Problem 93 - A Graphical Approach to College Algebra

Question

Each 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. $$f(x)=\frac{x-x^{2}}{x+2} ;[-9.4,9.4] \text { by }[-15,25]$$

Step-by-Step Solution

Answer

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

Step 1: Perform Division

To find the oblique asymptote of the rational function \(f(x) = \frac{x - x^2}{x + 2}\), we need to perform polynomial long division. Divide \(x - x^2\) by \(x + 2\). Start by dividing the leading term \(-x^2\) of the numerator by the leading term \(x\) of the denominator, giving \(-x\). Multiply \(-x\) by \(x + 2\) to get \(-x^2 - 2x\), and subtract this from \(x - x^2\) to obtain \(x + 2x = 3x\).

Step 2: Continue Division

Continue the long division process by dividing \(3x\) by \(x\), which gives \(3\). Multiply \(3\) by \(x + 2\) to get \(3x + 6\). Subtract this from \(3x\), obtaining the remainder \(-6\).

Step 3: Determine Oblique Asymptote

The quotient from the long division is \(-x + 3\), which represents the oblique asymptote \(y = -x + 3\). Since the division process stops here, \(y = -x + 3\) is the equation of the oblique asymptote.

Step 4: Graph the Function and Asymptote

Use a graphing calculator to plot the function \(f(x) = \frac{x - x^2}{x + 2}\) along with its oblique asymptote \(y = -x + 3\). Set the window to \([-9.4, 9.4] \times [-15, 25]\) to capture the relevant details of the graph.

Key Concepts

Oblique Asymptote

An oblique asymptote is a slanted line that a rational function approaches but never touches as the input values increase or decrease towards infinity. This occurs when the degree of the polynomial in the numerator is one more than the degree of the polynomial in the denominator. In the function given, \( f(x) = \frac{x-x^2}{x+2} \), the numerator is quadratic (degree 2), and the denominator is linear (degree 1), making an oblique asymptote possible. These asymptotes are sometimes called slant asymptotes and can reveal significant insights into the behavior of a function outside the central range of interest. Understanding the concept of oblique asymptotes enables you to predict how graphs of certain rational functions behave as \( x \) moves towards positive or negative infinity.

Polynomial Long Division

Polynomial long division is a mathematical technique used to divide two polynomials, similar to traditional long division with numbers. It helps find the quotient and remainder when dividing polynomials. For our function \( f(x) = \frac{x-x^2}{x+2} \), we use polynomial long division to determine the oblique asymptote. Here's how:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this result.
• Subtract this product from the original dividend.
• Repeat these steps with the new polynomial formed until a remainder is found.

With these steps, dividing \( -x^2 + x \) by \( x + 2 \), we find the quotient \( -x + 3 \), which directly gives us the equation of the oblique asymptote: \( y = -x + 3 \).

Graphing Calculator

Using a graphing calculator is an effective way to visualize complex functions like rational functions and their asymptotes. Graphing calculators help plot both the function \( f(x) = \frac{x-x^2}{x+2} \) and its asymptotes on the same coordinate plane, offering a clear picture of how the function behaves near these asymptotes. To graph the function correctly:

• Enter the rational function equation into the calculator.
• Set the viewing window to the specified range, here ([-9.4, 9.4] \times [-15, 25]).
• Simultaneously plot the equation of the oblique asymptote \( y = -x + 3 \).

This visualization shows the slant nature of the oblique asymptote and how it guides the end behavior of the rational function.

Asymptote Equation

The asymptote equation in the context of this problem refers to the formula we derive from the polynomial long division of the rational function. The asymptote equation indicates a line that serves as a guide for the behavior of the function when \( x \) values are very large or very small.To derive the equation for the oblique asymptote from the function \( f(x) = \frac{x-x^2}{x+2} \), we followed the division process, yielding the quotient \( y = -x + 3 \), which is a linear equation. This line is what the curve of \( f(x) \) follows as it stretches towards infinity in positive and negative directions. Understanding the asymptote equation helps in predicting and sketching the graph of the function efficiently, especially when using technology like a graphing calculator.