Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function

Transcription

1 SECTIONS 3.5: Rational Functions Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function I. Rational Functions A rational function is a function of the form f(x) = p(x) q(x) Where p and q are polynomial functions and q is not the zero polynomial. Domain: The domain of a rational function will be all real numbers except those numbers that make the denominator q(x) zero. Range: The range is the set of y-values on the graph on f(x). x-intercepts: Set f(x) = 0. Recall that f(x) = 0 when the numerator is 0, so just set the numerator p(x) = 0 and solve for x when the function is in most reduced form. Find f(0). That is, set x = 0 and solve for f(x). II. Domain and Intercepts of a Rational Function Find the domain and intercepts of the following functions. Ex1. f(x) = 4 x+4 Ex2. f(x) = x2 16 x 2 +4x Ex3. f(x) = x2 4x x 2 +4 Domain: x-int(s): y-int: 1

2 III. Vertical Asymptotes and Holes Vertical Asymptotes or Holes occur where a rational function is undefined. A hole will occur on the graph of a rational function when the function has a common reducible factor in both the numerator and denominator. To find the x-coordinate of a hole, set each reducible factor equal to 0 and solve for x. To find the y-coordinate of a hole, plug the x-value of any hole into the most reduced form of the function. Because functions are undefined at holes, the x-values where holes occur cannot be in the domain!!! A vertical asymptote is an imaginary vertical line that will occur on the graph of a rational function when the function has factors only in the denominator that do not reduce. If as x k, y approaches or, then the vertical line x = k is a vertical asymptote of the graph of a function. To find the equation of a vertical asymptote, set each irreducible factor in the denominator equal to 0 and solve for x. The graph of a function CANNOT cross a vertical asymptote. Because functions are undefined at vertical asymptotes, the x-values where vertical asymptotes occur cannot be in the domain!!! Find the vertical asymptotes and holes in the graph. Ex4. 2

3 Ex5. f(x) = x 2 x 3 Ex6. f(x) = x2 5x+6 x 3 Ex7. f(x) = x 2 +x x 2 2x 3 Ex8. f(x) = x 2 x 2 25 Ex9. f(x) = x 5 x

4 IV. Horizontal Asymptotes A horizontal asymptote is an imaginary horizontal line that will occur on the graph of a rational function when the degree of the denominator is greater than or equal to the degree of the numerator. If as x or x, the y-values approach some fixed number k, then the horizontal line y = k is a horizontal asymptote of the graph of a function. Horizontal asymptotes tell us about the end behavior of the graph of a rational function. To find the equation of a horizontal asymptote, use the comparison test: Comparison Test: Compare the leading term of the numerator to the leading term of the denominator. ax n bx m n < m n = m n > m HA: y = 0 HA: y = a b HA: None The graph of the function may or may not cross horizontal asymptotes. If degree of the numerator is one more than the degree of the denominator then there is an oblique (slant) asymptote. We will not oblique asymptotes in this course! Find the horizontal asymptotes. Ex10. 4

5 Ex11. f(x) = 5x 3 8 2x Ex12. f(x) = 3x+1 x 2 4x+3 Ex13. f(x) = x3 + 4x 9 7x+6x 2 V. Graphing Rational Functions Steps for Analyzing and Graphing a Rational Function 1. Use the comparison test to locate any horizontal asymptotes. 2. Factor the numerator and denominator. DO NOT REDUCE YET. Calculate the domain. 3. Locate the x-coordinates of any holes in the graph by considering the reducible factors (factors common to both the numerator and denominator). 4. REDUCE the function. 5. Locate any vertical asymptotes. 6. Calculate x-intercept(s) and the y-intercept. 7. Find the y-coordinates of any holes. 8. Use the above information to sketch the graph. Plot additional points, as needed, to get a complete graph. 9. Check your graph by using your graphing calculator. 5

6 Ex14. f(x) = x 4 2x 2 Ex15. f(x) = x 1 x 2 3x 6

7 Ex16. f(x) = x2 +x 6 x 2 Ex17. f(x) = 3x2 x 2 x 2 1 7