Problem 109 - 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. $$f(x)=\frac{2 x^{2}+3}{4-x} ;[-19.8,19.8] \text { by }[-50,25]$$

Step-by-Step Solution

Answer

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

Step 1: Identify the conditions for an oblique asymptote

An oblique asymptote occurs when the degree of the numerator polynomial is exactly one more than the degree of the denominator polynomial. Here, the degree of the numerator \(2x^2+3\) is 2, and the degree of the denominator \(4-x\) is 1. Thus, there is an oblique asymptote.

Step 2: Perform polynomial long division

To find the oblique asymptote, divide \(2x^2+3\) by \(4-x\). Rewrite \(4-x\) as \(-x+4\) and perform long division. You’ll divide \(2x^2+3\) by \(-x+4\), which gives:1. Divide \(2x^2\) by \(-x\) to get \(-2x\).2. Multiply \(-2x\) by \(-x+4\), giving \(2x^2 - 8x\).3. Subtract \(2x^2 - 8x\) from \(2x^2 + 3\) to get \(8x + 3\).4. Divide \(8x\) by \(-x\) to get \(-8\).5. Multiply \(-8\) by \(-x+4\) to get \(8x - 32\).6. Subtract \(8x - 32\) from \(8x + 3\) to get the remainder of \(35\).The quotient obtained is \(-2x - 8\).

Step 3: Write the oblique asymptote equation

From the long division, the quotient part \(-2x - 8\) represents the oblique asymptote. Therefore, the equation of the oblique asymptote is \(y = -2x - 8\).

Step 4: Graph the function and the asymptote using a graphing tool

Use a graphing calculator to plot \(f(x) = \frac{2x^2+3}{4-x}\) and the asymptote \(y = -2x-8\) over the interval \([-19.8, 19.8]\) for \(x\) and \([-50, 25]\) for \(y\). You will observe that as \(x\) approaches positive or negative infinity, the graph of \(f(x)\) gets closer to the line \(y = -2x - 8\).

Key Concepts

Rational Functions

Rational functions combine polynomial expressions through division, forming a relationship like \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). In our example, \( f(x) = \frac{2x^2 + 3}{4 - x} \) is a rational function because it involves dividing polynomials.
When analyzing rational functions, it's crucial to consider asymptotes, which indicate the behavior of the graph when \( x \) approaches extreme values. **Oblique asymptotes** arise when the numerator's degree exceeds the denominator's degree by exactly one. In this case, we have an oblique asymptote since the degree of \( 2x^2 + 3 \) (numerator) is 2, and the degree of \( 4 - x \) (denominator) is 1. This difference of one degree signifies that studying these functions requires understanding advanced graphing concepts like asymptotes.

• **Horizontal asymptotes:** Occur when the degrees of numerator and denominator are equal or the denominator is higher.
• **Vertical asymptotes:** Happen when the denominator is zero at certain points.
• **Oblique asymptotes:** Begin where the degree of the numerator is exactly one more than that of the denominator.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to long division with numbers. It helps in simplifying rational functions and locating oblique asymptotes. In our example, we divided \( 2x^2 + 3 \) by \( 4 - x \) to find the oblique asymptote. Here's how you carry out the process:1. Rewrite the divisor if necessary, e.g., rewrite \( 4 - x \) as \( -x + 4 \).2. Divide the first term of the dividend by the first term of the divisor, which gives the leading term of the quotient.3. Multiply the entire divisor by this term and subtract the result from the dividend.4. Repeat the process with the new dividend obtained after subtraction, until you reach a remainder smaller in degree than the divisor.In our problem, dividing \( 2x^2 \) by \( -x \) gave us \( -2x \), leading to \( -2x - 8 \) as the quotient. This quotient represents the oblique asymptote equation \( y = -2x - 8 \). Understanding polynomial long division is vital, as it provides insight into how a rational function behaves as \( x \) tends towards infinity.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical concepts, such as rational functions and their asymptotes. When faced with a complex function like \( f(x) = \frac{2x^2 + 3}{4 - x} \), using a graphing calculator can reveal key properties at a glance. Here’s how it can help:- **Visual Representation:** Plotting the function and its oblique asymptote \( y = -2x - 8 \) allows you to see where the function approaches this line as \( x \) goes to infinity.- **Setting the Window:** Ensure you choose the correct window settings, like \([-19.8, 19.8]\) for \( x \) and \([-50, 25]\) for \( y \), to capture the behavior accurately.- **Analyze Behavior:** Observe how the function behaves near asymptotes and how it stabilizes as \( x \) increases or decreases. Using a graphing calculator provides a clear view of mathematical behavior, making abstract concepts more tangible, especially for functions with complex behaviors like oblique asymptotes.