Problem 91 - 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. $$f(x)=\frac{2 x^{2}+3}{4-x} ;[-18.8,18.8] \text { by }[-50,25]$$

Step-by-Step Solution

Answer

The oblique asymptote is \(y = -2x - 8\).

Step 1: Identify division for finding oblique asymptote

The oblique asymptote of a rational function occurs when the degree of the numerator is exactly one more than the degree of the denominator. Here, the numerator is \(2x^2+3\) (degree 2), and the denominator is \(4-x\) (degree 1). Since the criteria is met, we'll use polynomial division to divide \(2x^2 + 3\) by \(4-x\).

Step 2: Perform polynomial long division

Start dividing \(2x^2\) by \(-x\), giving \(-2x\). Multiply \(-2x\) by \(4-x\) to get \(-2x(4-x) = -8x + 2x^2\), and subtract this from the original \(2x^2 + 3\) leading to \(8x + 3\). Then divide \(8x\) by \(-x\), giving \(-8\). Multiply \(-8\) by \(4-x\) to get \(-32 + 8x\), and subtract this from \(8x + 3\) to find 35, the remainder.

Step 3: Write the equation of the oblique asymptote

After division, you have \(-2x - 8 + \frac{35}{4-x}\). For the oblique asymptote, we consider what happens as \(x\) approaches infinity or negative infinity. The \(\frac{35}{4-x}\) becomes insignificant, so the equation of the oblique asymptote is \(y = -2x - 8\).

Step 4: Graph the function and the asymptote

Use a graphing calculator to input \(f(x)=\frac{2x^2+3}{4-x}\) and the asymptote \(y = -2x - 8\). Adjust the window settings to \([-18.8,18.8]\) for \(x\) and \([-50,25]\) for \(y\). You should see the curve of the rational function approaching the line of the asymptote as \(x\) goes to positive or negative infinity.

Key Concepts

Rational Functions

Rational functions form an essential class of mathematical functions. A rational function is generally expressed as the ratio

• Numerator: A polynomial
• Denominator: Another polynomial

In this format, you can see that rational functions are simply fractions involving polynomials. For example, in the function given in the original exercise, the numerator is a polynomial: \(2x^2 + 3\), and the denominator is another polynomial: \(4-x\).
Rational functions are valuable because they exhibit rich behavior, including asymptotic behavior, as observed in our exercise. Understanding these functions requires grasping the relationship between the degrees of the terms in the numerator and the denominator. As per the problem, the degree of the numerator (2) is one more than that of the denominator (1), which is critical for identifying oblique asymptotes.

Polynomial Division

To find oblique asymptotes, the polynomial division is an indispensable technique. Here, the important goal is to divide the numerator polynomial by the denominator polynomial and identify what remains. The division rendition of the problem features terms \(2x^2 + 3\) divided by \(4-x\). This process is often pursued through long division or synthetic division.
The process follows these steps:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this quotient result.
• Subtract the product from the current numerator.
• Repeat the process with any resulting remaining expression.

Through polynomial division in the given problem, we ascertain the components \(-2x - 8\) as our initial focus, ignoring any remainder for large \(x\). This simplifies constructing the oblique asymptote's equation.

Graphing Calculator

When it comes to visualizing complex functions, graphing calculators are an invaluable tool. They allow you to input functions, view graphs, and manipulate axes to observe intricate behaviors. In our original exercise, a graphing calculator plays a pivotal role in rendering the rational function \(f(x)=\frac{2x^2+3}{4-x}\) and its oblique asymptote \(y = -2x - 8\).
To effectively use a graphing calculator, follow these steps:

• Input the rational function as it is.
• Add in the equation of the asymptote separately.
• Set the window according to exercise requirements:
  • \(x\) from [-18.8, 18.8]
  • \(y\) from [-50, 25]
• Examine how the graph of the function approaches the asymptote as \(x\) increases or decreases.

Using these visual tools helps solidify understanding of asymptotic behavior as real life graphical representation of mathematical principles.

Asymptote Equation

Asymptotes are fundamental concepts in the analysis of rational functions' behaviors. Especially for rational functions, oblique asymptotes, also known as slant asymptotes, appear when the degree of the numerator is precisely one more than the denominator’s. This is exactly what we encounter in the problem provided.
The steps in determining the equation of the asymptote include performing polynomial division of the entire rational function. Upon division, the equation \(y = -2x - 8\) emerges as the oblique asymptote, representing the line that our rational function approaches infinitely. This discovery excludes the fraction remainder, as it vanishes toward zero at infinity, solidifying the line of interest.
Understanding these equations is crucial since they tell how the function behaves without calculating high numbers. Known graphs and plotted results then visually reinforce these observations. Asymptotes ensure we comprehend both local and global function tendencies.