Identify the Vertical Asymptote

Oct 13, 2025 by talynt.org 126 views

a. $y=\frac{3}{x-4}+2$

Horizontal Asymptote:

Vertical Asymptote:

Understanding Rational Functions Rational functions are a type of function that can be expressed as the ratio of two polynomials. They are an essential concept in mathematics, particularly in algebra and calculus. In this discussion, we will focus on graphing a specific rational function, $y=\frac{3}{x-4}+2$ . To graph this function, we need to understand its components and how they interact with each other.

Components of the Rational Function The given rational function consists of two main components: a numerator and a denominator. The numerator is a constant value of 3, while the denominator is a linear expression of $x-4$ . The constant term 2 is added to the function, which is known as the vertical shift.

Graphing the Rational Function To graph the rational function, we need to consider the following steps:

Identify the Vertical Asymptote: The vertical asymptote is the value of $x$ that makes the denominator equal to zero. In this case, the denominator is $x-4$ , which equals zero when $x=4$ . Therefore, the vertical asymptote is $x=4$ .
Identify the Horizontal Asymptote: The horizontal asymptote is the value of $y$ that the function approaches as $x$ becomes very large or very small. In this case, the numerator is a constant value of 3, while the denominator is a linear expression of $x-4$ . As $x$ becomes very large or very small, the denominator becomes very large, and the function approaches a value of 0. Therefore, the horizontal asymptote is $y=0$ .
Plot the Function: To plot the function, we need to consider the behavior of the function as $x$ approaches the vertical asymptote. As $x$ approaches 4 from the left, the function approaches negative infinity. As $x$ approaches 4 from the right, the function approaches positive infinity. The function also has a vertical shift of 2 units up.

Graphing the Rational Function Based on the above analysis, the graph of the rational function $y=\frac{3}{x-4}+2$ is a hyperbola with a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ . The function approaches negative infinity as $x$ approaches 4 from the left and positive infinity as $x$ approaches 4 from the right. The function also has a vertical shift of 2 units up.

Key Takeaways In conclusion, graphing a rational function involves identifying the vertical and horizontal asymptotes, plotting the function, and considering the behavior of the function as $x$ approaches the vertical asymptote. The rational function $y=\frac{3}{x-4}+2$ has a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ . The function approaches negative infinity as $x$ approaches 4 from the left and positive infinity as $x$ approaches 4 from the right. The function also has a vertical shift of 2 units up.

Real-World Applications Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics. For example, rational functions can be used to model the behavior of electrical circuits, mechanical systems, and economic systems. In physics, rational functions can be used to describe the motion of objects under the influence of gravity or other forces. In engineering, rational functions can be used to design and analyze electrical circuits, mechanical systems, and other complex systems.

Conclusion In conclusion, graphing a rational function involves identifying the vertical and horizontal asymptotes, plotting the function, and considering the behavior of the function as $x$ approaches the vertical asymptote. The rational function $y=\frac{3}{x-4}+2$ has a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ . The function approaches negative infinity as $x$ approaches 4 from the left and positive infinity as $x$ approaches 4 from the right. The function also has a vertical shift of 2 units up. Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics.

Horizontal Asymptote:

The horizontal asymptote of a rational function is the value of $y$ that the function approaches as $x$ becomes very large or very small. In the case of the rational function $y=\frac{3}{x-4}+2$ , the horizontal asymptote is $y=0$ .

Vertical Asymptote:

The vertical asymptote of a rational function is the value of $x$ that makes the denominator equal to zero. In the case of the rational function $y=\frac{3}{x-4}+2$ , the vertical asymptote is $x=4$ .

Key Takeaways:

The rational function $y=\frac{3}{x-4}+2$ has a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ .
The function approaches negative infinity as $x$ approaches 4 from the left and positive infinity as $x$ approaches 4 from the right.
The function also has a vertical shift of 2 units up.

Real-World Applications:

Conclusion:

In conclusion, graphing a rational function involves identifying the vertical and horizontal asymptotes, plotting the function, and considering the behavior of the function as $x$ approaches the vertical asymptote. The rational function $y=\frac{3}{x-4}+2$ has a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ . The function approaches negative infinity as $x$ approaches 4 from the left and positive infinity as $x$ approaches 4 from the right. The function also has a vertical shift of 2 units up. Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics.

a. $y=\frac{3}{x-4}+2$

Horizontal Asymptote:

Vertical Asymptote:

Graphing the Rational Function To graph the rational function, we need to consider the following steps:

Identify the Vertical Asymptote: The vertical asymptote is the value of $x$ that makes the denominator equal to zero. In this case, the denominator is $x-4$ , which equals zero when $x=4$ . Therefore, the vertical asymptote is $x=4$ .
Identify the Horizontal Asymptote: The horizontal asymptote is the value of $y$ that the function approaches as $x$ becomes very large or very small. In this case, the numerator is a constant value of 3, while the denominator is a linear expression of $x-4$ . As $x$ becomes very large or very small, the denominator becomes very large, and the function approaches a value of 0. Therefore, the horizontal asymptote is $y=0$ .
Plot the Function: To plot the function, we need to consider the behavior of the function as $x$ approaches the vertical asymptote. As $x$ approaches 4 from the left, the function approaches negative infinity. As $x$ approaches 4 from the right, the function approaches positive infinity. The function also has a vertical shift of 2 units up.

Q&A

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.

Q: What are the components of a rational function? A: The components of a rational function are the numerator and the denominator.

Q: What is the vertical asymptote of a rational function? A: The vertical asymptote of a rational function is the value of $x$ that makes the denominator equal to zero.

Q: What is the horizontal asymptote of a rational function? A: The horizontal asymptote of a rational function is the value of $y$ that the function approaches as $x$ becomes very large or very small.

Q: How do you graph a rational function? A: To graph a rational function, you need to identify the vertical and horizontal asymptotes, plot the function, and consider the behavior of the function as $x$ approaches the vertical asymptote.

Q: What are the real-world applications of rational functions? A: Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics.

Q: Can you give an example of a rational function? A: Yes, an example of a rational function is $y=\frac{3}{x-4}+2$ .

Q: What is the graph of the rational function $y=\frac{3}{x-4}+2$ ? A: The graph of the rational function $y=\frac{3}{x-4}+2$ is a hyperbola with a vertical asymptote at $x=4$ and a horizontal asymptote at $y=0$ .

Q: What is the behavior of the function as $x$ approaches the vertical asymptote? A: As $x$ approaches 4 from the left, the function approaches negative infinity. As $x$ approaches 4 from the right, the function approaches positive infinity.

Q: What is the vertical shift of the function? A: The function also has a vertical shift of 2 units up.

Key Takeaways

A rational function is a type of function that can be expressed as the ratio of two polynomials.
The components of a rational function are the numerator and the denominator.
The vertical asymptote of a rational function is the value of $x$ that makes the denominator equal to zero.
The horizontal asymptote of a rational function is the value of $y$ that the function approaches as $x$ becomes very large or very small.
To graph a rational function, you need to identify the vertical and horizontal asymptotes, plot the function, and consider the behavior of the function as $x$ approaches the vertical asymptote.
Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics.

Real-World Applications

Rational functions have numerous real-world applications in various fields, including physics, engineering, and economics. For example, rational functions can be used to model the behavior of electrical circuits, mechanical systems, and economic systems. In physics, rational functions can be used to describe the motion of objects under the influence of gravity or other forces. In engineering, rational functions can be used to design and analyze electrical circuits, mechanical systems, and other complex systems.