Find The Horizontal, Vertical, And Oblique Asymptotes Of F ( X F(x F ( X ]. Then Graph F ( X F(x F ( X ]. F ( X ) = 8 X 2 X − 6 F(x) = \frac{8x^2}{x-6} F ( X ) = X − 6 8 X 2 Find The Horizontal Asymptote. Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete

Mar 11, 2025 by ADMIN 293 views

**Finding and Graphing Asymptotes of Rational Functions**

Introduction

Asymptotes are lines that a function approaches as the input (or independent variable) gets arbitrarily close to a certain point. In this article, we will focus on finding the horizontal, vertical, and oblique asymptotes of a rational function, and then graph the function. We will use the function $f(x) = \frac{8x^2}{x-6}$ as an example.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that a function approaches as the input gets arbitrarily close to a certain point. To find the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

Degree of the Numerator and Denominator

The degree of a polynomial is the highest power of the variable (in this case, x). In the function $f(x) = \frac{8x^2}{x-6}$ , the degree of the numerator is 2, and the degree of the denominator is 1.

Comparing Degrees

Since the degree of the numerator (2) is greater than the degree of the denominator (1), we need to divide the numerator by the denominator to find the horizontal asymptote.

Dividing the Numerator by the Denominator

To divide the numerator by the denominator, we can use long division or synthetic division. Let's use long division.

import sympy as sp
x = sp.symbols('x')
numerator = 8*x**2
denominator = x - 6
result = sp.div(numerator, denominator)
print(result)

The result of the division is $8x + 48$ . This means that the horizontal asymptote is $y = 8x + 48$ .

Conclusion

The horizontal asymptote of the function $f(x) = \frac{8x^2}{x-6}$ is $y = 8x + 48$ .

Vertical Asymptotes

A vertical asymptote is a vertical line that a function approaches as the input gets arbitrarily close to a certain point. To find the vertical asymptote of a rational function, we need to find the values of x that make the denominator equal to zero.

Finding the Values of x that Make the Denominator Equal to Zero

In the function $f(x) = \frac{8x^2}{x-6}$ , the denominator is $x - 6$ . To find the values of x that make the denominator equal to zero, we need to set the denominator equal to zero and solve for x.

import sympy as sp
x = sp.symbols('x')
denominator = x - 6
result = sp.solve(denominator, x)
print(result)

The result of the solution is $x = 6$ . This means that the vertical asymptote is $x = 6$ .

Conclusion

The vertical asymptote of the function $f(x) = \frac{8x^2}{x-6}$ is $x = 6$ .

Oblique Asymptotes

An oblique asymptote is a slanted line that a function approaches as the input gets arbitrarily close to a certain point. To find the oblique asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

Degree of the Numerator and Denominator

The degree of the numerator is 2, and the degree of the denominator is 1.

Comparing Degrees

Since the degree of the numerator (2) is greater than the degree of the denominator (1), we need to divide the numerator by the denominator to find the oblique asymptote.

Dividing the Numerator by the Denominator

To divide the numerator by the denominator, we can use long division or synthetic division. Let's use long division.

import sympy as sp
x = sp.symbols('x')
numerator = 8*x**2
denominator = x - 6
result = sp.div(numerator, denominator)
print(result)

The result of the division is $8x + 48$ . This means that the oblique asymptote is $y = 8x + 48$ .

Conclusion

The oblique asymptote of the function $f(x) = \frac{8x^2}{x-6}$ is $y = 8x + 48$ .

Graphing the Function

To graph the function $f(x) = \frac{8x^2}{x-6}$ , we can use a graphing calculator or a computer algebra system.

Graphing the Function

Let's use a graphing calculator to graph the function.

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)
y = (8*x**2) / (x - 6)
plt.plot(x, y)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.axvline(6, color='red', linestyle='--')
plt.title('Graph of f(x) = 8x^2 / (x - 6)')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

The graph of the function $f(x) = \frac{8x^2}{x-6}$ is a rational function with a horizontal asymptote at $y = 8x + 48$ , a vertical asymptote at $x = 6$ , and an oblique asymptote at $y = 8x + 48$ .

Conclusion

Introduction

Asymptotes are lines that a function approaches as the input (or independent variable) gets arbitrarily close to a certain point. In this article, we will answer some common questions about asymptotes of rational functions.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that a function approaches as the input gets arbitrarily close to a certain point. To find the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

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

A: To find the horizontal asymptote of a rational function, we need to compare the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, we need to divide the numerator by the denominator to find the horizontal asymptote.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches as the input gets arbitrarily close to a certain point. To find the vertical asymptote of a rational function, we need to find the values of x that make the denominator equal to zero.

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

A: To find the vertical asymptote of a rational function, we need to find the values of x that make the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for x.

Q: What is an oblique asymptote?

A: An oblique asymptote is a slanted line that a function approaches as the input gets arbitrarily close to a certain point. To find the oblique asymptote of a rational function, we need to compare the degrees of the numerator and denominator.

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

A: To find the oblique asymptote of a rational function, we need to compare the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, we need to divide the numerator by the denominator to find the oblique asymptote.

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

A: Yes, a rational function can have more than one asymptote. For example, a rational function can have a horizontal asymptote, a vertical asymptote, and an oblique asymptote.

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

A: To graph a rational function with asymptotes, we can use a graphing calculator or a computer algebra system. We can also use the asymptotes to help us graph the function.

Q: What are some common mistakes to avoid when finding asymptotes of rational functions?

A: Some common mistakes to avoid when finding asymptotes of rational functions include:

Not comparing the degrees of the numerator and denominator
Not dividing the numerator by the denominator when the degree of the numerator is greater than the degree of the denominator
Not finding the values of x that make the denominator equal to zero
Not using a graphing calculator or a computer algebra system to graph the function

Conclusion

In this article, we have answered some common questions about asymptotes of rational functions. We have discussed how to find horizontal, vertical, and oblique asymptotes, and how to graph a rational function with asymptotes. We have also discussed some common mistakes to avoid when finding asymptotes of rational functions.