Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{2 x^{3}-x^{2}-2 x+1}{x^{2}+3 x+2}$$

Step-by-Step Solution

Answer

Vertical asymptotes are at \(x = -1\) and \(x = -2\), there are no horizontal asymptotes, the slant asymptote is \(2x -3\), and there are no holes in the graph.

Step 1: Calculation of Vertical Asymptotes

Set the denominator equal to zero and solve for x to find the vertical asymptotes. \[x^{2}+3 x+2=0\] Factorizing, we get \[(x+1)(x+2)=0\] The roots of the equation are -1 and -2, hence, we have two vertical asymptotes at \(x = -1\) and \(x = -2\)

Step 2: Calculation of Horizontal Asymptotes

If the degree of the numerator is equal to or smaller than the degree of the denominator, the function has a horizontal asymptote. Here, the degree of the numerator (3) is higher than the degree of the denominator (2), thus there are no horizontal asymptotes.

Step 3: Calculation of Slant Asymptotes

A slant (or oblique) asymptote exists whenever the degree of the top is exactly one greater than the degree of the bottom. Here, the degree of the numerator is one more than the degree of the denominator. So, we perform polynomial division to find the equation of the slant asymptote which is \(2x -3\).

Step 4: Finding out Holes in Graph

To find the holes in the graph, we have to factor the numerator and the denominator and find the common factors that cancel out. In this case, the numerator and denominator have no common factors so there are no holes in the graph.

Step 5: Graphing the Function

Use a graphing utility to sketch the graph of the function, marking the asymptotes and noting where the function approaches these asymptotes. Verify there are no holes in the graph and that the function approaches the asymptotes as calculated.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator equals zero, while the numerator is not zero, causing the function to approach infinity. To find them, set the denominator to zero and solve for x.
For the function \( f(x) = \frac{2x^3 - x^2 - 2x + 1}{x^2 + 3x + 2} \), the denominator is \( x^2 + 3x + 2 \). Set it equal to zero:

• \( x^2 + 3x + 2 = 0 \)
• Factor this to \( (x + 1)(x + 2) = 0 \)
• This gives roots \( x = -1 \) and \( x = -2 \)

Therefore, the vertical asymptotes are at \( x = -1 \) and \( x = -2 \). This means as \( x \) approaches -1 or -2, the function will spike up or down sharply.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \( x \) tends to positive or negative infinity. They indicate a constant value that the function approaches when neither the numerator nor denominator affects the growth differently.
To determine their presence, compare the degrees of the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = 0 \).
• If the degrees are equal, divide the leading coefficients for the asymptote \( y = \frac{a}{b} \).
• If the numerator's degree is greater, as in \( f(x) \), there is no horizontal asymptote.

Here, the numerator has degree 3 and the denominator degree 2, indicating no horizontal asymptote exists as both grow at different rates.

Slant Asymptotes

A slant asymptote, or oblique asymptote, is present when the degree of the numerator is exactly one higher than the degree of the denominator, causing the function to lean. To find it, perform polynomial division of numerator by denominator.
For \( f(x) = \frac{2x^3 - x^2 - 2x + 1}{x^2 + 3x + 2} \), note the top degree is 3 and the bottom is 2:

• Divide the numerator by the denominator.
• The result of the division gives the equation of the slant asymptote.
• Here, the slant asymptote is at \( y = 2x - 3 \).

This means the graph approaches the line \( y = 2x - 3 \) when \( x \) is very large in positive or negative directions.

Holes in Graph

A hole in the graph of a rational function represents a missing point usually caused by common factors in both numerator and denominator that cancel each other out. To check for holes:

• Factor both the numerator and denominator.
• See if any common factors exist and cancel.

In the function \( f(x) \), after factorizing both parts:

• No common factors appear.
• This indicates there are no holes in the graph.

Thus, every part of \( x \) trajectory remains defined except at vertical asymptotes, confirming a complete continuous path without interruptions.