Interpretation of the Horizontal Asymptote

Oct 13, 2025 by miodeya.com 14 views

Understanding the Function

The given function $y=\frac{25+8x}{x}$ represents the average cost per person, including transportation, to a museum. This function is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a linear polynomial $25+8x$ , and the denominator is a linear polynomial $x$ .

Graphing the Function

To graph the function, we need to understand its behavior as $x$ approaches positive infinity and negative infinity. As $x$ approaches positive infinity, the numerator $25+8x$ also approaches positive infinity, and the denominator $x$ approaches positive infinity. This means that the ratio $\frac{25+8x}{x}$ approaches a finite limit as $x$ approaches positive infinity.

Horizontal Asymptote

The horizontal asymptote of a rational function is the horizontal line that the function approaches as $x$ approaches positive infinity or negative infinity. In this case, the horizontal asymptote is $y=0$ . This means that as $x$ approaches positive infinity, the function approaches the horizontal line $y=0$ .

Interpretation of the Horizontal Asymptote

The horizontal asymptote $y=0$ means that as the number of people increases, the average cost per person approaches zero. This makes sense, because as the number of people increases, the fixed cost of transportation and other expenses becomes less significant, and the average cost per person approaches the cost of the museum visit itself.

Vertical Asymptote

A vertical asymptote is a vertical line that the function approaches as $x$ approaches a certain value. In this case, the vertical asymptote is $x=0$ . This means that as $x$ approaches zero, the function approaches positive infinity.

Interpretation of the Vertical Asymptote

The vertical asymptote $x=0$ means that as the number of people approaches zero, the average cost per person approaches positive infinity. This makes sense, because as the number of people approaches zero, the fixed cost of transportation and other expenses becomes more significant, and the average cost per person approaches positive infinity.

Conclusion

In conclusion, the graph of the function $y=\frac{25+8x}{x}$ has a horizontal asymptote $y=0$ and a vertical asymptote $x=0$ . The horizontal asymptote means that as the number of people increases, the average cost per person approaches zero. The vertical asymptote means that as the number of people approaches zero, the average cost per person approaches positive infinity.

Discussion

What is the significance of the horizontal asymptote in this context?
How does the vertical asymptote relate to the average cost per person?
What does the graph of the function tell us about the relationship between the number of people and the average cost per person?

Key Takeaways

The horizontal asymptote $y=0$ means that as the number of people increases, the average cost per person approaches zero.
The vertical asymptote $x=0$ means that as the number of people approaches zero, the average cost per person approaches positive infinity.
The graph of the function provides insight into the relationship between the number of people and the average cost per person.

Further Exploration

What happens to the average cost per person as the number of people increases or decreases?
How does the fixed cost of transportation and other expenses affect the average cost per person?
What are the implications of the horizontal and vertical asymptotes for the average cost per person?
The Function Representing Average Cost per Person to a Museum: Q&A ================================================================

Q: What is the significance of the horizontal asymptote in this context?

A: The horizontal asymptote $y=0$ means that as the number of people increases, the average cost per person approaches zero. This makes sense, because as the number of people increases, the fixed cost of transportation and other expenses becomes less significant, and the average cost per person approaches the cost of the museum visit itself.

Q: How does the vertical asymptote relate to the average cost per person?

A: The vertical asymptote $x=0$ means that as the number of people approaches zero, the average cost per person approaches positive infinity. This makes sense, because as the number of people approaches zero, the fixed cost of transportation and other expenses becomes more significant, and the average cost per person approaches positive infinity.

Q: What does the graph of the function tell us about the relationship between the number of people and the average cost per person?

A: The graph of the function provides insight into the relationship between the number of people and the average cost per person. As the number of people increases, the average cost per person approaches zero, and as the number of people approaches zero, the average cost per person approaches positive infinity.

Q: What happens to the average cost per person as the number of people increases or decreases?

A: As the number of people increases, the average cost per person approaches zero. As the number of people decreases, the average cost per person approaches positive infinity.

Q: How does the fixed cost of transportation and other expenses affect the average cost per person?

A: The fixed cost of transportation and other expenses affects the average cost per person by making it more significant as the number of people approaches zero. As the number of people increases, the fixed cost becomes less significant, and the average cost per person approaches the cost of the museum visit itself.

Q: What are the implications of the horizontal and vertical asymptotes for the average cost per person?

A: The horizontal and vertical asymptotes have significant implications for the average cost per person. The horizontal asymptote $y=0$ means that as the number of people increases, the average cost per person approaches zero, and the vertical asymptote $x=0$ means that as the number of people approaches zero, the average cost per person approaches positive infinity.

Common Misconceptions

Some people may think that the average cost per person will always increase as the number of people increases.
Others may think that the average cost per person will always decrease as the number of people increases.
However, the graph of the function shows that the average cost per person approaches zero as the number of people increases.

Real-World Applications

The function $y=\frac{25+8x}{x}$ can be used to model the average cost per person for a museum visit.
The horizontal and vertical asymptotes can be used to make predictions about the average cost per person based on the number of people.
This function can be used in a variety of real-world applications, such as budgeting for a museum visit or predicting the cost of a large group of people.

Conclusion

In conclusion, the function $y=\frac{25+8x}{x}$ represents the average cost per person for a museum visit. The horizontal asymptote $y=0$ means that as the number of people increases, the average cost per person approaches zero, and the vertical asymptote $x=0$ means that as the number of people approaches zero, the average cost per person approaches positive infinity. This function can be used in a variety of real-world applications, such as budgeting for a museum visit or predicting the cost of a large group of people.