Problem 306 - Calculus Volume 1

Question

For \(f(x)=\frac{P(x)}{Q(x)}\) to have an asymptote at \(y=2\) then the polynomials \(P(x)\) and \(Q(x)\) must have what relation?

Step-by-Step Solution

Answer

\(P(x)\) and \(Q(x)\) must have equal degrees, and the leading coefficient of \(P(x)\) must be twice that of \(Q(x)\).

Step 1: Understand Horizontal Asymptotes

Horizontal asymptotes for rational functions occur based on the degrees of the polynomial in the numerator and the polynomial in the denominator. If the degree of \(P(x)\) (numerator) is equal to the degree of \(Q(x)\) (denominator), the horizontal asymptote is \(y=\frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of \(P(x)\) and \(Q(x)\), respectively.

Step 2: Set Leading Coefficients Equal to Desired Asymptote

For the function \(f(x)=\frac{P(x)}{Q(x)}\) to have an asymptote at \(y=2\), the leading coefficients \(a\) of \(P(x)\) and \(b\) of \(Q(x)\) need to satisfy the equation \(\frac{a}{b}=2\). This implies that \(a=2b\).

Step 3: Verify Degree of Polynomials

In order to have the horizontal asymptote at \(y=2\), ensure that the degrees of \(P(x)\) and \(Q(x)\) are equal, since the asymptote calculation relies on comparing the leading coefficients of polynomials of the same degree.

Key Concepts

Rational Functions

Rational functions are devoid of mystery! Simply put, a rational function is a fraction where both the numerator and the denominator are polynomials. If you visualize a fraction like this: \[ f(x) = \frac{P(x)}{Q(x)} \]You can see the polynomials, \(P(x)\) and \(Q(x)\), occupying the top and bottom positions just as we would expect a fraction to have. Rational functions have distinct characteristics, such as vertical asymptotes, holes, and horizontal asymptotes. These appear due to the behavior and relationship between the polynomials in the numerator and denominator.

Horizontal asymptotes give us insights into the function's behavior as \(x\) approaches infinity or negative infinity. When dealing with them, it's critical to observe the degrees of \(P(x)\) and \(Q(x)\). This leads us to our next big topic: polynomials!

Polynomials

To make sense of rational functions, understanding polynomials is key. Polynomials are expressions involving variables and coefficients, composed of terms added up together. Each term consists of:

• A coefficient
• A variable raised to a non-negative integer power

Look at this polynomial example: \[ P(x) = 3x^4 + 2x^3 - x + 7 \]In it, 3, 2, and -1 are coefficients, and the terms with powers are essential to understanding other aspects of the polynomial, like its degree and leading coefficient.

It's the structure of polynomials that provides rational functions with their behavior patterns, including how asymptotes form.

Leading Coefficients

Leading coefficients often hold the key to understanding the end-behavior of polynomials or rational functions. The leading coefficient is the coefficient of the term with the highest power in a polynomial. In simple terms, it’s at the forefront when terms are ordered by their exponent in decreasing sequence.

In the polynomial \(P(x) = 3x^4 + 2x^3 - x + 7\), 3 stands as the leading coefficient because it is paired with the highest power, \(x^4\). While dealing with horizontal asymptotes in rational functions, these leading coefficients are compared, especially when the degrees of the polynomial in the numerator and denominator are equal.

In such cases, it’s their ratio that shapes the function's horizontal asymptote, guiding us in exploring how the graph of \(f(x)\) ultimately flattens out or stretches as \(x\) extends towards infinity.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable present. It's a simple concept but plays a significant role in determining the behavior of polynomials in rational functions. Let’s explore the polynomial \(Q(x) = 5x^4 + 2x^2 + 3\).

• Here, the degree is 4 because the highest power of \(x\) is \(x^4\).

The degree determines the number of potential intercepts and the long-term behavior of a polynomial. In the realm of rational functions, comparing the degree of the numerator polynomial to the denominator polynomial is crucial in figuring out horizontal positions or trends of asymptotes.

When the degrees are equal, the leading coefficients of both polynomials are compared to see where the horizontal asymptote lies. This nesting of concepts allows us to predict whether curves will level off, rise, or dip as we traverse the graph from one side to another.