Problem 41 - Single Variable Calculus

Question

In Exercises 39-42, find the slant asymptotes of the graphs of the function. Then sketch the graph of the function. $$ f(x)=\frac{x^{2}-2 x-3}{2 x-2} $$

Step-by-Step Solution

Answer

The slant asymptote of the function \(f(x) = \frac{x^2 - 2x -3}{2x-2}\) is given by the equation \(y = x - 1\), obtained by performing long division of the polynomials. To sketch the graph, plot the slant asymptote as a dashed line and then plot the function, observing that the graph of the function approaches the slant asymptote as the value of x increases or decreases.

Step 1: Perform long division of the polynomials

In this step, we will perform long division of the polynomials, as shown below: \( \begin{array}{c|cc cc} \multicolumn{2}{r}{x} & -1 \\ \cline{2-5} 2x-2 & x^2& -2x & -3 \\ \cline{2-3} \multicolumn{2}{r}{x^2} & -x & \\ \cline{2-3} \multicolumn{2}{r}{0} & -x & -3 \\ \cline{3-4} \multicolumn{3}{r}{-x} & +3 \\ \cline{3-4} \multicolumn{3}{r}{0} & -6 \\ \end{array} \)

Step 2: Write the quotient

From the long division, the quotient is \(x - 1\). This quotient represents the equation for the slant asymptote of the function.

Step 3: Find the slant asymptote

We now have the equation of the slant asymptote, which is: $$ y = x - 1 $$

Step 4: Sketch the graph of the function and the slant asymptote

Start by plotting the slant asymptote (\(y = x - 1\)) as a dashed line. Then, plot the function (\(f(x) = \frac{x^2 - 2x - 3}{2x - 2}\)). Notice that as the value of x increases or decreases, the graph of the function approaches the slant asymptote. Here's a summary of the process: 1. Perform long division of the polynomials. 2. Write the quotient. 3. Find the slant asymptote. 4. Sketch the graph of the function and the slant asymptote. By following these steps, you should be able to find the slant asymptote and sketch the graph of any given rational function.

Key Concepts

Understanding Long Division and Rational Functions

Long division is a crucial mathematical tool you use to simplify expressions, particularly when dealing with polynomials. It's similar to the long division process you're familiar with in basic arithmetic, but here, instead of numbers, you have polynomials to deal with. In essence, you're dividing one polynomial by another to simplify it.

When working with rational functions, which are simply fractions where the numerator and the denominator are polynomials, long division helps break them down. This is essential for finding slant asymptotes—a line that the graph of the function approaches but never actually touches.
To perform long division with polynomials:

• Write the dividend (numerator) under the division symbol and the divisor (denominator) outside.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this quotient and subtract it from the dividend.
• Bring down the next term and repeat the process.

By following these steps, you find a quotient and sometimes a remainder, where the quotient becomes especially important when determining the slant asymptote.

Exploring Rational Functions and Their Characteristics

Rational functions, as mentioned earlier, are ratios of two polynomial functions. They play a vital role in calculus and algebra because they can be used to model various real-world phenomena. These functions often have specific characteristics such as vertical and slant asymptotes and intercepts.

Understanding how these functions behave involves recognizing their asymptotic behavior. Vertical asymptotes occur where the denominator is zero, resulting in a function that approaches infinity. Slant asymptotes happen when the degree of the numerator is exactly one more than the degree of the denominator.

For our example function, the slant asymptote is found after performing long division. This indicates that as the input value increases, the function value will approach the line of the slant asymptote. Rational functions can also have regions where they are undefined, usually at points where the denominator equals zero. Notably, slant asymptotes can offer great insight into the end behavior of a function.

Graphing Functions and Identifying Slant Asymptotes

Graphing functions can be a rewarding process, as it offers a visual perspective on mathematical expressions. When graphing rational functions, key features like intercepts, asymptotes, and general behavior need to be considered. Often, the asymptotes guide the overall shape of the graph, painting a clearer picture.

Start by plotting any intercept points and marking any asymptotes. In the case of slant asymptotes, draw them as dashed lines to indicate that the function will get infinitely close but never quite touch them. Then sketch the function. Notice how it approaches the asymptotes as the x-values grow larger or smaller.

When graphing, it's also essential to check regions around critical points such as intercepts and intersections with the asymptotes. Observing how the function behaves near these key points gives you a deeper understanding and better accuracy in the sketch.

• Identify intercept points (where the graph crosses the axes).
• Draw vertical asymptotes as needed (where the denominator equals zero).
• Plot the slant asymptote using the quotient from long division.
• Sketch how the function approaches each asymptote.

This step-by-step visualization deepens your understanding of the behavior and characteristics of rational functions.