Select The Correct Answer.Function { R $}$ Is A Continuous Rational Function With A Horizontal Asymptote At { Y = -8 $}$.Which Statement Describes The Key Features Of { S(x) = R(x+2) - 1 $} ? A . F U N C T I O N \[ ?A. Function \[ ? A . F U N C T I O N \[ S
Introduction
In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. These functions can have various key features, including vertical asymptotes, horizontal asymptotes, and holes. In this article, we will explore the key features of a rational function and how to identify them.
Horizontal Asymptotes
A horizontal asymptote is a horizontal line that the graph of a function approaches as the absolute value of the x-coordinate gets larger and larger. In the case of the function { r $}$, we are given that it has a horizontal asymptote at { y = -8 $}$. This means that as x approaches positive or negative infinity, the value of the function approaches -8.
Transforming the Function
The function { s(x) = r(x+2) - 1 $}$ is a transformation of the original function { r $}$. To understand the key features of this new function, we need to analyze the transformation.
Vertical Shift
The function { s(x) = r(x+2) - 1 $}$ has a vertical shift of -1. This means that the graph of the function is shifted down by 1 unit. As a result, any vertical asymptotes or holes in the original function will also be shifted down by 1 unit.
Horizontal Shift
The function { s(x) = r(x+2) - 1 $}$ also has a horizontal shift of -2. This means that the graph of the function is shifted to the left by 2 units. As a result, any vertical asymptotes or holes in the original function will also be shifted to the left by 2 units.
Key Features of the Function
Based on the transformation, we can conclude that the function { s(x) = r(x+2) - 1 $}$ has the following key features:
- A horizontal asymptote at { y = -8 $}$
- A vertical shift of -1
- A horizontal shift of -2
Conclusion
In conclusion, the function { s(x) = r(x+2) - 1 $}$ has a horizontal asymptote at { y = -8 $}$, a vertical shift of -1, and a horizontal shift of -2. These key features can be identified by analyzing the transformation of the original function.
Answer
The correct answer is:
- A. Function { s $}$ has a horizontal asymptote at { y = -8 $}$, a vertical shift of -1, and a horizontal shift of -2.
Key Takeaways
- Rational functions can have various key features, including vertical asymptotes, horizontal asymptotes, and holes.
- Transformations of a function can affect its key features.
- Analyzing the transformation of a function can help identify its key features.
Frequently Asked Questions
Q: What is a horizontal asymptote?
A: A horizontal asymptote is a horizontal line that the graph of a function approaches as the absolute value of the x-coordinate gets larger and larger.
Q: What is a vertical shift?
A: A vertical shift is a transformation that moves the graph of a function up or down.
Q: What is a horizontal shift?
A: A horizontal shift is a transformation that moves the graph of a function left or right.
Q: How do I identify the key features of a rational function?
A: To identify the key features of a rational function, you need to analyze its transformation and look for any vertical asymptotes, horizontal asymptotes, or holes.
Q: What is the difference between a vertical asymptote and a hole?
A: A vertical asymptote is a point where the function approaches positive or negative infinity, while a hole is a point where the function is undefined.
Q: How do I determine the horizontal asymptote of a rational function?
A: To determine the horizontal asymptote of a rational function, you need to look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: How do I determine the vertical asymptote of a rational function?
A: To determine the vertical asymptote of a rational function, you need to look for any points where the denominator is equal to zero. These points are the vertical asymptotes of the function.
Q: How do I determine the holes of a rational function?
A: To determine the holes of a rational function, you need to look for any points where the numerator and denominator are both equal to zero. These points are the holes of the function.
Q: What is the difference between a rational function and a polynomial function?
A: A rational function is a function that can be expressed as the ratio of two polynomials, while a polynomial function is a function that can be expressed as a sum of terms, each of which is a constant times a power of x.
Q: How do I graph a rational function?
A: To graph a rational function, you need to identify its key features, including vertical asymptotes, horizontal asymptotes, and holes. You can then use this information to sketch the graph of the function.
Q: What is the importance of understanding rational functions?
Q: What is a rational function?
A: A rational function is a function that can be expressed as the ratio of two polynomials. It is a type of function that can be written in the form { f(x) = \frac{p(x)}{q(x)} $}$, where { p(x) $}$ and { q(x) $}$ are polynomials.
Q: What are the key features of a rational function?
A: The key features of a rational function include:
- Vertical asymptotes: These are points where the function approaches positive or negative infinity.
- Horizontal asymptotes: These are horizontal lines that the function approaches as the absolute value of the x-coordinate gets larger and larger.
- Holes: These are points where the function is undefined.
Q: How do I determine the vertical asymptotes of a rational function?
A: To determine the vertical asymptotes of a rational function, you need to look for any points where the denominator is equal to zero. These points are the vertical asymptotes of the function.
Q: How do I determine the horizontal asymptotes of a rational function?
A: To determine the horizontal asymptotes of a rational function, you need to look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: What is the difference between a vertical asymptote and a hole?
A: A vertical asymptote is a point where the function approaches positive or negative infinity, while a hole is a point where the function is undefined.
Q: How do I determine the holes of a rational function?
A: To determine the holes of a rational function, you need to look for any points where the numerator and denominator are both equal to zero. These points are the holes of the function.
Q: What is the difference between a rational function and a polynomial function?
A: A rational function is a function that can be expressed as the ratio of two polynomials, while a polynomial function is a function that can be expressed as a sum of terms, each of which is a constant times a power of x.
Q: How do I graph a rational function?
A: To graph a rational function, you need to identify its key features, including vertical asymptotes, horizontal asymptotes, and holes. You can then use this information to sketch the graph of the function.
Q: What is the importance of understanding rational functions?
A: Understanding rational functions is important because they are used to model many real-world phenomena, including population growth, chemical reactions, and electrical circuits.
Q: How do I simplify a rational function?
A: To simplify a rational function, you need to factor the numerator and denominator and cancel out any common factors.
Q: What is the difference between a rational function and a radical function?
A: A rational function is a function that can be expressed as the ratio of two polynomials, while a radical function is a function that can be expressed as a root of a polynomial.
Q: How do I determine the domain of a rational function?
A: To determine the domain of a rational function, you need to look for any points where the denominator is equal to zero. These points are not in the domain of the function.
Q: What is the difference between a rational function and a trigonometric function?
A: A rational function is a function that can be expressed as the ratio of two polynomials, while a trigonometric function is a function that can be expressed in terms of trigonometric functions, such as sine and cosine.
Q: How do I use rational functions in real-world applications?
A: Rational functions are used to model many real-world phenomena, including population growth, chemical reactions, and electrical circuits. They can be used to make predictions and analyze data.
Q: What are some common applications of rational functions?
A: Some common applications of rational functions include:
- Modeling population growth
- Analyzing chemical reactions
- Designing electrical circuits
- Predicting stock prices
- Analyzing data
Q: How do I use technology to graph rational functions?
A: You can use technology, such as graphing calculators or computer software, to graph rational functions. This can help you visualize the function and identify its key features.
Q: What are some common mistakes to avoid when working with rational functions?
A: Some common mistakes to avoid when working with rational functions include:
- Not factoring the numerator and denominator
- Not canceling out common factors
- Not identifying the vertical and horizontal asymptotes
- Not identifying the holes
- Not using technology to graph the function
Q: How do I choose the right technology to graph rational functions?
A: You can choose from a variety of technologies, including graphing calculators, computer software, and online tools. The right technology for you will depend on your needs and preferences.
Q: What are some common tools and software used to graph rational functions?
A: Some common tools and software used to graph rational functions include:
- Graphing calculators, such as the TI-83 or TI-84
- Computer software, such as Mathematica or Maple
- Online tools, such as Desmos or GeoGebra
Q: How do I use online tools to graph rational functions?
A: You can use online tools, such as Desmos or GeoGebra, to graph rational functions. These tools can help you visualize the function and identify its key features.
Q: What are some common resources for learning about rational functions?
A: Some common resources for learning about rational functions include:
- Textbooks, such as "Algebra and Trigonometry" by Michael Sullivan
- Online resources, such as Khan Academy or MIT OpenCourseWare
- Video tutorials, such as 3Blue1Brown or Crash Course
- Practice problems, such as those found on websites like Mathway or Wolfram Alpha
Q: How do I practice working with rational functions?
A: You can practice working with rational functions by completing practice problems, such as those found on websites like Mathway or Wolfram Alpha. You can also use online tools, such as Desmos or GeoGebra, to graph rational functions and identify their key features.