Understanding the Key Features of a Rational Function

Oct 13, 2025 by windailys.com 33 views

Understanding the Key Features of a Rational Function

In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. These functions are widely used in various fields, including algebra, calculus, and engineering. In this article, we will explore the key features of a rational function and how they are affected by transformations.

The original function, denoted as ${$ r $}$, is a continuous rational function with a horizontal asymptote at ${$ y = -8 $}$. This means that as the input values of the function approach positive or negative infinity, the output values approach -8. The horizontal asymptote is a horizontal line that the function approaches as the input values increase without bound.

The function ${$ s(x) = r(x+2) - 1 $}$ is a transformation of the original function ${$ r $}$. This transformation involves two main steps:

Horizontal Shift: The function is shifted horizontally by 2 units to the left. This means that the input values of the function are decreased by 2.
Vertical Shift: The function is shifted vertically by 1 unit down. This means that the output values of the function are decreased by 1.

The transformed function ${$ s $}$ has the following key features:

Continuity: The function ${$ s $}$ is continuous, just like the original function ${$ r $}$. This means that the function can be drawn without lifting the pencil from the paper, and there are no gaps or jumps in the graph.
Horizontal Asymptote: The horizontal asymptote of the function ${$ s $}$ is still at ${$ y = -8 $}$. This means that as the input values of the function approach positive or negative infinity, the output values still approach -8.
Domain: The domain of the function ${$ s $}$ is the same as the domain of the original function ${$ r $}$. This means that the function is still defined for all real numbers.
Range: The range of the function ${$ s $}$ is also the same as the range of the original function ${$ r $}$. This means that the function can still take on all real values.

In conclusion, the transformed function ${$ s $}$ has the same key features as the original function ${$ r $}$. The function is continuous, has a horizontal asymptote at ${$ y = -8 $}$, and has the same domain and range as the original function. The horizontal shift and vertical shift transformations do not affect the key features of the function.

The transformed function ${$ s $}$ is continuous and has a horizontal asymptote at ${$ y = -8 $}$.
The domain and range of the function ${$ s $}$ are the same as the domain and range of the original function ${$ r $}$.
The horizontal shift and vertical shift transformations do not affect the key features of the function.

In this article, we explored the key features of a rational function and how they are affected by transformations. We saw that the transformed function ${$ s $}$ has the same key features as the original function ${$ r $}$. This means that the function is still continuous, has a horizontal asymptote, and has the same domain and range. The horizontal shift and vertical shift transformations do not affect the key features of the function.
Q&A: Understanding the Key Features of a Rational Function

In our previous article, we explored the key features of a rational function and how they are affected by transformations. In this article, we will answer some frequently asked questions about rational functions and their transformations.

A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that has a numerator and a denominator, and the denominator is not equal to zero.

The horizontal asymptote of a rational function is a horizontal line that the function approaches as the input values increase without bound. It is a line that the function gets arbitrarily close to as the input values get arbitrarily large.

Horizontal shifts affect the domain of the function, while vertical shifts affect the range of the function. Horizontal shifts do not affect the horizontal asymptote, while vertical shifts do not affect the domain.

A horizontal shift to the left by a units results in a shift of the graph a units to the left. A horizontal shift to the right by a units results in a shift of the graph a units to the right.

A vertical shift up by b units results in a shift of the graph b units up. A vertical shift down by b units results in a shift of the graph b units down.

Yes, a rational function can have a vertical asymptote. A vertical asymptote occurs when the denominator of the function is equal to zero.

To determine the domain of a rational function, you need to find the values of x that make the denominator equal to zero. These values are not included in the domain of the function.

To determine the range of a rational function, you need to find the values of y that the function can take on. The range of a rational function is all real numbers except for the values that make the denominator equal to zero.

Yes, a rational function can have a hole in its graph. A hole occurs when there is a factor in the numerator and denominator that cancels out.

To find the horizontal asymptote of a rational function, you need to compare 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.

In this article, we answered some frequently asked questions about rational functions and their transformations. We hope that this article has helped you to better understand the key features of rational functions and how they are affected by transformations.

A rational function is a type of function that can be expressed as the ratio of two polynomials.
The horizontal asymptote of a rational function is a horizontal line that the function approaches as the input values increase without bound.
Horizontal shifts affect the domain of the function, while vertical shifts affect the range of the function.
A rational function can have a vertical asymptote, a hole in its graph, and a horizontal asymptote.

In this article, we explored some frequently asked questions about rational functions and their transformations. We hope that this article has helped you to better understand the key features of rational functions and how they are affected by transformations.