Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx

Rational Function

1. Domain, Range, Zeros, Intercepts and Asymptotes of Rational Function

2. Objectives: find the domain, range, zeroes and intercepts of rational functions determine the vertical and horizontal asymptotes of rational function.

3. The domain of a rational function is all the values of that will not make equal to zero. To find the range of rational function is by finding the domain of the inverse function. Another way to find the range of rational function is to find the value of horizontal asymptote.   Domain and Range of Rational Function

4. Example

5.   EXAMPLE 1:

6.     Focus on the denominator The domain of is the set of all real numbers except .   EXAMPLE 1:   To find the domain:  

7.   Change into y   EXAMPLE 1: To find the range:   Interchange the position of x and y   Simplify the rational expression     Solve for y in terms of x       Equate the denominator to 0.   The range of is the set of all real numbers except .    

8.   EXAMPLE 2:

9.     Focus on the denominator The domain of is the set of all real numbers except .   EXAMPLE 2:   To find the domain:  

10.   Change into y   EXAMPLE 2: To find the range:   Interchange the position of x and y   Simplify the rational expression     Solve for y in terms of x       Equate the denominator to 0.   The range of is the set of all real numbers except .        

11.   EXAMPLE 3:

12.     Focus on the denominator The domain of is the set of all real numbers except .   EXAMPLE 3:   To find the domain:    

13.   Change into y   EXAMPLE 3: To find the range:   Interchange the position of x and y   Simplify the rational expression   2   Solve for y in terms of x       Equate the denominator to 0.   The range of is the set of all real numbers except .          

14. Vertical and Horizontal Asymptotes of Rational Functions

15. They are the restrictions on the x – values of a reduced rational function. To find the restrictions, equate the denominator to 0 and solve for x. Finding the Vertical Asymptotes of Rational Functions

16. Let n be the degree of the numerator and m be the degree of denominator: If If where is the leading coefficient of the numerator and is the leading coefficient of the denominator. If , there is no horizontal asymptote.   Finding the Horizontal Asymptotes of Rational Functions

17. Find the Degree of Polynomial.              

18. Find the Degree of Polynomial.              

19. Example

20. EXAMPLE 1:  

21.   To find the vertical asymptote:     Focus on the denominator The vertical asymptote is .   EXAMPLE 1:

22.   To find the horizontal asymptote:   Focus on the degree of the numerator and denominator The horizontal asymptote is   1 EXAMPLE 1:

23.   EXAMPLE 2:

24.   To find the vertical asymptote:     Focus on the denominator The vertical asymptote is .   EXAMPLE 2:

25.   To find the horizontal asymptote:   Focus on the degree of the numerator and denominator The horizontal asymptote is   1 1 EXAMPLE 2:       a is the leading coefficient of 4x b is the leading coefficient of x

26.   EXAMPLE 3:

27.   To find the vertical asymptote:   Focus on the denominator The vertical asymptote are and .   EXAMPLE 3:            

28. 1 2 EXAMPLE 3:   To find the horizontal asymptote:   Focus on the degree of the numerator and denominator The horizontal asymptote is  

29.   EXAMPLE 4:

30.   To find the vertical asymptote:   Focus on the denominator The vertical asymptote are and .   EXAMPLE 4:            

31. 3 2 EXAMPLE 4: To find the horizontal asymptote:   Focus on the degree of the numerator and denominator The rational function has no horizontal asymptote    

32. Zeros of Rational Function

33. Finding the Zeros of Rational Functions Steps: Factor the numerator and denominator. Identify the restrictions. Identify the values of x that make the numerator equal to zero. Identify the zero of f(x).

34. Example

35.   EXAMPLE 1:

36.   Factor the numerator and denominator EXAMPLE 1:   Identify the restrictions.     Identify the values of x that will make the numerator equal to zero.         Identify the zeroes of f(x).    

37.   EXAMPLE 2:

38. Factor the numerator and denominator EXAMPLE 2:   Identify the restrictions.     Identify the values of x that will make the numerator equal to zero.         Identify the zeroes of f(x).              

39.   EXAMPLE 3:

40. Factor the numerator and denominator EXAMPLE 3:   Identify the restrictions.     Identify the values of x that will make the numerator equal to zero.         Identify the zeroes of f(x).            

41. Intercepts of Rational Functions

42. Intercepts are x and y – coordinates of the points at which a graph crosses the x-axis or y-axis, respectively. y-intercept is the y-coordinate of the point where the graph crosses the y-axis. x-intercept is the x-coordinate of the point where the graph crosses the x-axis. Note: Not all rational functions have both x and y intercepts. If the rational function has no real solution, then it does not have intercepts.

43. Rule to find the Intercepts To find the y-intercept, substitute 0 for x and solve for y or f(x). To find the x-intercept, substitute 0 for y and solve for x.

44. Example

45.   EXAMPLE 1:

46.   EXAMPLE 1: y - intercept           x - intercept            

47.   EXAMPLE 2:

48. EXAMPLE 1: y - intercept     x - intercept                    

49. TRY!

50. Find each of the following: Intercepts 1)  

51. Thank You