Select The Correct Answer.Which Statement Describes The Graph Of The Function H ( X ) = 4 X 2 − 100 8 X − 20 H(x)=\frac{4x^2-100}{8x-20} H ( X ) = 8 X − 20 4 X 2 − 100 ​ ?A. The Graph Has An Oblique Asymptote.B. There Is A Horizontal Asymptote At Y = 1 2 Y=\frac{1}{2} Y = 2 1 ​ .C. There Is A Horizontal Asymptote

Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. The graph of a rational function can have various characteristics, including asymptotes, holes, and intercepts. In this article, we will explore the graph of the function $h(x)=\frac{4x^2-100}{8x-20}$ and determine which statement describes its graph.

Asymptotes

An asymptote is a line that the graph of a function approaches as the input values get arbitrarily large or arbitrarily small. There are two types of asymptotes: vertical and horizontal. A vertical asymptote occurs when the denominator of a rational function is equal to zero, causing the function to approach infinity or negative infinity. A horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator, causing the function to approach a constant value.

Degree of the Numerator and Denominator

To determine the degree of the numerator and denominator, we need to look at the highest power of the variable in each polynomial. In the function $h(x)=\frac{4x^2-100}{8x-20}$, the numerator has a degree of 2, and the denominator has a degree of 1.

Horizontal Asymptote

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. However, we can find the horizontal asymptote by dividing the leading term of the numerator by the leading term of the denominator.

Oblique Asymptote

An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the degree of the numerator is 2, and the degree of the denominator is 1. Therefore, there is an oblique asymptote.

Finding the Oblique Asymptote

To find the oblique asymptote, we need to divide the numerator by the denominator using long division or synthetic division. The result of the division is $h(x)=x+\frac{10}{2x-5}$.

Graph of the Function

The graph of the function $h(x)=\frac{4x^2-100}{8x-20}$ has an oblique asymptote at $y=x$. The graph approaches the oblique asymptote as the input values get arbitrarily large or arbitrarily small.

Conclusion

In conclusion, the graph of the function $h(x)=\frac{4x^2-100}{8x-20}$ has an oblique asymptote at $y=x$. There is no horizontal asymptote, and the graph approaches the oblique asymptote as the input values get arbitrarily large or arbitrarily small.

Answer

The correct answer is:

A. The graph has an oblique asymptote.

Final Thoughts

Introduction

In our previous article, we explored the graph of the function $h(x)=\frac{4x^2-100}{8x-20}$ and determined that it has an oblique asymptote at $y=x$. In this article, we will answer some frequently asked questions about the graph of a rational function.

Q: What is a rational function?

A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that has a numerator and a denominator, and the denominator is not equal to zero.

Q: What is an asymptote?

An asymptote is a line that the graph of a function approaches as the input values get arbitrarily large or arbitrarily small. There are two types of asymptotes: vertical and horizontal.

Q: What is a vertical asymptote?

A vertical asymptote occurs when the denominator of a rational function is equal to zero, causing the function to approach infinity or negative infinity.

Q: What is a horizontal asymptote?

A horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator, causing the function to approach a constant value.

Q: What is an oblique asymptote?

An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator, causing the function to approach a linear value.

Q: How do I find the oblique asymptote?

To find the oblique asymptote, you need to divide the numerator by the denominator using long division or synthetic division.

Q: What is the difference between a rational function and a polynomial function?

A rational function is a function that has a numerator and a denominator, while a polynomial function is a function that has only a numerator.

Q: Can a rational function have a horizontal asymptote?

Yes, a rational function can have a horizontal asymptote if the degree of the numerator is equal to the degree of the denominator.

Q: Can a rational function have an oblique asymptote?

Yes, a rational function can have an oblique asymptote if the degree of the numerator is exactly one more than the degree of the denominator.

Q: How do I determine the type of asymptote a rational function has?

To determine the type of asymptote a rational function has, you need to compare the degree of the numerator and the degree of the denominator.

Q: What is the significance of the oblique asymptote?

The oblique asymptote is significant because it tells us the behavior of the function as the input values get arbitrarily large or arbitrarily small.

Conclusion

In conclusion, understanding the graph of a rational function requires knowledge of asymptotes, holes, and intercepts. By analyzing the degree of the numerator and denominator, we can determine the type of asymptote and the behavior of the graph. We hope this Q&A article has helped you understand the graph of a rational function better.

Final Thoughts

If you have any more questions about the graph of a rational function, feel free to ask. We are here to help you understand this important topic in mathematics.