Problem 36 - Precalculus with Calculus Previews

Question

Find the point where the graph of \(f\) crosses its slant asymptote. Use a graphing utility to obtain the graph of \(f\) and the slant asymptote in the same coordinate plane. $$ f(x)=\frac{x^{3}+2 x-4}{x^{2}} $$

Step-by-Step Solution

Answer

The graph crosses its slant asymptote at the point (2, 2).

Step 1: Determine the Slant Asymptote

To find the slant (or oblique) asymptote of a rational function, divide the numerator by the denominator if the degree of the numerator is exactly one degree higher than the degree of the denominator. For the function \(f(x) = \frac{x^3 + 2x - 4}{x^2}\), perform polynomial long division of \(x^3 + 2x - 4\) by \(x^2\). The result of the division is \(x + 0\), which means the slant asymptote is \(y = x\). Note that the remainder will be irrelevant for the asymptote.

Step 2: Set Up the Equation to Find Intersection

To find the point where the graph of \(f(x)\) crosses its slant asymptote, set \(f(x)\) equal to the asymptote equation. Thus, solve the equation \(\frac{x^3 + 2x - 4}{x^2} = x\).

Step 3: Solve the Equation for x

Multiply both sides by \(x^2\) to eliminate the fraction: \(x^3 + 2x - 4 = x^3\). Subtract \(x^3\) from both sides to simplify:\(2x - 4 = 0\).

Step 4: Solve for x and Find y-coordinate

Solve \(2x - 4 = 0\):Add 4 to both sides: \(2x = 4\).Divide by 2: \(x = 2\).To find the y-coordinate, substitute \(x = 2\) back into the asymptote equation \(y = x\):\(y = 2\).

Step 5: Conclusion – Intersection Point

The graph of \(f(x)\) crosses its slant asymptote at the point \((2, 2)\).

Key Concepts

Polynomial Long Division

Understanding polynomial long division is essential when dealing with rational functions, especially those where the numerator's degree is higher than the denominator's degree. In this exercise, we have the function \( f(x) = \frac{x^3 + 2x - 4}{x^2} \). Notice that the numerator, a cubic polynomial, has a degree one greater than the denominator, a quadratic polynomial.

Polynomial long division is a systematic method to divide these polynomials much like how we handle regular numbers. Here's how it works with the example:

• Divide the first term of the numerator \(x^3\) by the first term of the denominator \(x^2\), which gives \(x\).
• Multiply the entire divisor \(x^2\) by this result, \(x\), yielding \(x^3\).
• Subtract \(x^3\) from the original polynomial's numerator to get the next working polynomial term.
• Repeat the process until the degree of the remainder is less than the divisor's degree.

The result \(x + 0\) indicates our slant asymptote is \(y = x\). The remainder in polynomial long division does not contribute to the slant asymptote, but it can be part of the simplified rational expression.

Rational Functions

Let's delve deeper into what rational functions are and why they're important in mathematics. A rational function is essentially a fraction in which both the numerator and the denominator are polynomials. In our exercise, \( f(x) = \frac{x^3 + 2x - 4}{x^2} \), we deal with a rational function where the degree of the numerator (3) is greater than the degree of the denominator (2).

This particular setup results in a slant asymptote since the numerator's degree is exactly one more than the denominator's. Rational functions can have:

• Vertical asymptotes, determined by the zeros of the denominator that are not canceled by the numerator.
• Horizontal or slant (oblique) asymptotes, providing insights into the function's behavior as \(x\) approaches positive or negative infinity.

In this case, as \(x\) grows very large or very small, \(f(x)\) starts to behave like \(y = x\), the slant asymptote. This feature is pivotal in graphically analyzing rational functions.

Graphical Analysis

Graphical analysis is an invaluable tool in visualizing the behavior of rational functions and their respective asymptotes. By graphing \( f(x) = \frac{x^3 + 2x - 4}{x^2} \) along with its slant asymptote \(y = x\), we can better understand the function's trajectory as \(x\) tends towards infinity.

To determine where the graph intersects its slant asymptote:

• Equate the function to the formula of the asymptote, solving \( \frac{x^3 + 2x - 4}{x^2} = x \).
• Clear the fraction by multiplying both sides by \(x^2\), resulting in \(x^3 + 2x - 4 = x^3\).
• This simplifies to \(2x - 4 = 0\), allowing us to solve quickly for \(x\): \(x = 2\).

By substituting \(x = 2\) into the equation of the asymptote \(y = x\), we find the y-coordinate, which is \(y = 2\). Therefore, the intersection point is \((2, 2)\). This graphical intersection point provides a specific location where the function and its slant asymptote meet on the graph, offering deeper insight into the function's behavior.