Question

the function $y = \frac{16,500 + 1020x}{x}$ models $y$, the average annual cost in dollars of owning a car for $x$ years. which statement best explains why the graph of $y = \frac{16,500 + 1020x}{x}$ should have a vertical asymptote at $x = 0$?

the time the car is owned is greater than 0.

the value of the car decreases over time.

the initial cost of the car is greater than $0.

the value of the car will never dip below $0.

Explanation:

Step1: Analyze the function and vertical asymptote

The function is \( y = \frac{16500 + 1020x}{x} \). A vertical asymptote at \( x = 0 \) occurs when the denominator is zero (and the numerator is not zero at that point). In the context of owning a car, \( x \) represents the number of years the car is owned.

Step2: Evaluate each option

• Option 1: "The time the car is owned is greater than 0." If \( x>0 \), it means we can't own a car for 0 years (since you can't have a time of ownership of 0 and still calculate an average annual cost for that non - existent time), which relates to why \( x = 0 \) is a vertical asymptote (we can't use \( x = 0 \) in the function as it would be undefined in the context of calculating the average cost for owning a car for \( x \) years).
• Option 2: "The value of the car decreases over time." This is about the depreciation of the car's value, which has no direct relation to the vertical asymptote at \( x = 0 \) of the cost function.
• Option 3: "The initial cost of the car is greater than $0." The initial cost being non - zero doesn't explain the vertical asymptote at \( x = 0 \) directly. The vertical asymptote is about the domain of \( x \) (time) in the context of the function, not just the initial cost.
• Option 4: "The value of the car will never dip below $0." This is about the minimum value of the car, not related to the vertical asymptote at \( x = 0 \) of the cost function.

Answer:

The time the car is owned is greater than 0.