Problem 114 - A Graphical Approach to Precalculus with Limits

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 it intersects at \(\left( \frac{1}{2}, -\frac{9}{2} \right)\).

Step 1: Identify Degree of Polynomial

The degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, there is an oblique (slant) asymptote.

Step 2: Perform Polynomial Division

Perform long division of \(x^3 - 4x^2 + x + 6\) by \(x^2 + x - 2\). Divide the leading term of the numerator by the leading term of the denominator: \(x^3 / x^2 = x\). Multiply \(x\) by the entire divisor \(x^2 + x - 2\) to get \(x^3 + x^2 - 2x\). Subtract this from the numerator to get \(-5x^2 + 3x + 6\). Repeat the process.

Step 3: Continue Division Process

Divide \(-5x^2\) by \(x^2\) to get \(-5\). Multiply \(-5\) by \(x^2 + x - 2\) to get \(-5x^2 - 5x + 10\). Subtract this from \(-5x^2 + 3x + 6\) to get \(8x - 4\). The final quotient is \(x - 5\) and the remainder is \(8x - 4\).

Step 4: Write Equation of Oblique Asymptote

The oblique asymptote is given by the quotient of the polynomial division without the remainder. Thus, the equation of the oblique asymptote is \(y = x - 5\).

Step 5: Find Intersection Point

Set the function equal to the equation of the oblique asymptote and solve for \(x\). This means solving \(\frac{x^3 - 4x^2 + x + 6}{x^2 + x - 2} = x - 5\). Simplify and solve the equation to find the intersection point. Using a calculator, solve for \(x\) where \(\frac{8x - 4}{x^2 + x - 2} = 0\). This implies \(8x - 4 = 0\) which solves to \(x = \frac{1}{2}\). Substitute back into \(y = x - 5\), leading to \(y = -\frac{9}{2}\). Therefore, the graph intersects its oblique asymptote at \(\left( \frac{1}{2}, -\frac{9}{2} \right)\).

Key Concepts

Long Division

Long division is an essential technique when dealing with polynomials. It helps to simplify complex rational functions, making it possible to find asymptotes and intersections. In polynomial long division, you divide the polynomial in the numerator by the polynomial in the denominator. Like traditional long division, this process involves several key steps:

• Divide the leading terms: Start by dividing the leading term of the numerator by the leading term of the denominator. In this case, divide the highest degree term of the numerator \(x^3\) by the highest degree term of the denominator \(x^2\).
• Multiply and subtract: Multiply the result by the entire divisor, then subtract this product from the original polynomial. This step reduces the degree of the polynomial, inching closer to completing the division.
• Repeat: Continue dividing, multiplying, and subtracting until the remaining polynomial is of a lower degree than the divisor.

After completing these steps, you are left with a quotient, which often determines the behavior of the rational function. Additionally, understanding this process allows us to identify oblique asymptotes and other significant features of the graph.

Oblique Asymptote

The concept of an oblique asymptote becomes crucial when dealing with certain rational functions. An oblique, or slant, asymptote is a line that the graph of the function approaches as \(x\) heads towards infinity. This occurs when the numerator's degree exceeds the denominator's degree by precisely one.

For the given function, the oblique asymptote is derived from the quotient of the polynomial division, disregarding the remainder. After dividing \(x^3 - 4x^2 + x + 6\) by \(x^2 + x - 2\), we find a quotient of \(x - 5\), leading to the oblique asymptote equation \(y = x - 5\).

The graph of the function will get closer and closer to this line as the value of \(x\) increases or decreases substantially. Recognizing this asymptotic behavior assists in sketching the graph's general layout and understanding its long-term trends.

Polynomial Degree

The degree of a polynomial is determined by the highest power of \(x\) present in the expression. This concept is vital when analyzing rational functions, especially to deduce the types of asymptotes the function might have.

In our exercise, the degree of the numerator, \(x^3 - 4x^2 + x + 6\), is 3 due to the \(x^3\) term. The degree of the denominator, \(x^2 + x - 2\), is 2 because of the \(x^2\) term. Since the numerator's degree is greater than the denominator's by one, we determine that the function has an oblique asymptote.

Understanding polynomial degrees is essential in calculus to assess the behavior of functions. It affects the end behavior of graphs, where they intersect, and the number and types of asymptotes. Remember, locating the degree immediately provides valuable insight into the function's structure and helps guide further analysis.