Asymptotes of a Rational Function

Oct 13, 2025 by talynt.org 105 views

Introduction to Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this case, we have a rational function of the form $f(x)=\frac{x^2+2x-48}{g(x)}$ , where $g(x)$ is a polynomial. The graph of a rational function can be quite complex, but by understanding the properties of the function, we can determine its possible graph.

Asymptotes of a Rational Function

One of the key features of a rational function is its asymptotes. An asymptote is a line that the graph of the function approaches as the input (or x-value) gets arbitrarily large. In the case of a rational function, there are two types of asymptotes: vertical and horizontal.

Vertical Asymptotes

A vertical asymptote occurs when the denominator of the rational function is equal to zero. In this case, the denominator is $g(x)$ , so we need to find the values of $x$ that make $g(x)$ equal to zero. To do this, we can factor the numerator of the rational function, which is $x^2+2x-48$ . Factoring this expression, we get $(x+8)(x-6)$ . Therefore, the values of $x$ that make $g(x)$ equal to zero are $x=-8$ and $x=6$ .

Horizontal Asymptotes

A horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 2, and the degree of the denominator is also 2. Therefore, the horizontal asymptote is the line $y=1$ .

Possible Graphs of the Rational Function

Now that we have determined the asymptotes of the rational function, we can consider the possible graphs of the function. The graph of a rational function can be a combination of the following:

A vertical asymptote at $x=-8$
A vertical asymptote at $x=6$
A horizontal asymptote at $y=1$
A hole at $x=-8$ or $x=6$

Based on these features, we can consider the following possible graphs of the rational function:

Graph A

This graph has a vertical asymptote at $x=-8$ and a vertical asymptote at $x=6$ . It also has a horizontal asymptote at $y=1$ . There is no hole at either $x=-8$ or $x=6$ .

Graph B

This graph has a vertical asymptote at $x=-8$ and a hole at $x=6$ . It also has a horizontal asymptote at $y=1$ .

Graph C

This graph has a vertical asymptote at $x=6$ and a hole at $x=-8$ . It also has a horizontal asymptote at $y=1$ .

Graph D

This graph has a vertical asymptote at $x=-8$ and a vertical asymptote at $x=6$ . It also has a horizontal asymptote at $y=1$ and a hole at both $x=-8$ and $x=6$ .

Conclusion

In conclusion, the possible graphs of the rational function $f(x)=\frac{x^2+2x-48}{g(x)}$ are Graph A, Graph B, Graph C, and Graph D. Each of these graphs has a combination of vertical and horizontal asymptotes, as well as holes at certain points. By understanding the properties of the rational function, we can determine its possible graph.

References

[1] "Rational Functions" by Math Open Reference
[2] "Asymptotes of Rational Functions" by Purplemath
[3] "Graphing Rational Functions" by Mathway

Discussion

What are some other properties of rational functions that we can consider when graphing them? How can we use these properties to determine the possible graph of a rational function?

Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this article, we will answer some frequently asked questions about rational functions, including their properties, asymptotes, and graphing.

Q: What is a rational function?

A: A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function of the form $f(x)=\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Q: What are the properties of a rational function?

A: The properties of a rational function include:

Domain: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero.
Asymptotes: A rational function can have vertical and horizontal asymptotes.
Holes: A rational function can have holes at certain points.
Degree: The degree of a rational function is the highest degree of the numerator and denominator.

Q: What are vertical asymptotes?

A: Vertical asymptotes are lines that the graph of a rational function approaches as the input (or x-value) gets arbitrarily large. They occur when the denominator of the rational function is equal to zero.

Q: What are horizontal asymptotes?

A: Horizontal asymptotes are lines that the graph of a rational function approaches as the input (or x-value) gets arbitrarily large. They occur when the degree of the numerator is less than or equal to the degree of the denominator.

Q: How do I graph a rational function?

A: To graph a rational function, you need to:

Find the vertical asymptotes: Find the values of x that make the denominator equal to zero.
Find the horizontal asymptote: Find the line that the graph approaches as the input (or x-value) gets arbitrarily large.
Plot the graph: Plot the graph of the rational function using the asymptotes and any holes.

Q: What is a hole in a rational function?

A: A hole in a rational function is a point where the graph of the function is not defined. It occurs when the numerator and denominator of the rational function are both equal to zero.

Q: How do I find the degree of a rational function?

A: To find the degree of a rational function, you need to:

Find the highest degree of the numerator: Find the highest degree of the numerator of the rational function.
Find the highest degree of the denominator: Find the highest degree of the denominator of the rational function.
Compare the degrees: Compare the degrees of the numerator and denominator to determine the degree of the rational function.

Q: What are some common mistakes to avoid when graphing rational functions?

A: Some common mistakes to avoid when graphing rational functions include:

Not finding the vertical asymptotes: Failing to find the vertical asymptotes can result in an incorrect graph.
Not finding the horizontal asymptote: Failing to find the horizontal asymptote can result in an incorrect graph.
Not plotting the graph correctly: Failing to plot the graph correctly can result in an incorrect graph.

Conclusion

In conclusion, rational functions are a type of function that can be expressed as the ratio of two polynomials. They have several properties, including domain, asymptotes, holes, and degree. By understanding these properties, you can graph a rational function correctly and avoid common mistakes.

References

[1] "Rational Functions" by Math Open Reference
[2] "Asymptotes of Rational Functions" by Purplemath
[3] "Graphing Rational Functions" by Mathway

Discussion

What are some other properties of rational functions that we can consider when graphing them? How can we use these properties to determine the possible graph of a rational function?