Chapter 3.5: Rational Functions
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1 Chapter.5: Rational Functions A rational number is a ratio of two integers. A rational function is a quotient of two polynomials. All rational numbers are, therefore, rational functions as well. Let s get reacquainted with an old friend. Eample 1: 1 Sketch f. Find the domain and range. Find and label all discontinuities. Find the intervals over which the function is increasing and decreasing. Describe any symmetry. Evaluate the following: lim f (b) lim f (c) lim f (d) lim f (e) lim f (f) lim f 4 Definition of a Vertical Asymptote If lim() f or lim() f, then there eists a vertical asymptote at c c c. Definition of a Horizontal Asymptote lim() f L or lim() f L if and only if there eists a Horizontal Asymptote (HA) at y L Page 1 of 7
2 Eample : Sketch a graph of each rational function by a transformation of the parent function domain and range, all asymptotes, and all discontinuities. 6 (a) f 4 (b) g 5 1 y. Identify the Eample : Sketch the graph of each of the following functions. Identify the domain and range, all asymptotes, and all discontinuities a) r b) R c) f Page of 7
3 Asymptotic and Discontinuous behavior of Rational Functions Let R be the rational function n n1 an an 1 a1 a R m m1 bm bm 1 b1 b 1. The Vertical Asymptotes (non-removable infinite discontinuities) of R are the lines a, where a is a zero of the denominator but NOT the numerator. That is R() a.. The Holes (removable point discontinuities) of R are the points b,lim R BOTH the numerator and denominator. That is R b. NOTE: If R c 4. (a) if n m lim R and R has an HA at y., then b, then c is a zero/-intercept/root of R. an an (b) if n m, then lim R, and R has an HA at y bm bm (c) if n m, then lim R or lim R, and R has no HA., where b is a zero of A horizontal asymptote is an eample of an end-behavior model. There are other types of end-behavior models that can be found the same way analyzing the leading coefficients in the numerator and denominator. The behavior of the end-behavior model and the original function will be the same as and as, although the local behavior (for small -values) will be different. Eample 4: Identify the leading term in the end behavior model of the following rational functions. Based on the endbehavior model, determine lim f and lim f. (a) f (b) f (c) f Page of 7
4 Eample 5: Find the domain, end behavior, and all discontinuities. Sketch the function. Find the range. Use long division to find the equation of the end-behavior model. Verify each on the calculator, then zoom out to see the end behavior (a) f (b) f 6 8 Eample 6: Analyze the graphs of the following rational functions: 18 (a) f (b) h 4 (c) j k (d) p (e) Q 18 4 (f) g Page 4 of 7
5 Eample 7: Construct the equation (in factored form) of a holey graph with holes at 1, 5, and 4, - intercepts at and 6, a vertical asymptotes at and 6 with a horizontal asymptote at y. Here s a quick summary of how to analyze rational functions: 1. Factor: Factor both the numerator and denominator. Domain: Find the values that make the denominator zero. This will be domain restrictions.. Discontinuities: means VA. means hole. 4. Bad Guy: Divide out any bad guy factors causing a hole. Use the equation that remains for all further analysis, including the y-value of the hole. lim f and lim f. Graph all HA s and SA s. 5. End Behavior: Find 6. Intercepts: Find the -intercepts by determining the zeros of the numerator, and the y-intercept from the value of the function at. 7. Symmetry: even, odd or neither 8. Sketch the Graph: Graph all asymptotes first, intercepts net, then combine the other information to fill in the rest of the graph. 9. Smile: Pat yourself on the back for a job well done! Eample 8 Graph the rational function m Page 5 of 7
6 Eample 9: Graph the rational function L Eample 1: 1 Write an equation of a function with a VA at, an SA at y, a y-intercept at 4, and a hole at 5. Find the end behaviors of this function. As, what do the slopes of the function approach? Page 6 of 7
7 Eample 11: Suppose that the rabbit population on Mr. Korpi s Hairy Hare farm follows the formula t P t t 1 Where t is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population on the relevant domain. (b) According the model, what is the initial population of the rabbits? How can this be?? (c) Using a calculator, what will the rabbit population be after 5.5 months? (d) Using a calculator, after how many months will the rabbit population be at 166 rabbits? (e) According to the math model, what eventually happens to the rabbit population, in the long run? Page 7 of 7
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