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Chapter 4: Problem 3

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

Short Answer

Expert verified
Answer: No, the graph of a polynomial function cannot have vertical or horizontal asymptotes due to their continuous nature and unbounded growth as the input (x) approaches infinity or negative infinity.

Step by step solution

01

1. Definition of Asymptotes

Asymptotes are lines that a function's graph approaches but never touches. There are two types of asymptotes: vertical and horizontal. A vertical asymptote is a vertical line (x = a) where the function goes to infinity (or negative infinity) as it approaches 'a'. A horizontal asymptote, on the other hand, is a horizontal line (y = b) where the function approaches 'b' as the input (x) goes to infinity or negative infinity.
02

2. Definition of a Polynomial Function

A polynomial function is a function of the form P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where n is a non-negative integer, and a_0, a_1, ..., a_n are constants (a_n 鈮� 0). Polynomials are smooth, continuous, and differentiable functions.
03

3. Vertical Asymptotes in Polynomial Functions

Polynomial functions are continuous and differentiable across their entire domain (i.e., for all real numbers). For a function to have a vertical asymptote, there must be a point (x = a) at which the function is not defined or approaches infinity. Since the polynomial functions are defined and continuous for all real values of x, they cannot have vertical asymptotes.
04

4. Horizontal Asymptotes in Polynomial Functions

To determine if a polynomial function has a horizontal asymptote, we need to analyze the function's behavior as x goes to infinity or negative infinity. Polynomials with the highest power of x having a positive coefficient grow without bound as x goes to infinity or negative infinity. As a result, they do not approach a fixed value, and thus, no horizontal asymptotes are present for polynomial functions.
05

Conclusion

The graph of a polynomial function cannot have vertical or horizontal asymptotes due to their continuous nature and unbounded growth as the input (x) approaches infinity or negative infinity.

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