Problem 67 - Precalculus Mathematics for Calculus

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

Slant asymptote: \( y = x - 2 \), Vertical asymptote: \( x = 0 \).

Step 1: Identify the Slant Asymptote

A slant asymptote is found when the degree of the numerator is one more than the degree of 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. Therefore, there is a slant asymptote. Divide the numerator by the denominator using polynomial long division to find it. This gives \( x - 2 \).

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. Set the denominator of \( r(x) = \frac{x^2 - 2x - 8}{x} \) equal to zero: \( x = 0 \). So, there is a vertical asymptote at \( x = 0 \).

Step 3: Sketching the Graph

Plot the graph by marking the vertical asymptote at \( x = 0 \) and the slant asymptote of \( y = x - 2 \). To understand the curve's behavior, calculate a few points or consider limits as \( x \to 0^- \) and \( x \to 0^+ \), which will show the behavior approaching the vertical asymptote. The graph should show that the curve approaches the slant asymptote for large \( |x| \), diverging from the vertical asymptote.

Key Concepts

Slant Asymptote

The concept of a slant asymptote often appears when dealing with rational functions. A slant asymptote, also known as an oblique asymptote, occurs when the degree of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator. In our function, \( r(x) = \frac{x^2 - 2x - 8}{x} \),

• the degree of the numerator, \( x^2 - 2x - 8 \), is 2,
• the degree of the denominator, \( x \), is 1.

Hence, the function has a slant asymptote. The quick way to find this slant asymptote is by performing polynomial long division of the numerator by the denominator. Doing this division results in the line \( y = x - 2 \). This line is your slant asymptote. It represents how the graph behaves for very large or very small values of \( x \). The curve of the function will approach this line but never cross it as \( x \to \pm\infty \).
This property helps us understand the end behavior of the graph at the extremes of the \( x\)-axis.

Vertical Asymptote

Vertical asymptotes are a key feature in the graph of rational functions, and they occur at the values of \( x \) that make the denominator zero. They often indicate points of undefined behavior.For \( r(x) = \frac{x^2 - 2x - 8}{x} \), set the denominator equal to zero to find the vertical asymptote:

• Set \( x = 0 \).

The solution tells us there is a vertical asymptote at \( x = 0 \).
This means that as \( x \) approaches 0 from either direction, the value of \( r(x) \) will increase or decrease without bound, giving the graph a vertical line that the curve will not cross or touch.
Understanding vertical asymptotes helps in predicting how a graph behaves as it approaches these lines, showing extreme rise or fall rates on either side.

Graph Sketching

When graphing rational functions, combining information about asymptotes with plotted points gives insight into the curve's behavior. For the function \( r(x) = \frac{x^2 - 2x - 8}{x} \), take the following steps:

• Mark the vertical asymptote at \( x = 0 \)
• Draw the slant asymptote \( y = x - 2 \)

These guide the graph's general shape. To understand the overall behavior, you may evaluate the function at certain points around the vertical asymptote, such as just to the left and right of \( x = 0 \).
This analysis shows how the function behaves as it approaches and diverges from these key lines.
By using limits, you can determine the direction from which the graph approaches the vertical asymptote. Additionally, examining the asymptotic behavior as \( x \to \pm\infty \) reveals that the graph tends to lean towards the slant asymptote. This knowledge aids the drawing and understanding of the function's complete graph, offering a more intuitive picture of how the asymptotes influence the curve.