it does not represent the cost for 1 hour.
D. This function does not represent a direct variation because the function rule for the cost is to add $10, not multiply by a constant.
Answer (2)
Calculate the rate of change between consecutive data points: Δ T im e Δ C os t .
Verify that the rate of change is constant.
Check if the function passes through the origin (0, 0).
Conclude that the function represents a direct variation because it has a constant rate of change of $5 per hour and passes through the origin.
Explanation
Understanding Direct Variation We are given a table representing the cost of bicycle rental as a function of time in hours. We need to determine if this relationship represents a direct variation and choose the correct explanation. A direct variation occurs when the ratio between two variables is constant, and the function passes through the origin (0,0).
Calculating Rate of Change To check if the function represents a direct variation, we need to calculate the rate of change between consecutive data points. The rate of change is calculated as the change in cost divided by the change in time, i.e., Δ T im e Δ C os t .
Verifying Constant Rate of Change Let's calculate the rate of change between the given data points:
Between (0, 0) and (2, 10): 2 − 0 10 − 0 = 2 10 = 5 Between (2, 10) and (4, 20): 4 − 2 20 − 10 = 2 10 = 5 Between (4, 20) and (6, 30): 6 − 4 30 − 20 = 2 10 = 5 Between (6, 30) and (8, 40): 8 − 6 40 − 30 = 2 10 = 5
The rate of change is constant and equal to 5.
Conclusion Since the rate of change is constant and equal to 5, and the function passes through the origin (0, 0), the function represents a direct variation. This means that for every hour, the cost increases by $5.
Final Answer Therefore, the correct explanation is: This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.
Examples
Direct variation is a fundamental concept in many real-world scenarios. For example, the relationship between the number of hours worked and the amount earned, assuming a constant hourly wage, is a direct variation. If you earn $15 per hour, the more hours you work, the more money you earn, and the relationship is linear and proportional. Similarly, the distance traveled by a car moving at a constant speed is directly proportional to the time traveled. Understanding direct variation helps in predicting and analyzing such relationships.
The bicycle rental function does represent a direct variation because it passes through the origin and has a constant rate of change of $5 per hour, suggesting a linear relationship. Therefore, the correct option is A.
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