Problem 43 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(39-44,\) (a) find a power function end behavior model for \(f .\) (b) Identify any horizontal asymptotes. $$f(x)=\frac{4 x^{3}-2 x+1}{x-2}$$

Step-by-Step Solution

Answer

The power function end behavior model for \(f\) is that the y-values increase without bound as x approaches infinity or negative infinity. This function does not have a horizontal asymptote.

Step 1: Identify the Highest Degree Polynomial

In given function \(f(x)=\frac{4 x^{3}-2 x+1}{x-2}\) the highest degree polynomial is \(x^3\).

Step 2: Analyze the End Behavior

The degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, we don't have a horizontal asymptote, but a slant (or oblique) asymptote occurs. The end behavior of this function can be determined by considering the ratio of leading coefficients, which is 4 in this case, indicating that the y-values increase without bound (approach infinity) as x approaches infinity or negative infinity.

Step 3: Find Horizontal Asymptotes

No horizontal asymptote in this case. The degree of the numerator is higher than the degree of the denominator, creating a slant asymptote instead.

Key Concepts

Power Function

A power function is an expression of the form \(f(x) = ax^n\), where \(a\) is a coefficient and \(n\) is the exponent of \(x\). When we talk about the "end behavior" of a function, we often focus on how the function behaves as \(x\) approaches infinity or negative infinity. For the given function \(f(x)=\frac{4x^3-2x+1}{x-2}\), the highest degree term in the numerator is \(4x^3\). This term dictates the end behavior of the function as \(x\) becomes very large or very small.
The degree of the numerator is 3, while the denominator's degree is 1. Since the degree of the numerator (3) is higher than that of the denominator (1), the function's end behavior will resemble that of the power function \(4x^3\). This means as \(x\) goes to infinity or negative infinity, \(f(x)\) behaves like \(4x^3\). This is crucial for understanding whether the graph goes upwards or downwards as \(x\) extends to these extremes. This is what we term as the power function end behavior model of \(f(x)\).
Key takeaways:

• The leading term \(4x^3\) determines the end behavior of \(f(x)\).
• The graph rises to infinity or falls to negative infinity in both directions.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches as \(x\) approaches infinity or negative infinity. They are a type of asymptotic behavior, indicating a constant value that \(f(x)\) approaches far off in the positive or negative \(x\)-axis.
In the function \(f(x)=\frac{4x^3-2x+1}{x-2}\), a horizontal asymptote is examined by comparing the degrees of the numerator and denominator. By comparing these degrees, we see that the numerator's degree (3) is greater than the denominator's degree (1).
When the degree of the numerator is greater, the function does not have a horizontal asymptote, as \(f(x)\) will approach infinity or negative infinity rather than approaching a constant line. Instead, this scenario leads to the formation of a slant or oblique asymptote. Hence, in this function, there are no horizontal asymptotes.
Key points:

• The function \(f(x)\) does not have a horizontal asymptote because the numerator's degree is greater than the denominator's.
• No approach to a fixed value indicates infinite rising or falling behavior similar to the highest degree term.

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, occurs when the degree of the numerator of a rational function is exactly one greater than the degree of the denominator. Instead of leveling off to a horizontal line, the graph of the function follows a linear path that slants up or down.
In examining the function \(f(x)=\frac{4x^3-2x+1}{x-2}\), we see that the degree of the numerator is 3, and the degree of the denominator is 1. This difference of one degree implies a slant asymptote.
To determine this slant asymptote, we perform polynomial long division. Dividing \(4x^3 - 2x + 1\) by \(x - 2\) gives a quotient that represents the equation of the slant asymptote. This tells us how the function behaves for large absolute values of \(x\).
The presence of a slant asymptote suggests that as \(x\) grows larger or smaller, \(f(x)\) approximates this linear equation, not a horizontal one. It provides a deeper understanding of how the function approaches infinity.
Important insights:

• The slant asymptote is derived from dividing the polynomial numerator by the linear denominator.
• The function behavior aligns with this asymptote at extreme values of \(x\), indicating a non-horizontal linear pattern.