Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{4 x^{2}-10}{2 x-4} $$

Step-by-Step Solution

Answer

The slant asymptote is \(y = 2x + 4\).

Step 1: Determine if a Slant Asymptote Exists

A slant asymptote exists if the degree of the numerator is exactly one more than the degree of the denominator. For the function \(f(x) = \frac{4x^2 - 10}{2x - 4}\), the degree of the numerator \(4x^2 - 10\) is 2, and the degree of the denominator \(2x - 4\) is 1. Since the degree of the numerator is one more than the degree of the denominator, a slant asymptote exists.

Step 2: Perform Polynomial Long Division

To find the slant asymptote, divide the numerator by the denominator using polynomial long division. Divide the leading term of the numerator \(4x^2\) by the leading term of the denominator \(2x\), resulting in a quotient of \(2x\). Multiply \(2x\) by the entire divisor \(2x - 4\) to get \(4x^2 - 8x\). Subtract \(4x^2 - 8x\) from \(4x^2 - 10\), which yields \(8x - 10\).

Step 3: Continue the Division

Take the new polynomial \(8x - 10\) and divide the leading term \(8x\) by the leading term of the divisor \(2x\). This gives a quotient of \(4\). Multiply \(4\) by the divisor \(2x - 4\), resulting in \(8x - 16\). Subtract \(8x - 16\) from \(8x - 10\), giving a remainder of \(6\).

Step 4: Write the Division Result

The result of the division is \(2x + 4\) with a remainder of \(6\). For very large \(x\), \(\frac{6}{2x-4}\) approaches 0, meaning the slant asymptote is represented by the quotient, which is \(y = 2x + 4\).

Step 5: Confirm the Equation of Slant Asymptote

Thus, the slant asymptote of the function \(f(x)\) is the line \(y = 2x + 4\). This is confirmed by the steps of the division focusing on the leading parts, which dictate the asymptotic behavior.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials similar to the long division process with numbers. It helps you find the quotient, which can reveal significant information about the function, such as slant asymptotes. To perform polynomial long division:

• Identify the dividend (numerator) and the divisor (denominator).
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract it from the original dividend.
• Repeat the process with the new polynomial formed by the subtraction until you're left with a remainder that's smaller in degree than the divisor.

In the context of slant asymptotes, we're interested in the quotient without the remainder, as it describes the asymptotic behavior of the function.

Degrees of Polynomials

Understanding the degrees of polynomials is crucial when analyzing rational functions for slant asymptotes. The degree of a polynomial is the highest power of the variable in the polynomial. For the function in question, the degrees are:

• The degree of the numerator, \(4x^2 - 10\), is 2.
• The degree of the denominator, \(2x - 4\), is 1.

The degree of the numerator is one higher than the degree of the denominator, which is a key condition for the existence of a slant asymptote. This situation indicates that as \(x\) becomes very large, the function will behave closely to a line rather than a horizontal line or a constant value. Recognizing this condition quickly helps in solving and understanding rational functions efficiently.

Asymptotic Behavior

The term asymptotic behavior refers to how a function behaves at the extremes of the graph, particularly as \(x\) approaches infinity or negative infinity. Functions can have horizontal, vertical, or slant asymptotes based on their structure. Slant asymptotes occur when the function graph approximates a line called the asymptote at extreme \(x\) values.
In our exercise, we found the slant asymptote to be \(y = 2x + 4\). This means that as \(x\) increases or decreases without bound, the graph of \(f(x)\) will get closer and closer to the line \(y = 2x + 4\). This understanding is crucial for predicting the general shape and direction of the graph based on its asymptotic behavior, which is often required in calculus and advanced algebra.

Quotient in Division

The quotient in polynomial division not only completes the division process but also can represent the function's long-term behavior. In rational functions specifically, the quotient obtained often serves as the slant asymptote.

• To find the quotient, divide the leading term of the numerator by the leading term of the denominator.
• Continue the process until the remaining polynomial is of a lower degree than the divisor.

For the function provided, the quotient \(2x + 4\) represents the slant asymptote. As \(x\) becomes very large, the fraction of the remainder over the divisor approaches zero. Thus, the function \(f(x)\) is asymptotically similar to the line \(y = 2x + 4\). Recognizing the quotient's significance helps in analyzing other similar functions and identifying their slant asymptotes effortlessly.