Problem 33 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \(h(x) = \frac{12x^3}{3x^2 + 1}\) does not have a horizontal asymptote.

Step 1 - Determine the degree of the numerator and denominator

From the function \(h(x) = \frac{12x^3}{3x^2 + 1}\), the degree of the numerator (the highest exponents in the term) is 3 and the degree of the denominator is 2.

Step 2 - Compare their degrees and determine the horizontal asymptote

Since the degree of the numerator is higher than the degree of the denominator, the function does not have a horizontal asymptote.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that represent the division of two polynomials. In simpler terms, think of them as fractions where both the numerator (the top part) and the denominator (the bottom part) are polynomials.

A polynomial is an algebraic expression made up of terms in the form of \(a x^n\) where \(n\) is a whole number and \(a\) is a coefficient. For instance, in the function \(h(x) = \frac{12x^3}{3x^2 + 1}\), the expression \(12x^3\) is a polynomial of degree 3 - indicated by the highest power of \(x\).

Understanding rational functions is crucial because they appear in various areas of mathematics and applications, such as calculus, engineering, and economics. They can describe growth and decay rates, the behavior of electrical circuits, and much more.

Degree of Polynomial

The degree of a polynomial is found by locating the highest power of the variable \(x\) in its expression; this essentially provides us with the 'steepest' part of the polynomial's roller coaster. It's like identifying the tallest hill on a roller coaster to anticipate the potential ups and downs of the ride. In the context of rational functions, this concept helps us predict how the function will behave as \(x\) becomes very large or very small.

For example, in the function \(h(x) = \frac{12x^3}{3x^2 + 1}\), the highest power in the numerator is 3, while in the denominator, it is 2, making the degree of the numerator larger. Knowing this, mathematicians can determine the long-term behavior of the function, such as whether it has any horizontal asymptotes, by comparing the degrees of the numerator and denominator.

Graphing Rational Functions

Graphing a rational function is a step-by-step process just like following a recipe: you identify key components and their contributions to the overall graph. One of these components is the horizontal asymptote, a line that the graph of the function gets closer and closer to but never actually touches, as \(x\) approaches infinity.

How do we locate this asymptote? We compare the degrees of the polynomials in the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the \(x\)-axis, which is \(y = 0\).
• If they are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the numerator's degree is higher, like in our \(h(x)\), there's no horizontal asymptote.

This is only a slice of the process. Other parts of the graph, such as vertical asymptotes, intercepts, and behavior at infinity, must also be pieced together for the full picture.