What Are The Equations Of The Asymptotes Of The Graph Of $f(x)=\frac{3}{x-2}+9$?Select Each Correct Answer.A. $x=2$ B. $ X = − 2 X=-2 X = − 2 [/tex] C. $y=9$ D. $y=-9$

Mar 10, 2025 by ADMIN 179 views

$What are the Equations of the Asymptotes of the Graph of $f(x)=\frac{3}{x-2}+9$?$

Introduction

When dealing with rational functions, it's essential to understand the behavior of the graph as x approaches certain values. This is where asymptotes come into play. Asymptotes are lines or curves that the graph approaches as x or y gets arbitrarily large or small. In this article, we will explore the equations of the asymptotes of the graph of $f(x)=\frac{3}{x-2}+9$.

Vertical Asymptotes

Vertical asymptotes occur when the denominator of a rational function is equal to zero. In the given function $f(x)=\frac{3}{x-2}+9$, the denominator is x-2. To find the vertical asymptote, we set the denominator equal to zero and solve for x.

x - 2 = 0
x = 2

So, the vertical asymptote is x = 2.

Horizontal Asymptotes

Horizontal asymptotes occur when the degree of the numerator is less than the degree of the denominator. In the given function $f(x)=\frac{3}{x-2}+9$, the degree of the numerator is 0 (since it's a constant), and the degree of the denominator is 1. Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote.

To find the horizontal asymptote, we look at the ratio of the leading coefficients of the numerator and denominator. In this case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. The ratio is 3/1 = 3.

However, we also need to consider the constant term in the function. The constant term is 9, which is added to the fraction. This means that the horizontal asymptote is actually y = 9, not y = 3.

Slant Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In the given function $f(x)=\frac{3}{x-2}+9$, the degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the numerator is not exactly one more than the degree of the denominator, there is no slant asymptote.

Conclusion

In conclusion, the equations of the asymptotes of the graph of $f(x)=\frac{3}{x-2}+9$ are:

Vertical asymptote: x = 2
Horizontal asymptote: y = 9

The correct answers are:

A. x = 2 C. y = 9

The other options are incorrect.

Discussion

This problem is a great example of how to find the equations of the asymptotes of a rational function. By understanding the behavior of the graph as x approaches certain values, we can determine the equations of the asymptotes. In this case, we found that the vertical asymptote is x = 2, and the horizontal asymptote is y = 9.

Additional Examples

Here are a few additional examples of rational functions and their asymptotes:

$f(x)=\frac{2x}{x-1}$: vertical asymptote x = 1, horizontal asymptote y = 2$
$f(x)=\frac{x^2}{x-1}$: vertical asymptote x = 1, slant asymptote y = x + 1$
$f(x)=\frac{3x^2}{x^2-4}$: vertical asymptotes x = 2 and x = -2, horizontal asymptote y = 3$

These examples illustrate how to find the equations of the asymptotes of a rational function.

Final Thoughts

In conclusion, finding the equations of the asymptotes of a rational function is an essential skill in mathematics. By understanding the behavior of the graph as x approaches certain values, we can determine the equations of the asymptotes. In this article, we explored the equations of the asymptotes of the graph of $f(x)=\frac{3}{x-2}+9$ and found that the vertical asymptote is x = 2, and the horizontal asymptote is y = 9.

Introduction

In our previous article, we explored the equations of the asymptotes of the graph of $f(x)=\frac{3}{x-2}+9$. Asymptotes are lines or curves that the graph approaches as x or y gets arbitrarily large or small. In this article, we will answer some frequently asked questions about asymptotes of rational functions.

Q1: What is a vertical asymptote?

A1: A vertical asymptote is a line that the graph of a rational function approaches as x gets arbitrarily close to a certain value. It occurs when the denominator of the rational function is equal to zero.

Q2: How do I find the vertical asymptote of a rational function?

A2: To find the vertical asymptote of a rational function, set the denominator equal to zero and solve for x.

Q3: What is a horizontal asymptote?

A3: A horizontal asymptote is a line that the graph of a rational function approaches as x gets arbitrarily large or small. It occurs when the degree of the numerator is less than the degree of the denominator.

Q4: How do I find the horizontal asymptote of a rational function?

A4: To find the horizontal asymptote of a rational function, look at the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.

Q5: What is a slant asymptote?

A5: A slant asymptote is a line that the graph of a rational function approaches as x gets arbitrarily large or small. It occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Q6: How do I find the slant asymptote of a rational function?

A6: To find the slant asymptote of a rational function, divide the numerator by the denominator using long division or synthetic division.

Q7: Can a rational function have more than one asymptote?

A7: Yes, a rational function can have more than one asymptote. For example, the function $f(x)=\frac{x^2}{x-1}$ has a vertical asymptote at x = 1 and a slant asymptote at y = x + 1.

Q8: How do I determine the type of asymptote a rational function has?

A8: To determine the type of asymptote a rational function has, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote. If the degree of the numerator is exactly one more than the degree of the denominator, the function has a slant asymptote. If the denominator is equal to zero, the function has a vertical asymptote.

Q9: Can a rational function have no asymptotes?

A9: Yes, a rational function can have no asymptotes. For example, the function $f(x)=\frac{x^2}{x2}$ has no asymptotes because the degree of the numerator is equal to the degree of the denominator.

Q10: How do I graph a rational function with asymptotes?

A10: To graph a rational function with asymptotes, first find the vertical asymptotes by setting the denominator equal to zero and solving for x. Then, find the horizontal or slant asymptote by comparing the degrees of the numerator and denominator. Finally, plot the asymptotes on a graph and use them as a guide to draw the graph of the rational function.

Conclusion

In conclusion, asymptotes are an essential part of understanding the behavior of rational functions. By answering these frequently asked questions, we hope to have provided a better understanding of asymptotes and how to find them. Remember to always compare the degrees of the numerator and denominator to determine the type of asymptote a rational function has.