Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{6 x^{3}-5 x}{3 x^{2}+4} $$

Step-by-Step Solution

Answer

The slant asymptote is \(y = 2x\).

Step 1: Identify the Degree of the Numerator and Denominator

To find the slant asymptote, first check if the degree of the numerator is exactly one higher than the degree of the denominator. Here, the degree of the numerator \(6x^3 - 5x\) is 3, and the degree of the denominator \(3x^2 + 4\) is 2. Since 3 is exactly one more than 2, there is a slant asymptote.

Step 2: Perform Polynomial Long Division

Divide the numerator \(6x^3 - 5x\) by the denominator \(3x^2 + 4\). Start by dividing the first term of the numerator \(6x^3\) by the first term of the denominator \(3x^2\), which gives \(2x\). Multiply \(2x\) by \(3x^2 + 4\) to get \(6x^3 + 8x\), and subtract from the original numerator to get a new polynomial \(-13x\). Repeat the process with \(-13x\). Divide \(-13x\) by \(3x^2\), which results in \(-\frac{13}{3x}\), yielding a remainder. However, for slant asymptotes, we're only concerned with the quotient without the remainder.

Step 3: State the Slant Asymptote

The result from the polynomial division gives us a quotient of \(2x\). Therefore, the slant asymptote is the equation of the straight line \(y = 2x\).

Key Concepts

Polynomial Long Division

When dealing with slant asymptotes in rational functions, understanding polynomial long division is essential. Polynomial long division is a method similar to arithmetic long division with numbers. It is used when you want to divide two polynomials. This process helps in simplifying expressions, especially when the numerator has a higher degree than the denominator, which indicates the presence of a slant asymptote in some rational functions.

To perform polynomial long division:

• Start by dividing the leading term of the numerator by the leading term of the denominator.
• Multiply this result by the entire denominator and subtract it from the original numerator.
• Repeat the process with the new polynomial obtained after subtraction until the degree of the remaining polynomial is less than that of the denominator.

The quotient you obtain, excluding any remainder, will help you determine the slant asymptote of the function.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They are of the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\).

These functions exhibit different types of asymptotic behaviors depending on the degrees of the numerator and the denominator. Asymptotes in rational functions could be horizontal, vertical, or slant. Slant asymptotes specifically occur when the degree of the numerator is exactly one more than the degree of the denominator.

Rational functions can exhibit interesting asymptotic behavior, which tells us how the function behaves at extreme values of \(x\), either as \(x\) approaches infinity or a point where the function is undefined.

Degree of Polynomial

The degree of a polynomial is the highest power of \(x\) that appears in the polynomial. It is a crucial factor in determining the behavior of a rational function, specifically its asymptotic behavior.

• The degree of the numerator determines whether a slant asymptote will be present (one higher than the denominator's degree).
• For example, in the function \(f(x)=\frac{6x^{3}-5x}{3x^{2}+4}\), the degree of the numerator is 3 and for the denominator is 2.

Understanding the degree of the polynomials will help you determine the type of asymptote present.

Asymptotic Behavior

The asymptotic behavior of a function describes how the function behaves as \(x\) approaches infinity or certain critical points. There are several types of asymptotic behavior:

• Horizontal Asymptotes: When the degrees of the numerator and denominator are equal, or the denominator's degree is greater.
• Vertical Asymptotes: Occur where the denominator equals zero, assuming no common factors in numerator and denominator.
• Slant Asymptotes: Occur when the numerator's degree is exactly one degree higher than the denominator's.

For slant asymptotes, such as in our given function example, they tell us that as \(x\) tends to infinity, the graph of the function will resemble a line, which can be derived from polynomial long division like \(y = 2x\). The slant asymptote is not a boundary for the graph but a line that the graph gets closer to as it extends to infinity.