Problem 70 - Algebra and Trigonometry

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: \(x = \frac{1}{2}\) and \(x = -1\).

Step 1: Identify the Degree of Numerator and Denominator

First, look at the degrees of the numerator and the denominator of the function. The numerator is \(x^3+4\), which has a degree of 3, and the denominator is \(2x^2+x-1\), which has a degree of 2.

Step 2: Determine the Existence of a Slant Asymptote

Since the degree of the numerator (3) is one more than the degree of the denominator (2), a slant asymptote exists for the given rational function.

Step 3: Long Division to Find the Slant Asymptote

Perform polynomial long division of \(x^3 + 4\) by \(2x^2 + x - 1\). This division will give you the equation of the slant asymptote. The quotient is \(\frac{1}{2}x\) with a remainder. Thus, the slant asymptote is \(y = \frac{1}{2}x\).

Step 4: Set Denominator Equal to Zero to Find Vertical Asymptotes

To find the vertical asymptotes, set the denominator equal to zero: \[ 2x^2 + x - 1 = 0 \]Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 2, b = 1, c = -1\). This gives results: \(x = \frac{-1 \pm \sqrt{9}}{4}\), leading to vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\).

Step 5: Prepare to Sketch the Graph

Consider the slant asymptote, \(y = \frac{1}{2}x\), and the vertical asymptotes, \(x = \frac{1}{2}\) and \(x = -1\). The slant asymptote suggests the behavior as \(x\) approaches infinity and negative infinity. The vertical asymptotes indicate points where the function is undefined, and near which the function will increase or decrease without bound.

Step 6: Sketch the Rational Function

Based on all the details: sketch the function showing the slant asymptote approaching near infinity, and ensure the vertical asymptotes \(x = \frac{1}{2}\) and \(x = -1\) are where the function is undefined. Check a few points to see the curvature behavior around these asymptotes and draw the function accordingly.

Key Concepts

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, occurs in a rational function when the degree of the numerator is one more than that of the denominator. This means the graph of the function will approach a line that is neither horizontal nor vertical as x heads towards positive or negative infinity.

To find the slant asymptote for a given function, we perform polynomial long division on the function. For example, in the rational function \(r(x) = \frac{x^3+4}{2x^2+x-1}\), the numerator has a degree of 3, while the denominator has a degree of 2. By dividing \(x^3 + 4\) by \(2x^2 + x - 1\), we discovered that the slant asymptote is \(y = \frac{1}{2}x\). This line describes the end behavior of the function, indicating a linear trajectory as the function values grow large or small.

Vertical Asymptotes

Vertical asymptotes occur in a rational function wherever the denominator is zero and the numerator is non-zero at those points. They indicate that the function approaches positive or negative infinity as it gets closer to a certain x-value.

To identify vertical asymptotes in the function \(r(x) = \frac{x^3+4}{2x^2+x-1}\), we need to solve for when \(2x^2 + x - 1 = 0\). Using the quadratic formula, we find the solutions to be \(x = \frac{1}{2}\) and \(x = -1\). At these points, the function becomes undefined, and its value tends toward infinity.

• For rational functions, each vertical asymptote corresponds to a root of the denominator that does not cancel with the numerator.
• The curve on the graph will tend to rise or fall sharply as it approaches these x-values.

Polynomial Long Division

Polynomial long division is a technique used to divide polynomial expressions, similar to long division with numbers. It helps to find slant asymptotes or simplify expressions.

Let's use our example: divide \(x^3 + 4\) by \(2x^2 + x - 1\). Think of this as a way to "fit" the smaller polynomial into the larger one. We repeatedly divide the leading terms, subtract, and bring down the next terms, much like traditional long division.

The result for our function is \(\frac{1}{2}x\) with a remainder, suggesting that the non-remainder part forms the slant asymptote's equation: \(y = \frac{1}{2}x\).

• This quotient (\(\frac{1}{2}x\)) approximates the behavior of the function graph as \(x\) heads to infinity.
• The remainder in the division affects the approximation's accuracy but typically diminishes with larger x-values.

Rational Function Graphing

Graphing a rational function involves combining several aspects: the location of vertical and slant (or horizontal) asymptotes, x-intercepts, y-intercepts, and overall behavior.

For instance, with \(r(x) = \frac{x^3+4}{2x^2+x-1}\), we already discovered its slant asymptote \(y = \frac{1}{2}x\) and vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\). These guides indicate the boundaries and directional tendencies of the function.

• As \(x\) approaches each vertical asymptote, the graph should diverge to \(+\infty\) or \(-\infty\).
• As \(x\) approaches \(+\infty\) or \(-\infty\), the function approaches its slant asymptote.
• Sketching the graph helps visualize these asymptotic behaviors.
• Plot additional points near key features to verify the curve's shape and direction.