Problem 79 - 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. $$ f(x)=\frac{4 x^{2}}{2 x-1} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = \frac{1}{2}\); slant asymptote \(y = 2x + 1\). Graph shows behavior relative to these asymptotes.

Step 1: Simplify and Identify Vertical Asymptotes

Identify the values that make the denominator of the function zero, as these values result in vertical asymptotes. For the function \( f(x) = \frac{4x^2}{2x-1} \), set the denominator equal to zero: \(2x - 1 = 0\). Solving for \(x\), we find \(x = \frac{1}{2}\). Thus, there is a vertical asymptote at \(x = \frac{1}{2}\).

Step 2: Find the Slant Asymptote

Since the degree of the numerator (2) is one more than the degree of the denominator (1), there is a slant asymptote. Divide the numerator by the denominator using polynomial long division: \[ \frac{4x^2}{2x - 1} = 2x + 1 + \frac{1}{2x-1} \]. The slant asymptote is the quotient without the remainder: \(y = 2x + 1\).

Step 3: Plot the Asymptotes on a Graph

First, draw the vertical asymptote as a dashed line at \(x = \frac{1}{2}\). Then, draw the slant asymptote as a dashed line with the equation \(y = 2x + 1\) across the graph. This will guide the behavior of the function as \(x\) approaches infinity.

Step 4: Sketch the Graph of the Function

Plot multiple points of the function \( f(x) = \frac{4x^2}{2x-1} \) near the asymptotes, especially on either side of the vertical asymptote at \(x = \frac{1}{2}\). Observe that as \(x\) approaches \(\frac{1}{2}\) from the left, \(f(x)\) will drop towards \(-\infty\), and as \(x\) approaches \(\frac{1}{2}\) from the right, \(f(x)\) will climb to \(\infty\). As \(x\) goes to positive or negative infinity, the graph will approach the slant asymptote \(y = 2x + 1\).

Step 5: Finalize the Graph

Ensure all asymptotes are present on the graph as dashed lines, and the curve is plotted following the asymptote guidance from Steps 3 and 4. Double-check points on either side of the asymptotes to ensure accuracy.

Key Concepts

Slant Asymptote

A slant asymptote occurs when the degree of the numerator of a rational function is exactly one more than the degree of the denominator. This kind of asymptote results in a diagonal line, showing how the function behaves as it approaches infinity or negative infinity.
To find a slant asymptote, you use polynomial long division. Divide the numerator by the denominator and ignore the remainder for this purpose.

• In our exercise, the rational function is \( f(x) = \frac{4x^2}{2x-1} \).
• The degree of the numerator is 2, and the degree of the denominator is 1, indicating a slant asymptote.
• Performing the division results in \( y = 2x + 1 \).

The slant asymptote \( y = 2x + 1 \) helps us understand how the function will behave at extreme values of \( x \), providing a kind of tilted reference line for graphing the function.

Vertical Asymptote

In rational functions, vertical asymptotes occur where the function's value heads towards infinity or negative infinity. These are typically found at the points where the denominator equals zero, causing the fraction to be undefined.
It's important to ensure these points are indicated on a graph as dashed lines to remind us of the restrictions at those spots.

• For the function \( f(x) = \frac{4x^2}{2x-1} \), set \( 2x - 1 = 0 \) to find the vertical asymptote.
• Solving for \( x \), we find \( x = \frac{1}{2} \).

This tells us there's a vertical asymptote at \( x = \frac{1}{2} \), where the function either climbs to positive infinity from one side or drops to negative infinity from the other. This characteristic changes drastically on either side of this line, shaping our understanding of the function's graph.

Graphing Functions

Graphing rational functions involves plotting not just the main curve of the function, but also correctly identifying and drawing asymptotes. Asymptotes are usually indicated by dashed lines on the graph.
In our example, we would:

• Draw the vertical asymptote at \( x = \frac{1}{2} \).
• Draw the slant asymptote along the line \( y = 2x + 1 \).

Additionally, while plotting the graph, pinpoint various points of the function on both sides of the vertical asymptote to understand how the function behaves as it approaches this line. This includes noting that as \( x \) approaches \( \frac{1}{2} \), the function's value either climbs to infinity or plummets to negative infinity. Also, observe that as \( x \) goes towards infinity, the graph aligns more with the slant asymptote, \( y = 2x + 1 \), confirming its role as a guiding line.

Rational Functions

Rational functions consist of the ratio of two polynomials. They often have interesting characteristics, such as potential asymptotes and unbounded behavior, making their graphs unique when compared to simpler polynomial functions.
Understanding a rational function involves looking at its numerator and denominator.

• If the degree of the numerator is greater than that of the denominator, expect a slant asymptote.
• Vertical asymptotes occur where the denominator equals zero, causing the function to be undefined at those points.

Rational functions can be challenging due to these nuances, but they offer rich exploration in behavior. Mastering their graphing involves understanding these critical aspects like slant and vertical asymptotes, which offer a scaffold for depicting the graph's trajectory across the coordinate plane.