Problem 110 - A Graphical Approach to Precalculus with Limits

Question

Rational function has an oblique asymptote. Determine the equation of this asymptote. Then use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x^{2}+9}{x+3} ;[-13.2,13.2] \text { by }[-25,25]$$

Step-by-Step Solution

Answer

The oblique asymptote is \(y = x - 3\).

Step 1: Understand the Problem

The problem asks us to find the oblique asymptote of the rational function \(f(x) = \frac{x^2 + 9}{x + 3}\). Oblique (or slant) asymptotes occur when the degree of the numerator is one more than the degree of the denominator. Here, the degree of the numerator is 2 and the degree of the denominator is 1, so an oblique asymptote is expected.

Step 2: Perform Polynomial Long Division

To find the oblique asymptote, we perform polynomial long division of \(x^2 + 9\) by \(x + 3\).1. Divide the leading term of the numerator \(x^2\) by the leading term of the denominator \(x\), which gives \(x\).2. Multiply \(x\) by \(x + 3\) and subtract from the original numerator to get a new remainder: \(x^2 + 9 - (x^2 + 3x) = 0x + 9 - 3x = -3x + 9\).3. Divide the new leading term \(-3x\) by \(x\), which gives \(-3\).4. Multiply \(-3\) by \(x + 3\) and subtract from the existing remainder: \(-3x + 9 - (-3x - 9) = 0\).The quotient is \(x - 3\), which is our oblique asymptote.

Step 3: Write the Equation of the Oblique Asymptote

The equation of the oblique asymptote is given by the result of the polynomial long division, which is \(y = x - 3\).

Step 4: Graph the Function and Asymptote

Use a graphing calculator to graph the function \(f(x) = \frac{x^2 + 9}{x + 3}\) and its oblique asymptote \(y = x - 3\) over the viewing window indicated: \([-13.2, 13.2]\) for the x-axis and \([-25, 25]\) for the y-axis. Observe that the graph of the function approaches the line \(y = x - 3\) as \(x\) moves towards positive or negative infinity.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to how you divide numbers. It is particularly useful when we need to find an oblique asymptote for a rational function. An oblique asymptote appears when the degree of the numerator is exactly one higher than the degree of the denominator. In the rational function \(f(x) = \frac{x^2 + 9}{x + 3}\), the numerator has a degree of 2 and the denominator has a degree of 1, indicating the presence of an oblique asymptote.

The steps for polynomial long division are as follows:

• Divide the leading term of the numerator by the leading term of the denominator. For \(x^2 + 9\) divided by \(x + 3\), this starts with \(x\).
• Multiply this result by the entire divisor \(x + 3\), giving back \(x^2 + 3x\).
• Subtract this from the original numerator to find the new remainder, resulting in \(-3x + 9\).
• Repeat these steps with each new remainder until no remainder remains, or the degree of the remainder is less than the degree of the divisor.

The outcome of the division, \(x - 3\), represents the oblique asymptote equation \(y = x - 3\). Polynomial long division simplifies rational functions and helps find asymptotic behavior.

Rational Functions

Rational functions are expressions that can be written as the quotient of two polynomials. They have the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). These functions can exhibit various types of asymptotes like vertical, horizontal, or oblique, depending on the degrees of the polynomials involved.

In our function, \(f(x) = \frac{x^2 + 9}{x + 3}\), the degree of the numerator is greater than that of the denominator, signaling an oblique asymptote. Asymptotes arise in rational functions due to undefined behavior or growth as \(x\) approaches certain values or extends towards infinity.

• Vertical asymptotes occur when the denominator is zero, indicating points where the function is undefined.
• Horizontal asymptotes occur when the degree of the numerator is the same or lower than the degree of the denominator.
• Oblique asymptotes appear when the degree of the numerator is one greater than that of the denominator.

Understanding rational functions and their asymptotes allows you to grasp the behavior of these functions, essential for graphing and analysis.

Graphing Calculators

Graphing calculators are powerful tools that help visualize mathematical functions and their properties. When handling complex equations like rational functions, these calculators can quickly plot graphs, allowing us to observe asymptotic behavior and other characteristics easily.
For the function \(f(x) = \frac{x^2 + 9}{x + 3}\) and its oblique asymptote \(y = x - 3\), a graphing calculator helps investigate their behavior over a chosen interval. The steps of using a graphing calculator in this context include:

• Input the rational function into the calculator.
• Enter the oblique asymptote as a second equation.
• Specify the viewing window, such as \([-13.2, 13.2]\) for the x-axis and \([-25, 25]\) for the y-axis.
• Plot the graph to visualize how \(f(x)\) approaches the line \(y = x - 3\) as \(x\) moves towards positive or negative infinity.

Graphing calculators simplify the process of exploring and confirming the behavior of mathematical functions, enhancing understanding through visual representation. Understanding how to use these tools effectively is crucial for students learning about complex functions and their graphs.