Math 111 Lecture Notes
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1 A rational function is of the form R() = p() q() where p and q are polnomial functions. A rational function is undefined where the denominator equals zero, as this would cause division b zero. The zeros of a rational function occur when the numerator of the simplified form of R is equal to zero, as the function s value is zero where the value of the numerator is zero. The zeros will occur at a when the factor ( a) is in the numerator of the simplified form of R. The multiplicit of the factor ( a) affects the behavior of the function at a. A vertical asmptote occurs when the denominator of the simplified form of R is equal to zero. The vertical asmptote = b will occur when the factor ( b) is in the denominator of the simplified form of R. The multiplicit of the factor ( b) affects the behavior of the function near = b. A hole occurs when both the numerator and denominator equal zero for some value of c and the associated factor of ( c) cancels. Note that when this factor cancels, it results in a domain restriction for R. The long run behavior and horizontal asmptote of R can be determined b the ratio of leading terms of p and q
2 Eample 1. Graph the rational function R() = b completing the following: + Factor and simplif R(). State the domain and an holes. State the long-run behavior and an horizontal asmptote. Find the vertical intercept. Find an zeros and find an vertical asmptotes. State the behavior of the function around the zeros and vertical asmptotes (preferabl b making a table). Figure 1 Instructor: A.E.Car Page of 10
3 Eample. Graph the rational function R() = 8 b completing the following: Factor and simplif R(). State the domain and an holes. State the long-run behavior and an horizontal asmptote. Find the vertical intercept. Find an zeros and find an vertical asmptotes. State the behavior of the function around the zeros and vertical asmptotes (preferabl b making a table). Figure Instructor: A.E.Car Page 3 of 10
4 Eample 3. Graph the rational function R() = 3 b completing the following: + Factor and simplif R(). State the domain and an holes. State the long-run behavior and an horizontal asmptote. Find the vertical intercept. Find an zeros and find an vertical asmptotes. State the behavior of the function around the zeros and vertical asmptotes (preferabl b making a table). Figure 3 Instructor: A.E.Car Page of 10
5 Eample. Graph the rational function R() = + 3 b completing the following: 3 Factor and simplif R(). State the domain and an holes. State the long-run behavior and an horizontal asmptote. Find the vertical intercept. Find an zeros and find an vertical asmptotes. State the behavior of the function around the zeros and vertical asmptotes (preferabl b making a table). Figure Instructor: A.E.Car Page 5 of 10
6 How to find a possible formula for a rational function: State an zeros. Use these to determine factors and the multiplicit of each factor that appears in the numerator. State an vertical asmptotes. Use these to determine factors and the multiplicit of each factor that appears in the denominator. If a hole appears at = a, then put the factor ( a) in both the numerator and denominator. Use one other point to determine if there is a constant factor other than 1. Eample 5. Find a possible formula for the rational function graphed in Figure 5. Figure 5 (3, ) = 3 = Instructor: A.E.Car Page of 10
7 Eample. Find a possible formula for the rational function graphed in Figure. Figure = ( ) 0, 8 3 = 1 = 3 Instructor: A.E.Car Page 7 of 10
8 Eample 7. Find a possible formula for the rational function graphed in Figure 7. Figure 7 (0, 1) = 0 = - = Instructor: A.E.Car Page 8 of 10
9 Group Work 1. Find a possible formula for the rational function graphed in Figure 8. Figure 8 = -1 = - Group Work. Find a possible formula for the rational function graphed in Figure 9. Figure 9 = = -5 = Instructor: A.E.Car Page 9 of 10
10 Eample 8. Oblique Asmptotes The graph of R() = 5 looks like the function defined b f() = 1 in the long run. We 8 know that this function has an oblique asmptote as the degree of the numerator is 1 greater than the degree of the denominator. To determine the equation of the oblique asmptote, either polnomial long division or a graphing calculator are needed. Using the epand ke on a graphing calculator to perform polnomial long division, or long division in WolframAlpha we find: We use this to write: R() = 5 8 R() = The first term in the epanded R(), which is, is the remainder. The epression 1 1 is used to 8 determine the equation of the oblique asmptote, which is = 1 1. The function and its oblique asmptote are graphed in Figure 10 below. Figure 10 = 1 1 = 5 8 = Instructor: A.E.Car Page 10 of 10
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