Presentation on theme: "Section 5.2 Properties of Rational Functions"— Presentation transcript:
1 Section 5.2 Properties of Rational Functions ObjectivesFind the Domain of a Rational FunctionDetermine the Vertical Asymptotes of a Rational FunctionDetermine the Horizontal or Oblique Asymptotes of a Rational Function
2 A rational function is a function of the form where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator q is 0.
3 Find the domain of the following rational functions. All real numbers x except -6 and -2.All real numbers x except -4 and 4.All Real Numbers
5 Graph the function using transformations (1,1)(-1,-1)(3,1)(1,-1)(2,0)(3,2)(1,0)(2,0)(0,1)
6 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.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.In the previous example, there was a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.(3,2)(1,0)(2,0)(0,1)
7 Examples of Horizontal Asymptotes y = Ly = R(x)yxy = Ly = R(x)yx
9 Theorem: Locating Vertical Asymptotes If an asymptote is neither horizontal nor vertical it is called oblique.yxTheorem: Locating Vertical AsymptotesA rational function R(x) = p(x) / q(x), in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.
10 Vertical asymptotes: x = -1 and x = 1 Example: Find the vertical asymptotes, if any, of the graph of each rational function.Vertical asymptotes: x = -1 and x = 1No vertical asymptotesVertical asymptote: x = -4
11 Consider the rational function in which the degree of the numerator is n and the degree of the denominator is m.1. If n < m, then y = 0 is a horizontal asymptote of the graph of R.2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R.3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division.4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.
12 Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3 Example: Find the horizontal or oblique asymptotes, if any, of the graph ofHorizontal asymptote: y = 0Horizontal asymptote: y = 2/3