Let $f(x)=\frac{3x(x+4)}{(x-2)(x-2)}$.Find The Requested Information For $f$. Vertical Asymptote(s): (smaller Value First) $\square$ $\square$ Horizontal Asymptote: $\square$ $x$-intercept(s):

Feb 27, 2025 by ADMIN 198 views

**Vertical Asymptote(s) and Horizontal Asymptote of a Rational Function**

Introduction

In this article, we will explore the concept of vertical and horizontal asymptotes of a rational function. A rational function is a function that can be expressed as the ratio of two polynomials. The vertical asymptotes of a rational function are the values of x that make the denominator of the function equal to zero, while the horizontal asymptote is the value that the function approaches as x approaches infinity.

Vertical Asymptote(s)

To find the vertical asymptote(s) of the given rational function $f(x)=\frac{3x(x+4)}{(x-2)(x-2)}$ , we need to find the values of x that make the denominator equal to zero.

Step 1: Factor the denominator

The denominator of the function is $(x-2)(x-2)$ . We can factor this expression as $(x-2)^2$ .

Step 2: Set the denominator equal to zero

To find the vertical asymptote(s), we need to set the denominator equal to zero and solve for x.

$(x-2)^2 = 0$

Step 3: Solve for x

Solving for x, we get:

$x-2 = 0$

$x = 2$

Therefore, the vertical asymptote of the function is $x = 2$ .

Horizontal Asymptote

To find the horizontal asymptote of the function, we need to compare the degrees of the numerator and the denominator.

Step 1: Compare the degrees of the numerator and the denominator

The degree of the numerator is 2, while the degree of the denominator is 2. Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator.

Step 2: Find the ratio of the leading coefficients

The leading coefficient of the numerator is 3, while the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is:

$y = \frac{3}{1}$

$y = 3$

Therefore, the horizontal asymptote of the function is $y = 3$ .

x-intercept(s)

To find the x-intercept(s) of the function, we need to set the numerator equal to zero and solve for x.

Step 1: Set the numerator equal to zero

Setting the numerator equal to zero, we get:

$3x(x+4) = 0$

Step 2: Solve for x

Solving for x, we get:

$x = 0$

$x = -4$

Therefore, the x-intercept(s) of the function are $x = 0$ and $x = -4$ .

Conclusion

In conclusion, the vertical asymptote of the function is $x = 2$ , the horizontal asymptote is $y = 3$ , and the x-intercept(s) are $x = 0$ and $x = -4$ .

References

[1] "Rational Functions" by Math Open Reference
[2] "Vertical and Horizontal Asymptotes" by Khan Academy

Discussion

Introduction

In our previous article, we explored the concept of vertical and horizontal asymptotes of rational functions. In this article, we will answer some frequently asked questions about vertical and horizontal asymptotes.

Q: What is a vertical asymptote?

A: A vertical asymptote is a value of x that makes the denominator of a rational function equal to zero. In other words, it is a value of x that makes the function undefined.

Q: How do you find the vertical asymptote(s) of a rational function?

A: To find the vertical asymptote(s) of a rational function, you need to set the denominator equal to zero and solve for x.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is the value that a rational function approaches as x approaches infinity.

Q: How do you find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to compare the degrees of the numerator and the denominator. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator.

Q: What is the difference between a vertical asymptote and a horizontal asymptote?

A: A vertical asymptote is a value of x that makes the function undefined, while a horizontal asymptote is the value that the function approaches as x approaches infinity.

Q: Can a rational function have both a vertical and a horizontal asymptote?

A: Yes, a rational function can have both a vertical and a horizontal asymptote.

Q: How do you determine the number of vertical asymptotes a rational function has?

A: To determine the number of vertical asymptotes a rational function has, you need to count the number of distinct values of x that make the denominator equal to zero.

Q: Can a rational function have no vertical asymptotes?

A: Yes, a rational function can have no vertical asymptotes if the denominator is never equal to zero.

Q: How do you determine the number of horizontal asymptotes a rational function has?

A: To determine the number of horizontal asymptotes a rational function has, you need to compare the degrees of the numerator and the denominator. If the degrees are equal, the function has one horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote of y = 0. If the degree of the numerator is greater than the degree of the denominator, the function has no horizontal asymptote.

Q: Can a rational function have no horizontal asymptote?

A: Yes, a rational function can have no horizontal asymptote if the degree of the numerator is greater than the degree of the denominator.

Conclusion

In conclusion, vertical and horizontal asymptotes are important concepts in the study of rational functions. By understanding these concepts, you can better analyze and solve problems involving rational functions.

References

[1] "Rational Functions" by Math Open Reference
[2] "Vertical and Horizontal Asymptotes" by Khan Academy

Discussion

What are some real-world applications of rational functions? How do you think the concept of vertical and horizontal asymptotes can be used in other areas of mathematics?