Problem 3 - Calculus Early Transcendentals

Question

Can the graph of a polynomial have vertical or horizontal asymptotes? Explain.

Step-by-Step Solution

Answer

If so, under what conditions? Answer: The graph of a polynomial function can have a horizontal asymptote only if it is a constant or linear function. Polynomial functions cannot have vertical asymptotes.

Step 1: Definition of Asymptote:

An asymptote is a line that the graph of a function approaches but never touches as the input (x) approaches infinity or negative infinity. There are two types of asymptotes: horizontal and vertical. A horizontal asymptote is a horizontal line that the graph of the function approaches as the input (x) goes to infinity or negative infinity. A vertical asymptote is a vertical line that the graph of the function approaches but never touches or crosses.

Step 2: Properties of Polynomial Functions:

Polynomial functions have the form P(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a non-negative integer. Polynomial functions are continuous on the entire real line, which means that they do not have any breaks or jumps in their graphs. Also, they are smooth on the entire real line, meaning that they do not have any sharp corners or cusps. #tagName# Vertical Asymptote of Polynomial Functions:

Step 3: A vertical asymptote occurs when the denominator of a rational function approaches zero but the numerator does not. However, polynomial functions do not have denominators, as they are not rational functions. Therefore, polynomial functions cannot have vertical asymptotes. #tagName# Horizontal Asymptote of Polynomial Functions:

A function has a horizontal asymptote if its graph approaches a horizontal line as the input (x) goes to infinity or negative infinity. Polynomial functions of degree 0 or 1 (constant or linear functions) have horizontal asymptotes because they approach a constant value as x goes to infinity or negative infinity. Higher-degree polynomial functions do not have horizontal asymptotes because their graphs continuously increase or decrease as x goes to infinity or negative infinity, depending on their leading term. In conclusion, the graph of a polynomial function can have a horizontal asymptote only if it is a constant or linear function, and it cannot have a vertical asymptote.