The function $y=\frac{16,500+1020 x}{x}$ models $y$, the average annual cost in dollars of owning a car for $x$ years. Which statement best explains why the graph of $y=\frac{16,500+1020 x}{x}$ should have a vertical asymptote at $x=0$?
A. The value of the car decreases over time.
B. The value of the car will never dip below $0.
C. The time the car is owned is greater than 0.
D. The initial cost of the car is greater than $0.
Answer (2)
The function y = x 16 , 500 + 1020 x models the average annual cost of owning a car for x years.
A vertical asymptote occurs when the denominator of a rational function approaches zero.
In this case, the denominator is x , so as x approaches 0, there is a vertical asymptote.
Since x represents the number of years, it must be greater than 0, which explains the vertical asymptote at x = 0 .
The correct answer is: The time the car is owned is greater than 0.
Explanation
Understanding the Problem We are given the function y = x 16 , 500 + 1020 x which models the average annual cost y of owning a car for x years. We need to determine why the graph of this function has a vertical asymptote at x = 0 . A vertical asymptote occurs when the denominator of a rational function approaches zero. In this case, the denominator is x .
Analyzing the Asymptote As x approaches 0, the denominator x approaches 0. This causes the function value to approach infinity, indicating a vertical asymptote at x = 0 . Now, let's consider the physical meaning of x . Since x represents the number of years, x must be greater than 0. Therefore, the time the car is owned is greater than 0 explains why the function has a vertical asymptote at x = 0 .
Conclusion The statement 'The time the car is owned is greater than 0' best explains why the graph of y = x 16 , 500 + 1020 x should have a vertical asymptote at x = 0 . This is because you cannot own a car for a negative or zero amount of time.
Examples
Understanding asymptotes is crucial in various real-world scenarios. For instance, in economics, cost functions often have asymptotes that represent limits to production or growth. Similarly, in physics, certain force fields can be modeled with functions that have asymptotes, indicating points where the force becomes infinitely strong. By analyzing these asymptotes, we can make informed decisions and predictions about the behavior of these systems.
The vertical asymptote at x = 0 in the function y = x 16 , 500 + 1020 x occurs because the denominator x approaches zero, making the function undefined. Since x represents the time the car is owned, it must always be greater than 0. Therefore, the statement 'The time the car is owned is greater than 0' is the best explanation for this asymptote.
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