Problem 486 - Precalculus

Question

For the following exercises, find the slant asymptote of the functions. $$f(x)=\frac{81 x^{2}-18}{3 x-2}$$

Step-by-Step Solution

Answer

Slant asymptote: \(y = 27x + 18\).

Step 1: Identify the Degree of the Numerator and Denominator

To find the slant asymptote, first check the degrees of the numerator and the denominator. The degree of the numerator \(81x^2 - 18\) is 2, and the degree of the denominator \(3x - 2\) is 1. Since the degree of the numerator is exactly one greater than the degree of the denominator, a slant asymptote exists.

Step 2: Perform Polynomial Long Division

Divide \(81x^2 - 18\) by \(3x - 2\) using polynomial long division. Start by dividing the highest degree terms, \(81x^2\) by \(3x\), which gives \(27x\). Multiply \(27x\) by \(3x - 2\) to get \(81x^2 - 54x\). Subtract \(81x^2 - 54x\) from \(81x^2 - 18\) to get \(54x - 18\).

Step 3: Continue the Division Process

Divide \(54x\) by \(3x\) to get \(18\). Multiply \(18\) by \(3x - 2\) to get \(54x - 36\). Subtract \(54x - 36\) from \(54x - 18\) to obtain the remainder \(18\).

Step 4: Identify the Slant Asymptote Equation

The division process results in \(27x + 18 + \frac{18}{3x-2}\). The slant asymptote is determined by the linear part of the quotient without the remainder. Thus, the slant asymptote is \(y = 27x + 18\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division process you might use in arithmetic with numbers. It is particularly useful in finding slant asymptotes when working with rational functions.
To begin, organize the division problem by writing the dividend (the polynomial being divided) and the divisor (the polynomial dividing the dividend). For the function \( f(x) = \frac{81x^2 - 18}{3x - 2} \), the dividend is \( 81x^2 - 18 \) and the divisor is \( 3x - 2 \).

• Divide the highest degree term of the dividend by the highest degree term of the divisor.
• In our example, divide \( 81x^2 \) by \( 3x \) to get \( 27x \).
• Multiply the entire divisor \( 3x - 2 \) by \( 27x \).
• Subtract this result from the original dividend.
• Continue this process with the resulting polynomial.

This operation provides the quotient, which, in slant asymptote problems, helps identify the asymptote equation. The remainder, if present, is expressed as a fraction of the divisor.

Degree of a Polynomial

The degree of a polynomial is an important characteristic that helps in various algebraic calculations. It is the highest power of the variable in the polynomial expression. In the polynomial \( 81x^2 - 18 \), the degree is 2 because the highest power of \( x \) is 2.
The degree of the polynomial tells us a lot:

• It indicates the highest term that influences the graph's shape.
• It affects the number of roots a polynomial might have.
• In the context of rational functions, comparing the degree of the numerator and the denominator helps determine asymptotes.

When the degree of the numerator is one more than the degree of the denominator, as in this exercise (2 for the numerator and 1 for the denominator), the rational function will have a slant asymptote.

Asymptotes in Rational Functions

Asymptotes are lines that a graph approaches but never actually touches or crosses. In rational functions, these can be vertical, horizontal, or slant (oblique) asymptotes.
Slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. This is the situation in our exercise with the function \( \frac{81x^2 - 18}{3x - 2} \).

• Slant asymptotes are found through polynomial long division, which simplifies the rational function to a form where the slant asymptote can be easily recognized as the linear component of the resulting expression.
• Once the polynomial long division is performed, the quotient gives the equation of the slant asymptote. In our exercise, the slant asymptote is \( y = 27x + 18 \).
• The remainder does not affect the slant asymptote, as it becomes insignificant when \( x \) is very large.

Understanding asymptotes is crucial for graphing rational functions accurately and predicting their behavior at extreme values.