Problem 46 - Calculus of a Single Variable

Question

In Exercises 45–48, use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur? $$ g(x)=\frac{2 x^{2}-8 x-15}{x-5} $$

Step-by-Step Solution

Answer

The slant asymptote of the function \(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\) is \(y=2x+2\). When zoomed out, the graph of the function appears to get closer to this line due to the asymptotic behavior of rational functions.

Step 1: Graph the Function

The first step is to input the function \(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\) into a graphing utility and create the graph. Sketch it on paper.

Step 2: Find the Slant Asymptote

The slant asymptote of a rational function can be determined by performing long division or synthetic division of the numerator by the denominator. In this case, the slant asymptote will be \(y=2x+2\). This is because when we perform long division on the given function, we get \(2x+2+\frac{-25}{x-5}\). As x tends to infinity, the fraction part tends to 0, leaving us with the line equation \(y=2x+2\).

Step 3: Observe the Graph

Upon repeatedly zooming out on the graph, you will notice that the graph seems to get closer and closer to the slant asymptote line \(y=2x+2\). The graph of the function will appear to flatten out and approach this line as you zoom out.

Step 4: Explain the Observation

The asymptotic behavior of the graph, i.e. the graph tending towards the line \(y=2x+2\), is due to the properties of rational functions. When the degree of the numerator and the denominator are equal, the graph approaches the line represented by the ratio of the coefficients as x tends to infinity.

Key Concepts

Graphing Rational Functions

Understanding how to graph rational functions is fundamental for analyzing their behavior. A rational function is a fraction of two polynomials, and its graph can have various features, such as intercepts, asymptotes, and discontinuities. To begin graphing, we must first identify these critical elements by investigating the function algebraically. For example, with the function
\(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\),
you might look for the x-intercepts by finding values for which the numerator equals zero. Similarly, the x-value that makes the denominator zero is where you'll find vertical asymptotes or holes, depending on whether the factor cancels out with the numerator. In this exercise, after plotting the function, you observe how it behaves as you zoom out, revealing the larger-scale structure of its graph, namely the slant asymptote.

Long Division of Polynomials

When faced with a rational function where the degree of the numerator is one more than the degree of the denominator, a slant (or oblique) asymptote often exists. To find this asymptote, we use long division of polynomials. This method is similar to long division of numbers and is used to divide the numerator by the denominator. In the case of our function,
\(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\),
long division would yield
\(2x+2+\frac{-25}{x-5}\).
As \(x\) tends toward infinity, the remainder vanishes, leaving the equation of the slant asymptote
\(y=2x+2\). This line is what the graph of the function will tend toward as \(x\) grows larger in absolute value. Mastering polynomial long division is crucial to understanding and graphing rational functions correctly.

Asymptotic Behavior

Asymptotic behavior refers to the tendencies of a function as the input values become incredibly large or small. In the context of rational functions, this concept helps you predict how the function will behave and where the graph will 'settle' at extreme values of \(x\). With our function
\(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\),
the graph will increasingly approximate the slant asymptote \(y=2x+2\) as \(x\) approaches infinity or negative infinity. This behavior arises because the degree of the polynomial in the numerator dictates the end behavior of the function, and when the degrees of the numerator and denominator are close, the leading terms dominate the function's growth. Thus, the graph will tend towards a line, which is the expression of the leading terms' ratio.

Using Graphing Utilities

Graphing utilities are invaluable tools for visualizing functions and their properties. Using a graphing calculator or software, you can input the equation of a function, like
\(g(x)=\frac{2 x^{2}-8 x-15}{x-5}\),
and get a detailed picture of its graph. This visual aid makes it easier to identify important characteristics like intercepts, asymptotes, and the overall shape of the graph. In our exercise, zooming out on the graphing utility provides a clear visualization of how the function behaves at large scale. It illustrates the concept of asymptotic behavior, showing the graph's tendency to adhere to the slant asymptote \(y=2x+2\). Utilizing graphing utilities not only speeds up the process of graphing complex functions but also enhances your comprehension of how these algebraic expressions translate into graphical representations.