Problem 31 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$g(x)=\frac{12 x^{2}}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(g(x)=\frac{12 x^{2}}{3 x^{2}+1}\) is \(y=4\).

Step 1: Identify the Rational Function

From the given equation, we can see that the rational function \(g(x)\) is given as \(g(x)=\frac{12 x^{2}}{3 x^{2}+1}\)

Step 2: Identify the Degrees of the Polynomials

The degree of the polynomial in the numerator, P(x), is 2 and the degree of the polynomial in the denominator, Q(x), is also 2.

Step 3: Calculate The Ratio of the Leading Coefficients

Since the degrees of P(x) and Q(x) are equal, the horizontal asymptote will be the ratio of the leading coefficients of the polynomials which are 12 for P(x) and 3 for Q(x). Therefore, the ratio is \( \frac{12}{3} \).

Step 4: Determine the Horizontal Asymptote

The horizontal asymptote of \(g(x)\) is given by \(y = \frac{12}{3}\), simplifies to \(y=4\) which is the horizontal asymptote of the function.

Key Concepts

Understanding Rational Functions

Rational functions, like the one in the exercise with \(g(x)=\frac{12 x^{2}}{3 x^{2}+1}\), represent relationships where one polynomial is divided by another. Simply put, they have the form \(f(x) = \frac{P(x)}{Q(x)}\) where both \(P(x)\) and \(Q(x)\) are polynomials. An important feature of rational functions is their asymptotic behavior, which can often tell us about the function's end behavior without having to graph the entire function. Asymptotes are lines that the graph of the function approaches but never touches.

In our specific scenario, we are interested in finding a horizontal asymptote, which, if it exists, is a horizontal line that the graph of the function approaches as \(x\) approaches positive or negative infinity. Identifying any horizontal asymptotes is vital in understanding the long-term behavior of the function.

Recognizing the Role of Leading Coefficients

Leading coefficients play a pivotal role in determining the horizontal asymptotes for rational functions. The leading coefficient is simply the coefficient of the term with the highest power in a polynomial. For instance, in \(g(x)=\frac{12 x^{2}}{3 x^{2}+1}\), the leading coefficients are 12 for the numerator and 3 for the denominator.

When comparing the degrees of the polynomials in the numerator and the denominator of a rational function, if they are equal—as they are in our example—the horizontal asymptote can be found by taking the ratio of the leading coefficients. This concept is crucial as it provides us with a straightforward method to determine a key characteristic of the function’s graph, which in the case of the given exercise is \(y=\frac{12}{3}\), simplifying to \(y=4\).

Degrees of Polynomials in Asymptote Analysis

The degree of a polynomial is the highest power of \(x\) in its expression. When assessing rational functions for horizontal asymptotes, the degree of the numerator and denominator polynomials must be compared. There are three scenarios to consider:

• If the degree of \(P(x)\) is less than the degree of \(Q(x)\), the horizontal asymptote is always \(y=0\).
• If the degrees are equal, as in our exercise \(g(x)=\frac{12 x^{2}}{3 x^{2}+1}\), the horizontal asymptote is the ratio of their leading coefficients.
• If the degree of \(P(x)\) is greater than the degree of \(Q(x)\), there is no horizontal asymptote.

Understanding the relationship between the degrees of the polynomials provides a clear path to solving for horizontal asymptotes, making it easier for students to predict the behavior of the graph at infinity without actually plotting the entire function.