Problem 22 - Algebra 2

Question

Find the horizontal asymptote of the graph of each rational function. $$ y=\frac{x^{2}+2}{2 x^{2}-1} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the graph of the function \( y=\frac{x^{2}+2}{2 x^{2}-1} \) is \( y=\frac{1}{2} \)

Step 1: Identifying the degrees of the polynomials

The degree of a polynomial is the highest power of the variable in the polynomial. Here, the numerator \( x^{2}+2 \) and the denominator \( 2x^{2}-1 \) are both of degree 2.

Step 2: Finding the leading coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. Here, the leading coefficient of \( x^{2}+2 \) is 1 and the leading coefficient of \( 2x^{2}-1 \) is 2.

Step 3: Determining the horizontal asymptote

If the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. Here, the ratio is \( \frac{1}{2} \). Hence, \( y=\frac{1}{2} \) is the horizontal asymptote of the given function.

Key Concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. These functions take the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial expressions. Why are they called "rational"? Because they can be expressed as a "ratio," like a fraction of two polynomials. They show interesting behavior, such as asymptotes, which are lines that the graph of the function approaches but never actually touches.

In examining rational functions, we often want to find their asymptotes, especially the horizontal ones, as these indicate what happens to the function's graph as \( x \) goes to infinity. Understanding how to determine these asymptotes is crucial for deeper insights into the behavior of these functions.

Degrees of Polynomials

The degree of a polynomial is a fundamental concept in algebra. It is defined as the highest power of the variable in the polynomial. For instance, in the expression \( x^{2} + 2 \), the highest power is 2, so it is a polynomial of degree 2. Likewise, in \( 2x^{2} - 1 \), the degree is also 2.

Understanding degrees is crucial when working with rational functions. They help us compare the growth rates of the numerator and the denominator. This comparison is essential in determining horizontal asymptotes of rational functions. When the degrees of the polynomials in the numerator and denominator are equal, the behavior of the function at infinity is balanced by their leading coefficients.

Leading Coefficients

The leading coefficient is the coefficient of the term in a polynomial with the highest degree. It plays a significant role in defining the function's behavior, especially so when analyzing rational functions. For example, in the polynomial \( x^{2} + 2 \), the leading coefficient is 1, and in \( 2x^{2} - 1 \), it is 2.

These coefficients are especially important when the degrees of the numerator and the denominator of a rational function are equal. In such cases, the horizontal asymptote of the function is determined by the ratio of the leading coefficients. Thus, leading coefficients help describe the ultimate behavior of a rational function graph in terms of its horizontal asymptote as \( x \) approaches infinity.

Algebra 2 Concepts

Understanding algebra 2 concepts is crucial for solving problems involving rational functions. This level builds upon basic algebra by introducing more complex functions and their properties, like asymptotes.

When dealing with rational functions in Algebra 2, students learn to analyze the graph's behavior as \( x \) approaches positive or negative infinity. The concept of horizontal asymptotes is grounded in the comparative study of the degrees and leading coefficients of the polynomials in the numerator and denominator. By mastering these algebraic principles, students can predict and explain the end-behavior of these versatile functions.

• How to find horizontal asymptotes
• The importance of polynomial degrees in function behavior
• The role of leading coefficients in simplifying complex expressions

These concepts collectively enhance a student’s problem-solving toolkit for higher-level studies in mathematics.