Problem 50 - Essential Calculus Early Transcendentals

Question

\(49-52=\) The line \(y=m x+b\) is called a slant asymptote if \(f(x)-(m x+b) \rightarrow 0\) as \(x \rightarrow \infty\) or \(x \rightarrow-\infty\) because the vertical distance between the curve \(y=f(x)\) and the line \(y=m x+b\) approaches 0 as \(x\) becomes large. Find an equation of the slant asymptote of the function and use it to help sketch the graph. [ For rational functions, a slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator. To find it, use long division to write $$f(x)=m x+b+R(x) / Q(x) ]$$ $$y=\frac{1+5 x-2 x^{2}}{x-2}$$

Step-by-Step Solution

Answer

The slant asymptote is \(y = -2x + 1\).

Step 1: Identify the Degrees

The given function is \( f(x) = \frac{1 + 5x - 2x^2}{x - 2} \). The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is exactly one more than the degree of the denominator, there is a slant asymptote.

Step 2: Set Up Long Division

To find the slant asymptote, we will divide \(-2x^2 + 5x + 1\) by \(x - 2\). Set up the division by writing \(-2x^2 + 5x + 1\) under the division symbol and \(x - 2\) outside.

Step 3: Perform the Division

1. Divide the first term of the numerator by the first term of the denominator: \(-2x^2 / x = -2x\). 2. Multiply \(-2x\) by \(x - 2\): \((-2x)(x - 2) = -2x^2 + 4x\).3. Subtract \(-2x^2 + 4x\) from \(-2x^2 + 5x\), giving you \(x\).4. Bring down the next term (\(+1\)) to get \(x + 1\).5. Repeat: Divide \(x / x = 1\).6. Multiply: \((1)(x - 2) = x - 2\).7. Subtract: \((x + 1) - (x - 2) = 3\). Remaining 3 represents the remainder.

Step 4: Write the Function in Division Form

From the division, \(f(x)\) can be expressed as \(f(x) = -2x + 1 + \frac{3}{x - 2}\). Here, the terms \(-2x + 1\) form the equation of the slant asymptote, while \(\frac{3}{x - 2}\) becomes negligible as \(x\) approaches infinity.

Step 5: Identify the Slant Asymptote

The slant asymptote is the equation \(y = -2x + 1\), obtained directly from dividing the terms of the polynomial.

Step 6: Sketch the Function Graph

To sketch the graph of \(f(x)\), draw the line \(y = -2x + 1\) as the slant asymptote. The function \(f(x)\) will approach this line as \(x\) moves towards positive or negative infinity.

Key Concepts

Rational Functions

Rational functions are expressions formed by dividing two polynomials. These functions take the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). These types of functions are pivotal in many areas of algebra because they can model real-world situations and showcase various asymptotic behaviors that are not present in simpler polynomial functions.
For instance, in our exercise, the given function is \(f(x) = \frac{1 + 5x - 2x^2}{x - 2}\). Here, the numerator \(1 + 5x - 2x^2\) is of degree 2, and the denominator \(x - 2\) is of degree 1. This setup provides an interesting situation where the degree of the numerator exceeds that of the denominator by exactly one, suggesting the presence of a slant (or oblique) asymptote. Asymptotes, in general, help us understand the behavior of rational functions as they tend to infinity or negative infinity. This understanding makes rational functions crucial for analyzing limits and asymptotic behavior.

Polynomial Division

To identify asymptotes, especially slant ones, performing polynomial division is essential. From our exercise, the method used is long division, a technique similar to the division of numbers but applied to polynomials. It helps in simplifying the rational function into a form that reveals potential asymptotes.

• First, arrange the polynomials. The numerator \(-2x^2 + 5x + 1\) is divided by the denominator \(x - 2\).
• Divide the leading term of the numerator by the leading term of the denominator: \(-2x^2 / x = -2x\).
• Multiply the entire divisor \(x - 2\) by \(-2x\) and subtract the result from the original numerator.
• Bring down the next term and repeat the steps until you can't continue.

This division yields \(f(x) = -2x + 1 + \frac{3}{x - 2}\), where \(-2x + 1\) becomes the key to finding the slant asymptote. Division simplifies complex functions into more interpretable forms, shedding light on their asymptotic nature.

Graph Sketching

Graph sketching is visualizing the behavior of a function. In the context of rational functions, sketching includes finding all intercepts, asymptotic behavior, and any turning points to create a comprehensive outline of the graph's storyline.
In our exercise, after determining the slant asymptote \(y = -2x + 1\), this line should be drawn as the primary guide for the graph of the function. The function itself, \(f(x) = \frac{1 + 5x - 2x^2}{x - 2}\), will approach this line as \(x\) moves towards positive or negative infinity.

• Start by plotting the slant asymptote.
• Determine the intercepts by calculating where the function equals zero and where it intersects the axes.
• Consider the function's behavior near the vertical asymptote or any points where the denominator equals zero.

These steps will provide a clearer picture of where the graph travels and its behavior, giving a visual representation of the function's dynamics.

Asymptotic Behavior

Asymptotic behavior in functions describes how they behave as they approach specific limits, either towards infinity or near undefined points. Different kinds of asymptotes illustrate distinct behaviors of a function. For rational functions, common types include vertical, horizontal, and slant asymptotes.
In our exercise, a slant asymptote exists because the degree of the numerator is one more than that of the denominator. Slant asymptotes indicate that as \(x\) approaches infinity or negative infinity, the function will closely resemble a linear expression. The slant asymptote for \(f(x) = \frac{1 + 5x - 2x^2}{x - 2}\) is \(y = -2x + 1\).
This describes the growth pattern of the function as it tries to become the straight line \(y = -2x + 1\). Asymptotic analysis is crucial for understanding unbounded behavior, providing insight into how a function behaves at extremes without having to plot infinite points.