Problem 59 - Algebra and Trigonometry

Question

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{x^{2}-2 x-8}{x}\)

Step-by-Step Solution

Answer

Vertical asymptote at \(x=0\), slant asymptote \(y=x-2\).

Step 1: Identify the type of function

The function given is a rational function, as it is the quotient of two polynomials: the numerator \(x^2 - 2x - 8\) and the denominator \(x\).

Step 2: Find the Vertical Asymptote

Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points. Here, the denominator is zero at \(x = 0\). Since the numerator \(x^2 - 2x - 8\) is not zero when \(x = 0\), we have a vertical asymptote at \(x = 0\).

Step 3: Simplify the function

Simplify \(r(x) = \frac{x^2 - 2x - 8}{x}\) by dividing each term in the numerator by \(x\). This gives \(r(x) = x - 2 - \frac{8}{x}\).

Step 4: Identify the Slant Asymptote

Slant (or oblique) asymptotes are found when the degree of the numerator is one more than the degree of the denominator. Perform polynomial long division on \(x^2 - 2x - 8\) by \(x\). This results in \(x - 2\), indicating that \(y = x - 2\) is the slant asymptote.

Step 5: Sketch the Graph

The graph of the function includes the following features: a vertical asymptote at \(x=0\), a slant asymptote corresponding to the line \(y=x-2\), and the behavior of the function around these lines. The graph will approach but never touch these asymptotes.

Key Concepts

Vertical Asymptotes in Rational Functions

Vertical asymptotes are an important feature in the graph of rational functions. They represent lines where the function does not exist, meaning the function's value approaches infinity.
These asymptotes occur at the values of x for which the denominator of the rational function equals zero, as long as these are not canceled out by zeros in the numerator.

For the function given in the exercise, \(r(x) = \frac{x^2 - 2x - 8}{x}\), the vertical asymptote is determined by setting the denominator, \(x\), to zero.

• The solution to this is simply \(x = 0\).

This means there is a vertical asymptote at \(x = 0\), meaning the graph will have a discontinuity or undefined point at this location.

Understanding vertical asymptotes helps visualize the boundaries and behavior of the graph around these special locations.

Understanding Slant Asymptotes

Slant asymptotes, also known as oblique asymptotes, appear when the degree of the numerator is exactly one greater than that of the denominator in a rational function.
This makes them different from vertical asymptotes, which depend solely on zeros of the denominator.

• To find the slant asymptote, perform polynomial long division of the numerator by the denominator.

For the function \(r(x) = \frac{x^2 - 2x - 8}{x}\), the degree of the numerator is 2 and the degree of the denominator is 1.
This prompts us to use long division:
Divide \(x^2 - 2x - 8\) by \(x\), which simplifies to \(x - 2\).

• Thus, the slant asymptote is given by the line \(y = x - 2\).

On the graph, \(y = x - 2\) serves as a diagonal line that the function approaches but never reaches as \(x\) becomes very large or very small in magnitude.

Recognizing slant asymptotes on a graph provides insights into the overall shape and direction of the curve at extreme values.

Graphing Rational Functions

When graphing rational functions like \(r(x) = \frac{x^2 - 2x - 8}{x}\), a few key features are essential to consider:

• Vertical and slant asymptotes.
• The behavior of the graph near these asymptotes.

Begin by identifying any asymptotes:
- The vertical asymptote at \(x = 0\) means the graph will approach this line but never cross it.
- The slant asymptote \(y = x - 2\) becomes particularly relevant as \(x\) moves towards positive or negative infinity.

Next, sketch the graph by considering every part of the function:

• The function simplifies to \(x - 2 - \frac{8}{x}\), which indicates typical behaviors to expect.
• As \(x\) approaches large values, the term \(-\frac{8}{x}\) diminishes, and the function approaches the line \(y = x - 2\).

Understanding the characteristic turns and shifts near the asymptotes can help draw an accurate representation.

Remember, the plot will approach but never touch the asymptotes. This understanding provides a fuller sense of the function's behavior and how best to depict it accurately on a graph, allowing for precise and informative visual interpretations.