Problem 96 - A Graphical Approach to College Algebra

Question

Consider the rational function $$f(x)=\frac{x^{3}-4 x^{2}+x+6}{x^{2}+x-2}$$ Divide the numerator by the denominator and use the method of Example 3 to determine the equation of the oblique asymptote. Then, determine the coordinates of the point where the graph of \(f\) intersects its oblique asymptote. Use a calculator to support your answer.

Step-by-Step Solution

Answer

The oblique asymptote is \( y = x - 5 \) and intersects \( f(x) \) at \( \left( \frac{1}{2}, -\frac{9}{2} \right) \).

Step 1: Identify the Division

We need to perform polynomial long division on the function \( f(x) = \frac{x^3 - 4x^2 + x + 6}{x^2 + x - 2} \). The aim is to find the quotient, which represents the equation of the oblique asymptote.

Step 2: Perform Polynomial Long Division

Begin dividing the numerator \( x^3 - 4x^2 + x + 6 \) by the denominator \( x^2 + x - 2 \).- Divide the first term of the numerator \( x^3 \) by the first term of the denominator \( x^2 \) to get \( x \).- Multiply \( x \) by \( x^2 + x - 2 \) to get \( x^3 + x^2 - 2x \).- Subtract \( x^3 + x^2 - 2x \) from the original numerator \((x^3 - 4x^2 + x + 6)\) to find the new polynomial: \(-5x^2 + 3x + 6 \).- Repeat the process: Divide \(-5x^2\) by \(x^2\) to get \(-5\).- Multiply \(-5\) by \(x^2 + x - 2\) to obtain \(-5x^2 - 5x + 10\).- Subtract \(-5x^2 - 5x + 10\) from \(-5x^2 + 3x + 6\) to get \(8x - 4\).The quotient, \( x - 5 \), determines the oblique asymptote: \( y = x - 5 \).

Step 3: Find Intersection with Oblique Asymptote

To find the point where the graph intersects its oblique asymptote, set \( f(x) = x - 5 \). This means solving:\[ \frac{x^3 - 4x^2 + x + 6}{x^2 + x - 2} = x - 5 \]Multiply both sides by \( x^2 + x - 2 \) to clear the fraction:\[ x^3 - 4x^2 + x + 6 = (x - 5)(x^2 + x - 2) \]Since we just did the division, the remainder was \(8x - 4\). Therefore, intersection occurs when the remainder equals zero: \[ 8x - 4 = 0 \]Solve for \( x \):\[ 8x = 4 \]\[ x = \frac{1}{2} \]

Step 4: Calculate Y-Coordinate of Intersection

Substitute \( x = \frac{1}{2} \) back into the asymptote equation \( y = x - 5 \):\[ y = \frac{1}{2} - 5 = -\frac{9}{2} \]Thus, the coordinates of the intersection point are \( \left( \frac{1}{2}, -\frac{9}{2} \right) \).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide two polynomials, similar to how you might divide two numbers. This technique is particularly useful when dealing with rational functions, where the numerator is divided by the denominator. The goal is to express the division in the form of a quotient and a remainder.

### Steps for Polynomial Long Division To perform polynomial long division, follow these steps:

• Divide the leading term of the numerator by the leading term of the denominator. This gives the first term of the quotient.
• Multiply the entire denominator by this term, and subtract the result from the original numerator.
• The remainder will be a polynomial of a lower degree than the original numerator.
• Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

Upon completion, you'll have a quotient, which can include a constant term and sometimes a variable term as well. The remainder can be ignored when identifying the asymptote, which is our next concept to explore.

Oblique Asymptote

An oblique asymptote, also known as a slant asymptote, occurs when the degree of the polynomial in the numerator is exactly one higher than the degree of the polynomial in the denominator. It represents the behavior of the graph of a rational function as the input values become very large or very small.

### Understanding Oblique Asymptotes To find an oblique asymptote, use the quotient obtained from polynomial long division:

• The oblique asymptote is given by the linear part of the quotient (ignoring the remainder).
• In our example, dividing the cubic numerator by the quadratic denominator provides a quotient of a linear polynomial, which is the oblique asymptote.

For the function provided, **f(x)**, the quotient was calculated as **x - 5**, thus the oblique asymptote is the line **y = x - 5**.

As the values of **x** move towards infinity, the graph of the function approaches the line of the oblique asymptote closely, demonstrating the function's direction.

Intersection Point

The intersection point refers to the specific coordinates where the graph of the function crosses its asymptote. Finding this point shows a special feature of the function's behavior against its asymptotic behavior.

### Calculating the Intersection PointTo determine the intersection point, set the expression for the rational function equal to the equation of the asymptote:

• Clear the fraction by multiplying through by the denominator of the rational function.
• Recognize when the remainder from the division is equal to zero. This condition is used to find the x-coordinate of the intersection.
• Substitute this x-value into the asymptote equation to find the corresponding y-coordinate.

In our exercise, this process gave us the point \( \left( \frac{1}{2}, -\frac{9}{2} \right) \), indicating the graph crosses the asymptote line **y = x - 5** at this specific coordinate. Understanding the intersection point gives insight into how the graph relates to its asymptotic path within the coordinate plane.