Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

What characteristics might the graph of a rational function (a polynomial divided by a polynomial) have that the graph of a polynomial will not have?

Step-by-Step Solution

Answer

The graph of a rational function can display vertical, horizontal or slant asymptotes, and holes, which are characteristics not present in the graph of a polynomial function.

Step 1: Understanding Polynomial Graphs

A polynomial function graph is a smooth curve, which means there are no sharp corners or gaps. The behaviour of a polynomial function can be understood by its degree. If the degree is odd, the ends of the graph will go in opposite directions. If the degree is even, both ends of the graph will go in the same direction.

Step 2: Understanding Rational Function Graphs

The graph of a rational function can also be smooth, but it can have vertical asymptotes, horizontal asymptotes, or slant asymptotes. Vertical asymptotes occur when the denominator of the rational function equals zero. Horizontal and slant asymptotes pertain to the end behavior of the rational function, depending on the degrees of the polynomials in the numerator and the denominator.

Step 3: Identifying the Characteristics Unique to Rational Functions

By comparing the traits of both polynomial and rational functions, it can be determined that rational functions can have the following characteristics not shared by polynomial functions: 1. Vertical asymptotes. 2. Horizontal or slant asymptotes. 3. Holes in the graph, when the same factor is found in both the numerator and the denominator

Key Concepts

Polynomial Function Graphs

When studying the characteristics of function graphs, it's essential to begin with the basics of polynomial function graphs. These graphs are known for their smooth, continuous curves without breaks, sharp corners, or gaps. Importantly, the overall shape of a polynomial graph is determined by the function's degree.

Effects of Degree on Polynomial Graphs

For a polynomial with an odd degree, the graph will extend to infinity in opposite directions. In contrast, a polynomial of even degree will have both ends rising or falling together towards infinity. A quintessential example is the parabola, which is a second-degree polynomial, always opening upward or downward based on the leading coefficient.

Vertical Asymptotes

Diving into the unique characteristics of rational functions, vertical asymptotes stand out. Unlike polynomial graphs that do not have any asymptotes, a vertical asymptote in a rational function graph indicates where the function approaches infinity.

How Vertical Asymptotes Occur

These asymptotes occur when the denominator of the rational function is zero (provided that the numerator is non-zero at that point), leading to undefined values for the function. They are represented as vertical lines on the graph, hinting at these 'off-limits' areas that the function will never touch, but will approach infinitely closely.

Horizontal Asymptotes

Apart from vertical asymptotes, rational functions might also have horizontal asymptotes which are not a feature of polynomial graphs. These horizontal lines on a graph represent the value that the function output (or y-value) approaches as the input (or x-value) heads towards infinity.

Determining Horizontal Asymptotes

The existence and position of a horizontal asymptote are influenced by the degree of the numerator and the degree of the denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be the x-axis (y=0). If the degrees are equal, the asymptote will be the ratio of the leading coefficients.

Slant Asymptotes

Slant asymptotes are another feature unique to rational functions and do not appear in polynomial graphs. A slant asymptote, also known as an oblique asymptote, occurs when the function graph approaches a line that isn't horizontal or vertical but has a slope.

Understanding Slant Asymptotes

This type of asymptote is typically found when the degree of the numerator is exactly one more than the degree of the denominator. As the x-values go to infinity, the rational function begins to behave like a line with a slope, defining the end behavior of the graph. It's important to note that unlike vertical asymptotes, a function can cross a slant asymptote at one or more points.