Problem 36 - Precalculus: Functions and Graphs

Question

Find the oblique asymptote, and sketch the graph of \(f\) $$f(x)=\frac{x^{3}+1}{x^{2}-9}$$

Step-by-Step Solution

Answer

The oblique asymptote is the line \( y = x \).

Step 1: Determine the Degree of the Numerator and Denominator

First, observe the degrees of the numerator and denominator: the numerator has degree 3 \( (x^3)\) and the denominator has degree 2 \( (x^2)\). Since the degree of the numerator is exactly one more than that of the denominator, there will be an oblique asymptote.

Step 2: Perform Polynomial Long Division

To find the oblique asymptote, divide the numerator by the denominator. \( \frac{x^3 + 1}{x^2 - 9} \).Perform polynomial long division: 1. Divide \( x^3 \) by \( x^2 \), getting \( x \).2. Multiply \( x \) by \( x^2 - 9 \), getting \( x^3 - 9x \).3. Subtract \( (x^3 + 1) - (x^3 - 9x) = 9x + 1 \).4. Divide \( 9x \) by \( x^2 \), obtaining no further terms since 9x is lower degree than x^2.

Step 3: Identify the Oblique Asymptote

The quotient from the division gives the equation of the oblique asymptote: \( y = x \). There are no further terms in this case since \( 9x + 1 \) divided by \( x^2 \) does not contribute.

Step 4: Sketch the Oblique Asymptote

On the graph, draw the line \( y = x \) as a dashed line to represent the oblique asymptote. This line will describe the end-behavior of the function; as \( x ightarrow \pm\infty \), the graph of \( f(x) \) will approach this line.

Key Concepts

Polynomial Long Division

Polynomial long division is an essential technique when dealing with rational functions, especially for finding oblique asymptotes. It resembles the long division you might remember from elementary school but applied to polynomial expressions. Here's how it works for finding an oblique asymptote:

• Identify the degrees of both the numerator and the denominator.
• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire denominator by the result and subtract from the numerator.
• Repeat this process with the new polynomial formed after subtraction until the degree of the remaining polynomial is less than the degree of the denominator.

In the given problem, we started with dividing \(x^3 + 1\) by \(x^2 - 9\). The division produced a quotient of \(x\), which determined the oblique asymptote \(y = x\). This is because the remainder term, \(9x + 1\), has a lower degree than the denominator and does not affect the asymptote.

Graph Sketching

Graph sketching is a fundamental skill that allows you to visualize the behavior of a function. To accurately sketch the graph of a rational function like \(f(x) = \frac{x^3 + 1}{x^2 - 9}\), one must take into account asymptotes, intercepts, and curvature.

• Identify Asymptotes: Begin by drawing the oblique asymptote, \(y = x\), as a guiding line across the graph. It's often drawn dashed to indicate it's a boundary the function approaches.
• Find Intercepts: Check where the function intersects the axes. For the y-intercept, evaluate \(f(0)\). The x-intercepts occur where the numerator equals zero, here they are more specifically found by solving \(x^3 + 1 = 0\).
• Examine Symmetry: Determine if the function has any symmetry, which can simplify the sketching process.

As you sketch, keep in mind the overall shape dictated by the asymptote and intercepts. The function's end-behavior, as described below, ensures that the graph approaches the oblique asymptote as \(x\to \pm \infty\).

End Behavior of Functions

Understanding the end behavior of functions provides insight into how the graph behaves as \(x\) becomes very large or very small. For rational functions, this behavior is greatly influenced by the degrees of the numerator and the denominator.

• Comparative Degrees: If the degree of the numerator is higher than the denominator by exactly one, the graph of the function will have an oblique asymptote, as is the case with our example \(f(x) = \frac{x^3 + 1}{x^2 - 9}\).
• Approaching Asymptotes: As \(x\to \pm \infty\), the function \(f(x)\) will grow closer to the line \(y = x\). This means that no matter how far you zoom out, the graph will seem to trace the path of the line \(y = x\).

Being aware of this end behavior helps in making accurate predictions about the graph's structure beyond the typical range of x and ensures a smoother integration of asymptotes into graph sketches.