Problem 70 - Precalculus Mathematics for Calculus

Question

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+4}{2 x^{2}+x-1}$$

Step-by-Step Solution

Answer

Slant asymptote: \(y = \frac{1}{2}x\); Vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\).

Step 1: Identify the degree of the polynomial

The degree of the numerator polynomial, \(x^3 + 4\), is 3, and the degree of the denominator polynomial, \(2x^2 + x - 1\), is 2. Since the degree of the numerator is one higher than the degree of the denominator, there is a slant asymptote in the graph.

Step 2: Find the Slant Asymptote

Perform polynomial long division of \(x^3 + 4\) by \(2x^2 + x - 1\).1. Divide the leading term of the numerator \(x^3\) by the leading term of the denominator \(2x^2\) to get \(\frac{1}{2}x\).2. Multiply \(\frac{1}{2}x\) by the divisor \(2x^2 + x - 1\), subtract the resulting polynomial \(x^3 + \frac{1}{2}x^2 - \frac{1}{2}x\) from the original polynomial \(x^3 + 4\).3. The result is \(\frac{1}{2}x^2 + \frac{1}{2}x + 4\).4. Repeat the process: divide the leading term \(\frac{1}{2}x^2\) by \(2x^2\) to find the next term of the quotient, and continue until the degree of the remainder is less than that of the divisor.The slant asymptote is given by \(\frac{1}{2}x\).

Step 3: Find the Vertical Asymptotes

Set the denominator equal to zero and solve for \(x\):\[2x^2 + x - 1 = 0\]This is a quadratic equation. Solve it using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where \(a = 2\), \(b = 1\), and \(c = -1\):\[x = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4}\]The solutions are \(x = \frac{1}{2}\) and \(x = -1\).Therefore, the vertical asymptotes are \(x = \frac{1}{2}\) and \(x = -1\).

Step 4: Sketching the Graph

When sketching the graph:1. Draw vertical dashed lines at \(x = \frac{1}{2}\) and \(x = -1\) to indicate the vertical asymptotes.2. Draw a slant asymptote line with the equation \(y = \frac{1}{2}x\).3. Plot sample points including some near and around the asymptotes to understand the behavior of the function.4. The graph should approach the vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\) and follow the slant asymptote as \(x\) approaches positive or negative infinity.

Key Concepts

Polynomial Division

Polynomial division is a crucial tool when dealing with rational functions, particularly for identifying slant asymptotes. Just like numerical long division, we focus on dividing the terms with the highest degree first.

In our example, we start with the numerator and the denominator:

• The numerator is an expression of the third degree: \(x^3 + 4\).
• The denominator is of the second degree: \(2x^2 + x - 1\).

Since the degree of the numerator (3) is exactly one more than the degree of the denominator (2), we know that a slant (or oblique) asymptote will occur.
To find this, perform polynomial long division:

• Divide the leading term \(x^3\) of the numerator by the leading term \(2x^2\) of the denominator.
• The first term of the quotient is \(\frac{1}{2}x\).
• Continue the division process until the degree of the remainder is less than that of the divisor.

The result is the equation of the slant asymptote, \(y = \frac{1}{2}x\). This represents a straight line that the graph of the rational function will approach as \(x\) moves towards positive or negative infinity.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at values of \(x\) that make the denominator zero, as long as the numerator is not zero at these values. These asymptotes represent locations where the function heads towards infinity.

For the function \(r(x)=\frac{x^{3}+4}{2x^{2}+x-1}\), we need to find the roots of the denominator \(2x^2 + x - 1 = 0\). Solving this equation will reveal the points where vertical asymptotes occur.
Using the quadratic formula:

• Identify \(a = 2\), \(b = 1\), and \(c = -1\).
• Calculate the discriminant: \(b^2 - 4ac = 1 + 8 = 9\).
• Solving gives the vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\).

These values show where \(r(x)\) is undefined, and the graph approaches either positive or negative infinity. When sketching, draw dashed vertical lines at these \(x\)-values to represent these asymptotes.

Graphing Rational Functions

Graphing rational functions can be straightforward once you understand the asymptotic behaviors, zeros, and general shape of the graph. The rational function \(r(x)=\frac{x^{3}+4}{2x^{2}+x-1}\) is exemplified by its slant and vertical asymptotes.

Steps to graph the function:

• Start by plotting the vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\). These are represented by dashed vertical lines.
• Draw the slant asymptote \(y = \frac{1}{2}x\), which is a diagonal line indicating the function's end behavior as \(x\) moves towards infinity.
• Choose strategic points around the asymptotes to plot, allowing us to understand the behavior near these critical lines.
• Ensure the graph approaches the asymptotes, getting infinitely closer without ever crossing them.

Plotting points near the asymptotes and understanding when the function heads up or down helps create an accurate representation of the function's behavior, providing an insightful view into the nature of rational functions.