Problem 36 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{-3 x+7}{5 x-2}\) is \(y=-3/5\).

Step 1: Identify the Degree of the Polynomials

Notice that both the numerator and the denominator are first degree polynomials. This implies that there is a horizontal asymptote.

Step 2: Determine the Coefficient of the Highest Degree Term

In the numerator and denominator, identify the coefficients of the highest degree terms. In the numerator, the coefficient of \(x\) is -3. In the denominator, the coefficient of \(x\) is 5.

Step 3: Calculate the Horizontal Asymptote

The horizontal asymptote of a rational function, when the degrees of the numerator and denominator are equal, is the ratio of the coefficients of the highest degree terms. In this case, that would be \(-3/5\). Therefore, the horizontal asymptote of this function is at \(y=-3/5\).

Key Concepts

Rational Functions

Rational functions form an essential part of algebra, particularly when analyzing asymptotic behavior. A rational function is a fraction where both the numerator and the denominator are polynomials. These functions can describe various real-world phenomena, such as chemical reactions or population dynamics. The general form of a rational function is:

• \( f(x) = \frac{P(x)}{Q(x)} \)

where \(P(x)\) and \(Q(x)\) are polynomials.

Understanding these functions involves examining the horizontal and vertical asymptotes, which aid in predicting the function's behavior at extreme values of \(x\). Horizontal asymptotes illustrate the long-term behavior of the function as \(x\) approaches infinity or negative infinity.

Degree of Polynomials

The degree of a polynomial is a fundamental concept that tells us about the highest power of the variable present in the polynomial. For instance, in the polynomial \(-3x + 7\), the degree is 1 since the term with the highest power of \(x\) is \(-3x^1\). This is relevant when determining the horizontal asymptotes of rational functions.

In the original exercise, both the numerator and the denominator have a degree of 1. Therefore, they are known as first-degree polynomials. This equality in degrees plays a crucial role in defining the horizontal asymptote of the function—indicating that we must compare the coefficients of these highest degree terms.

Coefficients

Coefficients are the numerical factors in polynomial terms. In the context of our exercise, understanding coefficients is key to finding horizontal asymptotes. The coefficient of a polynomial term is simply the number in front of the variable.

• In the numerator \(-3x + 7\), the coefficient of \(x\) is \(-3\).
• In the denominator \(5x - 2\), the coefficient of \(x\) is \(5\).

These coefficients are used to determine the horizontal asymptote of a rational function by forming their ratio when the degrees of the polynomials are equal. Thus, the horizontal asymptote is \(y = \frac{-3}{5}\).

This ratio simplifies the function's behavior analysis at extreme \(x\) values.

Asymptotic Behavior

Asymptotic behavior describes how a function acts as the variable \(x\) moves towards extreme values, either infinitely large or small. Horizontal asymptotes are particularly valuable because they show the trend of a graph as \(x\) approaches infinity or negative infinity.

For the rational function \(f(x) = \frac{-3x + 7}{5x - 2}\), identifying asymptotic behavior involves looking at the function when the degrees of the numerator and denominator are the same. In such cases, as \(x\) tends to infinity, the function approaches a horizontal line, the equation of which is given by the ratio of the leading coefficients. As found in the exercise, this line is \(y = -\frac{3}{5}\).

Thus, the horizontal asymptote helps in sketching the function's graph and predicting its values at large or small \(x\), providing an insight into behavior trends without calculating each point.