Problem 80 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{4 x^{2}+x-2}{4 x-3} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = \frac{3}{4}\), slant asymptote: \(y = x\).

Step 1: Identify Vertical Asymptotes

The vertical asymptote occurs where the denominator is zero. Solve \(4x - 3 = 0\).\\[4x - 3 = 0 \rightarrow 4x = 3 \rightarrow x = \frac{3}{4}\] \Thus, there is a vertical asymptote at \(x = \frac{3}{4}\).

Step 2: Determine the Slant Asymptote

Since the degree of the numerator (2) is higher than the degree of the denominator (1), a slant asymptote exists. Perform long division on \(\frac{4x^2 + x - 2}{4x-3}\).\to get \(y = x + \frac{\text{constant}}{4x-3}\). The slant asymptote is \(y = x\).

Step 3: Graph the Function

Plot the function \(y = \frac{4x^2 + x - 2}{4x - 3}\). Include the vertical asymptote \(x = \frac{3}{4}\) and the slant asymptote \(y = x\) in the graph. Mark asymptotes as dashed lines and ensure the curve approaches but never intersects with these lines.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are vertical lines that a graph approaches but never touches or crosses. They occur in rational functions where the denominator equals zero, as this makes the expression undefined. For the function \( f(x) = \frac{4x^2 + x - 2}{4x - 3} \), we find these asymptotes by setting the denominator equal to zero and solving for \( x \). This gives us:

• \(4x - 3 = 0\)
• \(4x = 3\)
• \(x = \frac{3}{4}\)

So, the vertical asymptote is at \( x = \frac{3}{4} \). On the graph, the function will approach \( x = \frac{3}{4} \) from both sides but won't actually meet the line. This is crucial information as it helps us understand the behavior of the function near that point.
Vertical asymptotes indicate a potential discontinuity in the function and often signify a boundary for plotted values.

Slant Asymptotes

Slant asymptotes, also known as oblique asymptotes, appear when the degree of the polynomial in the numerator is exactly one higher than that of the denominator. In our function \( f(x) = \frac{4x^2 + x - 2}{4x - 3} \), the numerator is of degree 2, and the denominator is of degree 1, thus confirming the existence of a slant asymptote.
To find the equation of the slant asymptote, we perform polynomial long division. The result of the division is a linear expression, which serves as the slant asymptote. In this example, dividing \( 4x^2 + x - 2 \) by \( 4x - 3 \) gives a quotient of \( x \), indicating that the slant asymptote is the line \( y = x \).
When graphing the function, this line will be shown as a diagonal dashed line. As \( x \) tends towards infinity, the graph of the rational function will tend to follow the slant asymptote line closely, highlighting a fundamental behavior characteristic of slant asymptotes.

Long Division of Polynomials

Long division of polynomials is a method used to divide polynomials similar to how you would divide numbers. It is notably useful when you need to find slant asymptotes in rational functions. To perform the division, you follow the sequence of dividing the terms, multiplying, subtracting, and bringing down the next term, repeated until all terms are processed.
For the function \( f(x) = \frac{4x^2 + x - 2}{4x - 3} \), we express it in the form of division:

• Divide the first term of the numerator \( 4x^2 \) by the first term of the denominator \( 4x \), yielding \( x \).
• Multiply \( x \) by the entire divisor \( 4x - 3 \), and subtract the result from the original numerator.
• Repeat this process with the new dividend until reaching a remainder polynomial of lower degree than the divisor.

The result \( x \) corresponds to the slant asymptote, and any remaining expression forms the remainder, expressed as a rational function. Understanding long division of polynomials aids in mastering rational expressions and analyzing their behaviors effectively.