Problem 42 - Algebra and Trigonometry

Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ h(x)=\frac{15 x^{3}}{3 x^{2}+1} $$

Step-by-Step Solution

Answer

The given function \( h(x) = \frac{15x^3}{3x^2 + 1} \) has no horizontal asymptote, as the degree of the numerator is greater than the degree of the denominator.

Step 1: Identify the Degree of the Numerator and the Denominator

The degree of a polynomial is the highest power of x. In the numerator, the highest power of x is 3; hence, the degree of the numerator is 3. In the denominator, the highest power of x is 2; hence, the degree of the denominator is 2.

Step 2: Compare the Degree of the Numerator and the Denominator

In this case, the degree of the numerator is bigger than the degree of the denominator. According to the rules for horizontal asymptotes, if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Key Concepts

Rational Function Asymptotes

An asymptote is a line that a graph approaches as the input or output values get infinitely large or small. For rational functions, which are functions represented as the ratio of two polynomials, asymptotes can be vertical, horizontal, or oblique.

Specifically focusing on horizontal asymptotes, these occur when the values of a function approach a constant value as the input grows larger or smaller without bound. The horizontal asymptote serves as a boundary line where the graph levels off. The determination of horizontal asymptotes depends largely on the comparison between the degrees of the polynomials in the numerator and the denominator.

Three rules simplify this determination:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis, or the line y = 0.
• If the degrees are equal, the horizontal asymptote is the line y = the ratio of the leading coefficients of the numerator and denominator.
• If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote.

For the exercise's rational function h(x) = \(\frac{15x^3}{3x^2+1}\), we observe the rules above to see that, indeed, there is no horizontal asymptote because the numerator's degree is higher.

Polynomial Degree

Understanding the degree of a polynomial is crucial when graphing functions and determining their asymptotes. The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 4x^2 + 7x + 2, the degree is 2 because the highest power of x is 2.

The degree of a polynomial has significant implications:

• It determines the number of possible roots or zeros of the polynomial.
• It influences the shape of the graph of the polynomial function, including the end behavior.
• It helps to identify potential asymptotes when the polynomial is part of a rational function.

For the given function h(x) = \(\frac{15x^3}{3x^2+1}\), the numerator, 15x^3, has a degree of 3, and the denominator, 3x^2+1, has a degree of 2. This tells us about the growth rate of the function's numeration relative to its denotation and impacts the presence or absence of horizontal asymptotes.

Graphing Rational Functions

The process of graphing rational functions involves several steps that help to visualize the behavior of these complex expressions. When graphing, it's helpful to identify key features of the function:

• Intercepts where the function crosses the axes.
• Vertical asymptotes where the function approaches infinity.
• Horizontal asymptotes which indicate the value that the function approaches as x goes to infinity.
• Any points of discontinuity or holes where the function is not defined.

The process usually starts with factoring and simplifying the function, if possible. Then determining intercepts and asymptotes and analyzing the polynomial degrees, as mentioned earlier.

By plotting these key points and asymptotes, we begin to understand the overall shape of the graph. Increasing and decreasing intervals, and behavior at infinity are critical aspects in drawing accurate graphs of rational functions. With the function h(x) = \(\frac{15x^3}{3x^2+1}\), we would plot the calculated intercepts and use our determination that a horizontal asymptote does not exist to sketch the behavior of the graph at extreme values of x.