Problem 73 - 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. $$ f(x)=\frac{2 x^{2}+6 x+6}{x+3}, g(x)=2 x $$

Step-by-Step Solution

Answer

The vertical asymptote is at \(x = -3\), and both functions have the same end behavior.

Step 1: Factor the numerator

We have the function \( f(x) = \frac{2x^2 + 6x + 6}{x+3} \). Start by factoring the numerator: \( 2x^2 + 6x + 6 = 2(x^2 + 3x + 3) \). This expression cannot be factored further using identifiable simple techniques.

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Set the denominator equal to zero: \(x+3=0\). Thus, \( x = -3 \) is a vertical asymptote.

Step 3: Determine End Behavior

For end behavior, simplify \( f(x) \) by dividing the numerator by the denominator: \( f(x) = \frac{2x^2 + 6x + 6}{x+3} \). Divide: \( 2x^2 + 6x + 6 \) by \( x + 3 \) results in \( 2x - 0 \). The remainder does not affect the end behavior, so \( f(x) \approx 2x \).

Step 4: Compare with g(x)

The function \( g(x) = 2x \) is a straight line. Since \( f(x) \approx 2x \) for large values of \( x \) or \( -x \), both functions have the same end behavior as \( x \to \infty \) or \( x \to -\infty \).

Step 5: Graph f(x) and g(x)

Graph \( f(x) = \frac{2x^2 + 6x + 6}{x+3} \) and \( g(x) = 2x \) on a large viewing rectangle. Observe that \( f(x) \) has a vertical asymptote at \( x = -3 \) and both \( f(x) \) and \( g(x) \) have similar behavior as \( x \to \infty \) or \( x \to -\infty \).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where a rational function becomes unbounded, meaning it shoots up to infinity or down to negative infinity. To find these asymptotes, set the denominator equal to zero and solve for the variable. In the function \[f(x) = \frac{2x^2 + 6x + 6}{x+3},\]the denominator is \(x+3\). Setting \(x+3=0\) gives \(x=-3\) as the solution. Hence, there is a vertical asymptote at \(x = -3\). Vertical asymptotes are crucial in understanding how the rational function behaves near these undefined points. The function will dramatically increase or decrease as \(x\) approaches the asymptote.

End Behavior

The end behavior of a function describes how the function behaves as the input values become very large (positively or negatively). For rational functions, it often involves simplifying the function by division to understand how it grows or decreases far from the origin. In this example, we simplify \[f(x) = \frac{2x^2 + 6x + 6}{x+3}\]through polynomial long division, resulting in \(f(x) \approx 2x\). Thus, as \(x\) becomes large, whether positive or negative, \(f(x)\) mimics the behavior of the straight line \(g(x) = 2x\). Both functions point in the same direction, confirming they have the same end behavior.

Factoring Polynomials

Factoring polynomials can often simplify rational functions. It's the process of decomposing a polynomial into a product of simpler polynomials, making it easier to analyze and solve. In this case, the numerator \[2x^2 + 6x + 6\]is factored as \[2(x^2 + 3x + 3).\]The lack of further factorization indicates the numerator does not contribute to the vertical asymptotes beyond what the denominator dictates. Factoring becomes especially useful when it leads to cancelling factors, simplifying the function further.

Graphical Representation

Graphing is a visual method to understand the behavior of a rational function. For our functions \(f(x)\) and \(g(x)\), plotting reveals essential characteristics. In \(f(x)\), the graph will show a steep incline or decline near the vertical asymptote \(x = -3\). This is where the function is undefined and can appear as a break in the graph. On the other hand, \(g(x) = 2x\) is a simple linear graph without any asymptotes. By comparing both graphs on the same axes, you observe that both follow a similar trajectory at their ends, confirming their shared end behavior. Graphical representation provides clarity on abstract algebraic findings, bringing insight to the eye and aiding understanding of the mathematical relationship.