Problem 20 - Algebra 2

Question

Find the horizontal asymptote of the graph of each rational function. $$ y=\frac{x+2}{2 x^{2}-4} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(y=\frac{x+2}{2x^{2}-4}\) is \(y=0\).

Step 1: Identify Degree of Numerator and Denominator

First, identify the degree of both the numerator and denominator of the function \(y=\frac{x+2}{2x^{2}-4}\). The degree of a polynomial is the highest power of the variable x. Here, the degree of the numerator (the polynomial x+2) is 1 and the degree of the denominator (the polynomial \(2x^{2}-4\)) is 2.

Step 2: Compare the Degrees

Secondly, comparing the degrees of both the numerator and the denominator, we see that the degree of the numerator is less than the degree of the denominator (1 < 2).

Step 3: Determine the Horizontal Asymptote

Since the degree of the numerator is less than the degree of the denominator, the x-axis (line \(y=0\)) is the horizontal asymptote of the function.

Key Concepts

Rational Functions and Their Characteristics

Rational functions are expressions formed by dividing two polynomials. For example, the function \( y = \frac{x+2}{2x^2 - 4} \) is a rational function. These functions are characterized by:

• A numerator, which is the top part of the fraction, in this case, \(x+2\).
• A denominator, which is the bottom part, here, \(2x^2 - 4\).

Rational functions can exhibit interesting behaviors when graphed, such as asymptotes. Asymptotes are lines that the graph approaches but never fully touches. Recognizing these characteristics is crucial in understanding the overall shape and behavior of the graph.

Understanding the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable within that polynomial. Knowing the degree helps in comparing the relative strength of the polynomial terms, especially when looking at rational functions.For the example \( y = \frac{x+2}{2x^2 - 4} \):

• The numerator, \(x+2\), has a degree of 1 because the highest power of \(x\) is \(x^1\).
• The denominator, \(2x^2 - 4\), has a degree of 2 since the highest power is \(x^2\).

Understanding the degrees of both the numerator and the denominator is essential for determining other features of the function, like horizontal asymptotes.

Comparing Numerator and Denominator

When considering asymptotes, it’s important to compare the degrees of the numerator and the denominator:- If the degree of the numerator is less than that of the denominator, the horizontal asymptote is the x-axis, \(y=0\).- If the degrees are equal, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator, respectively.- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.In the provided function \( y = \frac{x+2}{2x^2 - 4} \), since the degree of the numerator (1) is less than the degree of the denominator (2), the x-axis, \(y=0\), is the horizontal asymptote. Understanding how to compare these degrees helps predict the behavior of the function as \(x\) approaches very large or very small values.