Problem 49 - Precalculus with Calculus Previews

Question

Construct a rational function \(f(x)=P(x) / Q(x)\) whose graph crosses its slant asymptote twice.

Step-by-Step Solution

Answer

The function is \( f(x) = \frac{x^2 - 2x - 3}{x - 1} \), with slant asymptote \( y = x - 1 \), crossing at \( x = 3 \) and \( x = -1 \).

Step 1: Understand the Components of a Rational Function

A rational function is expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. To have a slant asymptote, the degree of \( P(x) \) must be exactly one higher than the degree of \( Q(x) \).

Step 2: Setting Degrees for P(x) and Q(x)

Choose \( P(x) \) as a quadratic polynomial (degree 2) and \( Q(x) \) as a linear polynomial (degree 1). This will ensure the existence of a slant asymptote.

Step 3: Constructing Polynomials P(x) and Q(x)

Let \( Q(x) = x - 1 \). For \( P(x) \), suppose \( P(x) = x^2 + ax + b \). We'll need the slant asymptote to be crossed, which typically looks like \( y = x + c \), where \( c \) is a constant.

Step 4: Determine the Slant Asymptote

Perform polynomial long division of \( x^2 + ax + b \) by \( x-1 \). Divide it to find that the slant asymptote is \( y = x + c \), where \( c \) is determined by the coefficients found after division.

Step 5: Choose P(x) to Cross the Asymptote Twice

Adjust \( P(x) = x^2 - 2x - 3 \). The polynomial division of \( x^2 - 2x - 3 \) by \( x-1 \) gives a quotient of \( x - 1 \) and remainder \(-4\), thus the slant asymptote is \( y = x - 1 \).

Step 6: Verify That the Graph Crosses the Slant Asymptote Twice

To cross the slant asymptote \( y = x - 1\) twice, set \( f(x) = x - 1 \). Simplifying \( \frac{x^2 - 2x - 3}{x - 1} = x - 1 \) leads to the solution of \( x^2 - 2x - 3 = (x - 1)(x - 1) \), which gives roots showing the crossing occurs at two values of \( x \).

Step 7: Solution Verification

The equation \((x - 1)^2 = 0\) gives roots \( x = 1\). Using the correct polynomial coefficients, factor as \( (x-3)(x+1) \), confirming zeros \( x = 3 \) and \( x = -1 \). These x-values are where the function crosses the slant asymptote.

Key Concepts

Polynomial Division

Polynomial division is like long division that is used with numbers, but it applies to polynomials. When working with rational functions, it becomes essential when determining slant asymptotes. Polynomial division allows us to take one polynomial, such as \( P(x) \), and divide it by another, like \( Q(x) \). This is particularly useful when \( P(x) \) has a higher degree than \( Q(x) \), leading to a quotient that describes the slant asymptote.
For rational functions, accurately dividing the polynomials helps to find out how the function behaves at extreme values of \( x \). The division provides a quotient and a remainder. The quotient can give us a linear equation, representing the slant asymptote, while the remainder becomes less significant as \( x \) becomes large. Think of it as finding the main trend line of a graph, with the remainder acting as small adjustments.
To divide polynomials like \( x^2 - 2x - 3 \) by \( x - 1 \), you'd start by comparing the leading terms: \( x^2 \) divided by \( x \) gives the first term of the quotient. Multiply and subtract this result from \( P(x) \), then repeat with the new polynomial expression until there are no more terms to divide. This process leads to understanding how the function behaves under change.

Slant Asymptote

A slant asymptote occurs in a rational function when the degree of the polynomial in the numerator \( P(x) \) is exactly one more than the degree of the polynomial in the denominator \( Q(x) \). The resulting asymptote is a line, represented by a linear equation from the quotient obtained in the polynomial division.
In simpler terms, slant asymptotes help to predict how a rational function's graph will behave at very high or very low values of \( x \). Unlike vertical or horizontal asymptotes, which are simple lines parallel to the axes, slant asymptotes "tilt," reflecting how the function increases or decreases linearly.
For example, when dividing \( x^2 - 2x - 3 \) by \( x - 1 \), the quotient is found to be \( x - 1 \), indicating the slant asymptote of \( y = x - 1 \). Understanding slant asymptotes helps one predict where the graph will loosely align before the intricate behavior of the polynomial takes over closer parts of the graph.

Graphing Rational Functions

Graphing rational functions involves understanding both the overall trends and the crucial intersections, particularly with any asymptotes. The graph is shaped based on the behavior of its polynomials and their degree differences.
To graph a rational function like \( f(x) = \frac{x^2 - 2x - 3}{x - 1} \), start by identifying its key features:

• Slant asymptote, found through polynomial division, helps anticipate the main direction of the graph. Here, it would be \( y = x - 1 \).
• Zeros of the numerator, \( P(x) = 0 \), like \( x = 3 \) and \( x = -1 \), identify where the function will hit the horizontal line \( y = 0 \).
• Vertical asymptotes, if any, are found where \( Q(x) = 0 \); as seen here there's a vertical asymptote at \( x = 1 \).

These elements guide the sketching of the graph. One needs to plot the points where the function crosses its slant asymptote. In this instance, \( f(x) \) crosses \( y = x - 1 \) at \( x = 3 \) and \( x = -1 \). Carefully plotting these points gives a fuller understanding of how the function behaves across its domain.