1. SECTION 3.2 RATIONAL FUNCTIONS RATIONAL FUNCTIONS

2. RATIONAL FUNCTIONS Rational functions take on the form: Rational functions take on the form: Where p(x) and q(x) are polynomials and q(x)  0

3. EXAMPLES:

4. DOMAIN OF RATIONAL FUNCTIONS ANY VALUE WHICH ZEROS OUT THE DENOMINATOR MUST BE EXCLUDED FROM THE DOMAIN. ANY VALUE WHICH ZEROS OUT THE DENOMINATOR MUST BE EXCLUDED FROM THE DOMAIN. Do Example 1 Do Example 1

5. GRAPHS OF RATIONAL FUNCTIONS ASYMPTOTE - a line which the graph will approach but will never reach. ASYMPTOTE - a line which the graph will approach but will never reach.

6. HORIZONTAL ASYMPTOTES When the degree in the numerator is equal to the degree in the denominator. When the degree in the numerator is equal to the degree in the denominator. Example: Graph Example: Graph

7. By studying the graph, we can describe what is happening using the following symbols: By studying the graph, we can describe what is happening using the following symbols:  As x  , f(x)  3  As x  - , f(x)  3 We say f(x) has a horizontal asymptote of y = 3. We say f(x) has a horizontal asymptote of y = 3.

8. To see the algebraic reasoning behind this, divide the rational expression through by x 2 and examine what happens as x gets huge.

9. The numerator goes to 3 and the denominator goes to 1.

10. EXAMPLE:

11. Again, algebraically, we can see that the numerator goes to 4 and the denominator goes to 1 as x gets huge. Thus, this function has a horizontal asymptote y = 4.

12. We can get even more specific with our symbols when describing the graph of the function and say the following: We can get even more specific with our symbols when describing the graph of the function and say the following:  As x  , f(x)  4 -  As x  - , f(x)  4 +

13. QUESTION: Can you find an easy way of looking at the symbolic form of a rational function in which the degree in the numerator is equal to the degree in the denominator to find the horizontal asymptote?

15. Answer: Answer: Just divide the leading coefficient in the numerator by the leading coefficient in the denominator. Just divide the leading coefficient in the numerator by the leading coefficient in the denominator.

16. EXAMPLE: Determine the equation of the horizontal asymptote for the graph of : Determine the equation of the horizontal asymptote for the graph of : Horizontal Asymptote: y = 3/2

17. VERTICAL ASYMPTOTES: Vertical lines are always given by an equation in the form x = c. Here, the graph of the rational function will approach a vertical line, yet never quite reach it. This means that this is a value x will never equal. Vertical lines are always given by an equation in the form x = c. Here, the graph of the rational function will approach a vertical line, yet never quite reach it. This means that this is a value x will never equal.

18. Vertical asymptotes are nothing more than domain restrictions, or values for the variable that will cause the denominator to equal 0. Vertical asymptotes are nothing more than domain restrictions, or values for the variable that will cause the denominator to equal 0.

19. EXAMPLE: x = 2 is not in the domain of f(x) because replacing x with 2 would cause the denominator to equal 0. Graph f(x).

20. Here, we can describe what is happening to the graph of the function by using the following language: Here, we can describe what is happening to the graph of the function by using the following language:  As x  2 -, f(x)   As x  2 +, f(x) 

21. Thus, the vertical asymptote for this function is x = 2. Thus, the vertical asymptote for this function is x = 2. Is this the only asymptote for this function? Is this the only asymptote for this function? No! This function also has a horizontal asymptote given by the equation y = 0 No! This function also has a horizontal asymptote given by the equation y = 0

22. As x gets huge, the numerator goes to zero and the denominator goes to 1. As x gets huge, the numerator goes to zero and the denominator goes to 1.

23. In fact, any rational function in which the degree in the numerator is less than the degree in the denominator will have a horizontal asymptote of y = 0 (or the x-axis). In fact, any rational function in which the degree in the numerator is less than the degree in the denominator will have a horizontal asymptote of y = 0 (or the x-axis).

24. EXAMPLE: We can either examine the graph to determine the asymptotes, or we can study the symbolic form, using the tools we’ve learned.

25. The vertical asymptotes will be the domain restrictions. What values will zero out the denominator? The vertical asymptotes will be the domain restrictions. What values will zero out the denominator? Thus, the only vertical asymptote is x = 2.

26. Now, for the horizontal asymptotes. Now, for the horizontal asymptotes. The degree in the numerator is equal to the degree in the denominator. The degree in the numerator is equal to the degree in the denominator. Thus, the horizontal asymptote is y = 3. Graph the function. Thus, the horizontal asymptote is y = 3. Graph the function.

27. EXAMPLE: Determine all asymptotes: Determine all asymptotes: H.A. y = 4 V.A.x = - 4 x = 2

28. SLANT ASYMPTOTES When the degree in the numerator is exactly one more than the degree in the denominator. When the degree in the numerator is exactly one more than the degree in the denominator.

29. To determine the slant asymptote, we simply do the division implied by the fraction line. We can use synthetic division.

30. 21- 3 6 1 2 - 1 - 2 4 Thus, f(x) can be written as:

31. As x gets huge, the fractional part tends toward 0 and the entire function tends toward the linear function y = x - 1. As x gets huge, the fractional part tends toward 0 and the entire function tends toward the linear function y = x - 1. Graph the function. Graph the function.

32. EXAMPLE: Determine all asymptotes of the function below: Determine all asymptotes of the function below: Vertical Asymptotes: x = - 3x = 2

33. Slant Asymptote: Slant Asymptote: 2x 2x 3 + 2x 2 - 12x x 2 + 12x x 2 + 12x + 1 x 2 + x - 6 x 2 + x - 6 11x + 6 11x + 6

34. Slant Asymptote: y = 2x + 1 Graph the function

35. CONCLUSION OF SECTION 3.2 CONCLUSION OF SECTION 3.2