Problem 65 - Algebra and Trigonometry

Question

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. \(f(x)=\frac{2 x^{2}+6 x+6}{x+3}, \quad g(x)=2 x\)

Step-by-Step Solution

Answer

The vertical asymptote is at \(x = -3\). Both functions have the same end behavior as \(x\) approaches infinity.

Step 1: Identify Possible Vertical Asymptotes

Identify values of the denominator that make the function undefined. For the function \(f(x) = \frac{2x^2 + 6x + 6}{x + 3}\), set the denominator equal to zero: \[ x + 3 = 0 \]Solve this equation to find \(x = -3\). Thus, there is a potential vertical asymptote at \(x = -3\).

Step 2: Perform Polynomial Long Division for Simplification

Conduct polynomial long division to simplify \(f(x)\). Divide the numerator \(2x^2 + 6x + 6\) by the denominator \(x + 3\): Performing division, we get:\(2x - 0\) with a remainder of 6. Thus, the simplified form of \(f(x)\) is:\[ f(x) = 2x + \frac{6}{x+3} \]

Step 3: Determine Vertical Asymptotes

From Step 1, we concluded there could be a vertical asymptote at \(x = -3\). Since the divisor \(x + 3\) is in the simplified form's denominator, \(f(x)\) has a vertical asymptote at \(x = -3\) as long as \(x = -3\) does not cancel out. It does not cancel, confirming a true vertical asymptote at \(x = -3\).

Step 4: Graph the Functions on a Large Scale

Graph \(f(x) = 2x + \frac{6}{x+3}\) and \(g(x) = 2x\) on the same coordinate system. Make sure to use a sufficiently large viewing rectangle to observe their behavior as \(x\) approaches positive and negative infinity. Note how both functions exhibit approximately the same slope and thus the same end behavior as \(x\) approaches infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are values of \( x \) where a function goes to infinity or negative infinity. These occur when the denominator of a rational function is equal to zero, making the function undefined at that point.
For the function \( f(x) = \frac{2x^2 + 6x + 6}{x + 3} \), the denominator \( x + 3 \) equals zero when \( x = -3 \). Therefore, a potential vertical asymptote is at \( x = -3 \). This point is confirmed as \( f(x) \) doesn’t cancel at \( x = -3 \), solidifying \( x = -3 \) as a vertical asymptote.
In a graph, the vertical asymptote will appear as a line that the function approaches but never crosses, indicating rapid growth or decline.

Polynomial Long Division

Polynomial long division is a method to divide polynomials much like the long division you do with numbers. It helps to simplify complex rational functions.
For \( f(x) = \frac{2x^2 + 6x + 6}{x + 3} \), divide the numerator \( 2x^2 + 6x + 6 \) by the denominator \( x + 3 \). This simplifies \( f(x) \) to \( 2x + \frac{6}{x+3} \).
The process shows step-by-step how each quotient term is found:

• Divide the highest degree term of the numerator by the highest degree term of the divisor.
• Multiply the whole divisor by this result and subtract from the original polynomial.
• Repeat with the new polynomial formed.

This technique is vital for simplifying rational functions to identify key features like asymptotes and behavior.

End Behavior

End behavior describes how a function behaves as \( x \) approaches positive or negative infinity. For rational functions, it’s essential to compare the degrees of the numerator and denominator.
In this case, the function \( f(x) = 2x + \frac{6}{x+3} \) has the highest degree terms \( 2x \), similar to \( g(x) = 2x \).
As \( x \) goes to infinity, \( \frac{6}{x+3} \) approaches 0, making \( f(x) \approx 2x \). Therefore, the end behavior of \( f(x) \) and \( g(x) \) is the same, with both having approximately linear behavior, growing similarly in either direction along the y-axis.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to show a visual representation of the function. For better understanding, use a large scale to capture asymptotes and end behavior effectively.
For \( f(x) = 2x + \frac{6}{x+3} \), graph this function together with \( g(x) = 2x \). Note how closely they resemble each other as \( x \) tends towards infinity, despite the asymptote influencing \( f(x) \) specifically.
Here’s what to do while graphing:

• Identify critical points, including where the function is undefined, and locate vertical asymptotes.
• Determine any horizontal or slant asymptotes based on end behavior.
• Use points determined from both functions to render accurate graphs.

This will provide clear insights into their behavior, especially how they approach similar lines at the graphs' ends.