Question

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? this function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour. this function represents a direct variation because it has a positive, constant rate of change of $10 per hour. this function does not represent a direct variation because it does not represent the cost for 1 hour. this function does not represent a direct variation because the function rule for the cost is to add $10, not multiply by a constant.

Explanation:

Step1: Recall Direct Variation Definition

Direct variation is a relationship \( y = kx \), where \( k \) is constant, and the graph passes through the origin \((0,0)\). The rate of change (slope) \( k=\frac{y}{x}\) must be constant.

Step2: Check the Table Data

• At \( x = 0 \), \( y = 0 \) (passes through origin).
• Calculate the rate of change between points:
• From \( (0,0) \) to \( (2,10) \): \( k=\frac{10 - 0}{2 - 0}=5 \).
• From \( (2,10) \) to \( (4,20) \): \( k=\frac{20 - 10}{4 - 2}=5 \).
• From \( (4,20) \) to \( (6,30) \): \( k=\frac{30 - 20}{6 - 4}=5 \).
• From \( (6,30) \) to \( (8,40) \): \( k=\frac{40 - 30}{8 - 6}=5 \).

Step3: Analyze Each Option

• Option 1: States it passes through origin and rate of change is $5 per hour. This matches our calculation (direct variation \( y = 5x \), since \( 0*5=0 \), \( 2*5=10 \), etc.).
• Option 2: Rate of change is claimed as $10, but we calculated $5. Incorrect.
• Option 3: Direct variation doesn’t require representing 1 hour specifically; the constant rate (and origin) is enough. Incorrect.
• Option 4: The function rule is multiplying by 5 (constant), not just adding. Incorrect.

Answer:

This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.