Homework 3.3 and 3.4

Finding a Horizontal or Oblique Asymptote of a Rational Function

1. If n<m (the degree of the numerator is less than the degree of the denominator), the line y=0 is a horizontal asymptote. 2. If n=m (the degree of the numerator equals the degree of the denominator), the line y=a/nbm is a horizontal asymptote. (That is, the horizontal asymptote equals the ratio of the leading coefficients.) 3. If n=m+1 (the degree of the numerator is one more than the degree of the denominator), the line y=ax+b is an oblique asymptote, which is the quotient found using long division. 4. If n≥m+2 (the degree of the numerator is two or more greater than the degree of the denominator), there are no horizontal or oblique asymptotes. The end behavior of the graph will resemble the power function y=an/bm(xn−m). Note: A rational function will never have both a horizontal asymptote and an oblique asymptote. A rational function may have neither a horizontal nor oblique asymptote.

Corollary

A polynomial function f of odd degree with real coefficients has at least one real zero. Proof: Because complex zeros occur as conjugate pairs in a polynomial function witrh real coefficients, there will always be an even number of zeros that are not real numbers.

Locating Vertical Asymptotes

A rational function R left (x) = p(x) / q(x)​, in lowest terms​, will have a vertical asymptote x=r if r is a real zero of the denominator q. That​ is, if x-r is a factor of the denominator q of a rational function R(x)=p(x) / q(x) ​, in lowest​ terms, R will have the vertical asymptote x=r.

Fundamental Theorem of Algebra

Every polynomial function with degree greater than zero has at least one complex zero.

Finding Horizontal and Oblique Asymptotes of a Rational Function R

If n≤m, then R is improper. Here long division is used. ​(a) If n = ​m, the quotient obtained will be the number an / bm​, and the line y= an / bm is a horizontal asymptote. ​(b) If n = m+1 the quotient obtained is of the form ax+b ​(a polynomial of degree​ 1), and the line ax+b is an oblique asymptote. ​(c) If n≥m+2, the quotient obtained is a polynomial of degree 2 or​ higher, and R has neither a horizontal nor an oblique asymptote. In this​ case, |x| ​unbounded, the graph of R will behave like the graph of the quotient.

Conjugate Pairs Theorem

Let f(x) be a polynomial function whose coefficients are real numbers. If r=a+bi is a zero of f, the complex conjugate ̄r=a−bi is also a zero of f.

The Domain

The Domain is all values of x, such that x≠0

Domain of a Rational Function

The domain of a rational function R(x)= p(x)/q(x) consists of all real numbers except those for which the denominator q is zero.

Vertical Asymptote

​If, as x approaches some number​ c, the values ⌈R(x)⌋ → ∞, then the line x=c is a vertical asymptote of the graph of R. The graph of R never intersects a vertical asymptote.

Horizontal Asymptote

​If, as x → −∞ or as x → ∞, the values of​ R(x) approach some fixed number​ L, then the line y=L is a horizontal asymptote of the graph of R.