the table represents a bicycle rental cost in dollars as a function of time in hours. bicycle rental time…

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.

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.

Answer

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 ( 05=0 ), ( 25=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.