Problem 35 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{-2x + 1}{3x + 5}\) is \(y = -\frac{2}{3}\).

Step 1: Determine the Degrees of the Polynomials

Firstly, identify the polynomials in the rational function \(f(x) = \frac{-2x + 1}{3x + 5}\). These are \(p(x) = -2x + 1\) in the numerator and \(q(x) = 3x + 5\) in the denominator. The degrees of these polynomials are both 1 because they're both linear functions, and the degree of a linear function is 1.

Step 2: Check Degree Comparison

The degrees of both polynomials are equal, which means the rule applicable is: if the degree of \(p(x)\) is equal to the degree of \(q(x)\), the horizontal asymptote is the ratio of the leading coefficients of \(p(x)\) and \(q(x)\). In this case, the leading coefficients are -2 and 3 respectively.

Step 3: Find the Ratio of Leading Coefficients

According to the rule which applies when the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients of \(p(x)\) and \(q(x)\). Therefore, calculate the ratio of -2 to 3 as \(-\frac{2}{3}\). This means the horizontal asymptote of the given function is \(y = -\frac{2}{3}\).

Key Concepts

Understanding Rational Functions

Rational functions are mathematical expressions that represent the division of two polynomials. Essentially, a rational function has the form: \[ f(x) = \frac{p(x)}{q(x)} \],where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \) is not the zero polynomial. The domain of a rational function consists of all real numbers except those that make the denominator zero because division by zero is undefined.One interesting feature of rational functions is their asymptotes, which can be vertical, horizontal, or oblique. Horizontal asymptotes provide information about the function's end behavior—how the function behaves as \( x \) approaches infinity or negative infinity. In the given example, \( f(x) = \frac{-2x + 1}{3x + 5} \), the function is rational because it represents the quotient of two linear polynomials. The concept of horizontal asymptotes is closely tied to the degrees of these polynomials and their leading coefficients, which we'll discuss in the following sections.

Polynomial Degrees and Their Significance

The degree of a polynomial is the highest power of the variable \( x \) in the polynomial expression. It provides a sense of how the polynomial will behave, especially as the value of \( x \) grows large. Polynomials are classified according to their degree—linear for degree 1, quadratic for degree 2, and so on.For the rational function discussed earlier, both the numerator \( p(x) = -2x + 1 \) and denominator \( q(x) = 3x + 5 \) are linear polynomials, meaning they have a degree of 1. The degrees of the polynomials offer a clue about the existence and nature of horizontal asymptotes. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (\( y = 0 \)). If the degrees are the same, the horizontal asymptote is found by the ratio of leading coefficients, which is the case in the example. Lastly, if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, and one might need to consider oblique asymptotes.

The Importance of Leading Coefficients

Leading coefficients are the coefficients of the highest degree terms in a polynomial. They play a vital role in determining the features of a polynomial's graph and are particularly crucial in understanding the horizontal asymptotes of rational functions.When the degrees of the numerator and denominator polynomials of a rational function are equal, the horizontal asymptote can be found simply by taking the ratio of their leading coefficients. In a step-by-step solution to our example, the leading coefficients of \( p(x) = -2x + 1 \) and \( q(x) = 3x + 5 \) are -2 and 3, respectively. The horizontal asymptote is thus \( y = -\frac{2}{3} \). Keep in mid that if the degree in the numerator had been higher, there wouldn't be a horizontal asymptote, and if it had been lower, the asymptote would have been the x-axis.Understanding leading coefficients is not only helpful when studying asymptotes, but also when predicting end behavior of the polynomial and its graph. This interplay between the degree and leading coefficients defines much of the polynomial's characteristics.