Problem 23 - Algebra 2

Question

Find the horizontal asymptote of the graph of each rational function. $$ y=\frac{5 x^{3}+2 x}{2 x^{5}-4 x^{3}} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(y = \frac{5x^{3} + 2x}{2x^{5}-4x^{3}}\) is \(y=0\).

Step 1: Identify the Degree of the Polynomials

An important aspect in determining the horizontal asymptote is recognizing the degree of the polynomials in the numerator and the denominator. The degree is the exponent of the highest degree term, that is the term with the largest exponent. In this case, the function is \( y = \frac{5x^{3} + 2x}{2x^{5}-4x^{3}} \). The degree of the numerator is 3 (from \(5x^{3}\)) and the degree of the denominator is 5 (from \(2x^{5}\)). Thus, the degree of the denominator is greater than the degree of the numerator.

Step 2: Determine the Horizontal Asymptote

In rational functions, if the degree of the denominator is greater than the degree of the numerator, then the horizontal asymptote is at y=0. This is because as \(x\) tends to positive or negative infinity, the function tends to zero since the higher degree term in the denominator dominates the numerator.

Key Concepts

Horizontal Asymptote in Rational Functions

A horizontal asymptote in a rational function gives us insight into how the graph behaves as the x-values move towards positive or negative infinity. It essentially represents a horizontal line that the function approaches but never quite reaches.

To determine the horizontal asymptote, focus on comparing the degrees of the polynomials in the numerator and denominator of the rational function. Depending on their degrees, the horizontal asymptote will be different:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at the line \(y = 0\).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but rather an oblique or slant asymptote.

In our example, the degree of the denominator (5) is greater than the degree of the numerator (3), so the horizontal asymptote is \(y = 0\). This highlights the behavior of the function as it flattens out along the x-axis as it heads towards infinity or negative infinity.

Understanding Polynomial Degree

The degree of a polynomial is a key concept that influences the behavior of rational functions, especially when determining horizontal asymptotes. The degree of a polynomial is identified by the highest exponent of its variables. This is crucial for understanding the comparative growth rates of the polynomials in the function.

In the rational function \(y = \frac{5x^{3} + 2x}{2x^{5} - 4x^{3}}\), let's break it down:

• The numerator, \(5x^{3} + 2x\), has a degree of 3 because the highest exponent is 3, from \(5x^{3}\).
• The denominator, \(2x^{5} - 4x^{3}\), has a degree of 5 because the highest exponent is 5, from \(2x^{5}\).

The polynomial degree influences the end behavior and asymptotic nature of the rational function. In this case, since the degree of the denominator exceeds that of the numerator, it confirms that the horizontal asymptote is at \(y=0\). The higher power in the denominator ensures that as \(x\) becomes large, the impact of the numerator lessens, effectively flattening the graph towards the horizontal axis.

Graph Behavior of Rational Functions

Rational functions exhibit a variety of behaviors on their graphs, largely determined by the degrees of the polynomials in the numerator and denominator. Understanding the degree enables one to predict how the graph approaches its horizontal asymptote.

As \(x\) approaches very large positive or negative numbers, the terms with the highest degrees dominate the behavior of the function:

• For our rational function \(y = \frac{5x^{3} + 2x}{2x^{5} - 4x^{3}}\), since the denominator’s degree (5) is greater than the numerator’s degree (3), the graph approaches the horizontal asymptote \(y = 0\).
• The function's graph thus flattens and runs parallel to the x-axis, without crossing it, as it stretches towards infinity in either direction.

This behavior is crucial when sketching graphs or predicting long-term trends in functions. Knowing the degrees of the polynomials allows you to anticipate the function's steepness and how quickly it levels off, thus revealing its general shape without plotting every single point.