Problem 23 - Calculus Early Transcendentals

Question

Sketch a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal, vertical, oblique, and curvilinear asymptotes and label them with their equations. Label point(s), if any, where the graph crosses an asymptote. Check your work with a graphing utility. $$ \frac{x^{3}-4 x-8}{x+2} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = -2\); oblique asymptote: \(y = x^2 - 2x + 4\). Verify by graph.

Step 1: Find the Domain

The domain of the rational function is all real numbers except where the denominator is zero. Here, the denominator is \(x+2\), which is zero at \(x=-2\). Hence, the domain is \(x eq -2\).

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur where the function is undefined. Since we found \(x = -2\) makes the denominator zero, there is a vertical asymptote at \(x = -2\).

Step 3: Find Horizontal and Oblique Asymptotes

The degree of the numerator (3) is greater than the degree of the denominator (1), so no horizontal asymptote exists. An oblique asymptote can be found by dividing \(x^3 - 4x - 8\) by \(x + 2\) using polynomial long division. The result is \(x^2 - 2x + 4\), making the oblique asymptote \(y = x^2 - 2x + 4\).

Step 4: Find Intercepts

To find the x-intercept set the numerator equal to zero: \(x^3 - 4x - 8 = 0\). Using synthetic division to check possible roots and factor gives us the intercepts. There is an x-intercept at approximately \((2.5615, 0)\) based on solving using a graphing utility. The y-intercept is found by evaluating \(\frac{x^3 - 4x - 8}{x+2}\) at \(x=0\), which yields \((-4, 0)\).

Step 5: Determine Stationary and Inflection Points

Stationary points occur where the first derivative is zero. Differentiate \(\frac{x^3 - 4x - 8}{x+2}\) to find \(f'(x)\) using the quotient rule, set it equal to zero, and solve for \(x\). Inflection points are found at changes in concavity by setting the second derivative \(f''(x)\) to zero and solving.

Step 6: Sketch the Graph

Using all the asymptotes, intercepts, and points found (stationary and inflection), sketch the graph on paper. The function approaches the vertical asymptote at \(x = -2\) and follows the curve closer to the oblique asymptote \(y = x^2 - 2x + 4\) as \(x\) moves towards infinity.

Step 7: Verify Using a Graphing Utility

Use a graphing calculator or software to plot: \(y = \frac{x^3 - 4x - 8}{x+2}\). Ensure the calculated asymptotes, intercepts, and post-calculated points are consistent with the graph.

Key Concepts

Understanding Asymptotes in Rational Functions

Asymptotes are invisible lines that a graph approaches but never touches. They help define the behavior of a rational function. There are different types of asymptotes you might encounter:

• Vertical Asymptotes: These occur where the rational function is undefined, typically where the denominator equals zero. For the function \( \frac{x^{3}-4x-8}{x+2} \), the vertical asymptote is at \( x = -2 \).
• Horizontal Asymptotes: They show the end behavior of the graph. If the degree of the numerator is less than or equal to the degree of the denominator, a horizontal asymptote exists. Here, since the degree of the numerator (3) is greater than the denominator (1), there isn't a horizontal asymptote.
• Oblique Asymptotes: When the numerator's degree is exactly one more than the denominator's, you find an oblique asymptote. By performing long division on the function \( \frac{x^3 - 4x - 8}{x+2} \), we get the asymptote \( y = x^2 - 2x + 4 \).

These lines are crucial in sketching accurate graphs and predicting function behavior as \( x \) approaches extreme values.

Finding Intercepts of Rational Functions

Intercepts are points where the graph crosses the x-axis and y-axis.

• X-Intercepts: Find these by solving the equation where the numerator equals zero. For the function, set \( x^3 - 4x - 8 = 0 \). Solving this can be complex, and sometimes graphing tools help find approximate roots, such as at \((2.5615, 0)\) for this specific problem.
• Y-Intercepts: These happen when \( x = 0 \). For the function \( \frac{x^3 - 4x - 8}{x+2} \), substitute \( x = 0 \) to find \( (-4, 0) \). This value helps in understanding where the function crosses the y-axis.

Knowing intercepts helps in graphing the function and understanding its range and start points.

Exploring Derivatives to Find Stationary and Inflection Points

Derivatives give insights into function behavior, letting us locate stationary and inflection points.

• Stationary Points: Found where the first derivative equals zero. For rational functions, differentiate using the quotient rule: \( f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2} \). Setting \( f'(x) = 0 \) helps find points where the function's slope is flat.
• Inflection Points: Occur where the second derivative changes sign, indicating changes in concavity. Differentiate the first derivative to get \( f''(x) \), then solve \( f''(x) = 0 \).

These points are essential for accurately sketching graphs, as they highlight key behavioral shifts in the function's curve. Calculating them effectively can determine where the graph bends and flattens.