Horizontal or Oblique Asymptote

Oct 13, 2025 by talynt.org 129 views

Graphing and Analyzing the Function $f(x)=\frac{x^2-5x-24}{x-6}$

In this article, we will explore the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ using a graphing tool. We will identify the characteristics of the graph, including the type of asymptote, the range of the function, and the end behavior.

To graph the function $f(x)=\frac{x^2-5x-24}{x-6}$ , we can use a graphing tool such as a graphing calculator or a computer algebra system. When we graph the function, we get a graph that has a vertical asymptote at $x=6$ .

Vertical Asymptote

The vertical asymptote at $x=6$ indicates that the function is not defined at this point. This is because the denominator of the function is equal to zero when $x=6$ , which makes the function undefined.

Horizontal or Oblique Asymptote

To determine whether the graph has a horizontal or oblique asymptote, we need to examine the behavior of the function as $x$ approaches infinity. We can do this by dividing the numerator and denominator of the function by the highest power of $x$ in the denominator.

import sympy as sp
x = sp.symbols('x')
f = (x**2 - 5*x - 24) / (x - 6)
f_simplified = sp.simplify(f)
print(f_simplified)

When we simplify the function, we get:

f(x) = x + 4 + \frac{20}{x-6}

As $x$ approaches infinity, the term $\frac{20}{x-6}$ approaches zero. Therefore, the graph of the function has a horizontal asymptote at $y=x+4$ .

Range of the Function

To determine the range of the function, we need to examine the behavior of the function as $x$ approaches infinity and negative infinity.

As $x$ approaches infinity, the function approaches the horizontal asymptote $y=x+4$ . Therefore, the range of the function is all real numbers greater than or equal to $x+4$ .

As $x$ approaches negative infinity, the function approaches the horizontal asymptote $y=x+4$ . Therefore, the range of the function is all real numbers less than or equal to $x+4$ .

End Behavior

The end behavior of the function is determined by the behavior of the function as $x$ approaches infinity and negative infinity.

As $x$ approaches infinity, the function approaches the horizontal asymptote $y=x+4$ . Therefore, the end behavior of the function is that it approaches infinity as $x$ approaches infinity.

As $x$ approaches negative infinity, the function approaches the horizontal asymptote $y=x+4$ . Therefore, the end behavior of the function is that it approaches negative infinity as $x$ approaches negative infinity.

In conclusion, the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ has a vertical asymptote at $x=6$ , a horizontal asymptote at $y=x+4$ , and a range of all real numbers greater than or equal to $x+4$ . The end behavior of the function is that it approaches infinity as $x$ approaches infinity and negative infinity as $x$ approaches negative infinity.

Vertical Asymptote: The graph has a vertical asymptote at $x=6$ .
Horizontal or Oblique Asymptote: The graph has a horizontal asymptote at $y=x+4$ .
Range of the Function: The range of the function is all real numbers greater than or equal to $x+4$ .
End Behavior: The end behavior of the function is that it approaches infinity as $x$ approaches infinity and negative infinity as $x$ approaches negative infinity.

Here is the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ :

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)
y = (x**2 - 5*x - 24) / (x - 6)
plt.plot(x, y)
plt.axvline(x=6, color='red', linestyle='--')
plt.axhline(y=4, color='green', linestyle='--')
plt.title('Graph of the Function  $f(x)=\frac{x^2-5x-24}{x-6}$ ')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

This graph shows the vertical asymptote at $x=6$ , the horizontal asymptote at $y=x+4$ , and the range of the function.
Q&A: Graphing and Analyzing the Function $f(x)=\frac{x^2-5x-24}{x-6}$

In our previous article, we explored the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ using a graphing tool. We identified the characteristics of the graph, including the type of asymptote, the range of the function, and the end behavior. In this article, we will answer some frequently asked questions about the graph of the function.

Q: What is the type of asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ has a vertical asymptote at $x=6$ .

Q: What is the horizontal or oblique asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ has a horizontal asymptote at $y=x+4$ .

Q: What is the range of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The range of the function $f(x)=\frac{x^2-5x-24}{x-6}$ is all real numbers greater than or equal to $x+4$ .

Q: What is the end behavior of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The end behavior of the function $f(x)=\frac{x^2-5x-24}{x-6}$ is that it approaches infinity as $x$ approaches infinity and negative infinity as $x$ approaches negative infinity.

Q: How do I graph the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: You can graph the function $f(x)=\frac{x^2-5x-24}{x-6}$ using a graphing tool such as a graphing calculator or a computer algebra system.

Q: What is the significance of the vertical asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The vertical asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ indicates that the function is not defined at the point $x=6$ .

Q: What is the significance of the horizontal asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: The horizontal asymptote in the graph of the function $f(x)=\frac{x^2-5x-24}{x-6}$ indicates that the function approaches a certain value as $x$ approaches infinity.

Q: How do I determine the range of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: You can determine the range of the function $f(x)=\frac{x^2-5x-24}{x-6}$ by examining the behavior of the function as $x$ approaches infinity and negative infinity.

Q: How do I determine the end behavior of the function $f(x)=\frac{x^2-5x-24}{x-6}$ ?

A: You can determine the end behavior of the function $f(x)=\frac{x^2-5x-24}{x-6}$ by examining the behavior of the function as $x$ approaches infinity and negative infinity.