2.4 Polynomial and Rational Functions

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1 Polnomial Functions Given a linear function f() = m + b, we can add a square term, and get a quadratic function g() = a 2 + f() = a 2 + m + b. We can continue adding terms of higher degrees, e.g. we can add a cube term and get h() = c 3 + g() = c 3 + a 2 + m + b, and so on. f(), g(), and h() are all special cases of a polnomial function. Definition (Polnomial Function) A polnomial function is a function that can be written in the form f() = a n n + a n 1 n a 1 + a 0 for n a nonnegative integer, called the degree of the polnomial. The coefficients a n, a n 1,..., a 1, a 0 are real numbers with a n 0. Note that although a n 0, the remaining coefficients a n 1, a n 2,..., a 1, a 0 can ver well be 0. Domain of Polnomial Function The domain of a polnomial function is R, the set of all real numbers. The domain of f() = n is R regardless the value of n (an nonnegative integer), and so is the domain of g() = a n, where a is some real number. Clearl, if ou add, sa k, such functions with different degrees (n) the domain of the resulting function will still be R.

2 Consider a function f() = ( 1)( 2)( 3). It could be rewritten as f() = ( 1)( 2)( 3) = = ( 1)( ) = = ( 1)( ) = = = = So, f() is a polnomial function of degree 3. Question: How man intercepts does f() have? Answer: Onl one, = f(0) = 6. An function can have at most one intercept, otherwise it will not pass the vertical line test. Intercept of a Polnomial Function If f() = a n n + a n 1 n a 1 + a 0 is a polnomial function, it has eactl one intercept = a 0. Question: How man intercepts does f() have? Answer: f() has 3 intercepts. 0 = ( 1)( 2)( 3) = = 1 or = 2 or = 3. Intercept of a Polnomial Function A polnomial of degree n can have, at most, n linear factors. Therefore, the graph of a polnomial function of positive degree n can intersect the ais at most n times. The intercepts of f() = a n n + a n 1 n a 1 + a 0 could be found b solving a n n + a n 1 n a 1 + a 0 = 0. 2

3 7 6 5 f() Consider a function h() = ( 2 + 1)( 2)( 3). h() = ( 2 + 1)( 2)( 3) = = ( 2 + 1)( ) = = ( 2 + 1)( ) = = = = h() is a polnomial function of degree, but has just 2 intercepts, because the equation 0 = ( 2 +1)( 2)( 3) has just 2 roots (zeros), which are = 2 and = 3. 3

4 20 h() Note that f() = has degree 3, which is an odd number. It starts negative, ends positive, and crosses the ais odd number of times. h() = has degree, which is an even number. It starts positive, ends positive, and cross the ais even number of times. Consider m() = f() = ( ) = , and n() = g() = ( ) = m() n()

5 Definition (Leading Coefficient) Given a polnomial function f() = a n n +a n 1 n a 1 +a 0, the coefficient a n of the highest-degree term is called the leading coefficient of a polnomial function f(). Graph of a Polnomial Function Given a polnomial function f() = a n n +a n 1 n a 1 +a 0 : (a) if a n > 0 and n is odd, then the graph of f() starts negative, ends positive, and crosses the ais odd number of times but at least once; (b) if a n < 0 and n is odd, then the graph of f() starts positive, ends negative, and crosses the ais odd number of times but at least once; (c) if a n > 0 and n is even, then the graph of f() starts positive, ends positive, and crosses the ais even number of times or does not cross it at all; (d) if a n < 0 and n is even, then the graph of f() starts negative, ends negative, and crosses the ais even number of times or does not cross it at all. Note: (c) is a reflection in the ais of (a), and (d) is a reflection in the ais of (b). Also note that a polnomial function alwas either increases or decreases without bound as goes to either negative or positive infinit. 5

6 Continuit and Smoothness of Polnomial Function Consider f() = 2. f() has a discontinuous break at = 0. 2 f()

7 Consider g() = 2. g() is continuous, but not smooth due to a sharp corner at (0, 2). 2 g() Consider h() = 2 1. h() has a discontinuous break at = h() Graph of a Polnomial Function The graph of a polnomial function is continuous, with no holes or breaks. That is, the graph can be drawn without removing a pen from the paper. Also, the graph of a polnomial is smooth, i.e. has no sharp corners. 7

8 Rational Functions Just as rational numbers are defined in terms of quotients of integers, rational functions are defined in terms of quotients of polnomials. Definition (Rational Function) A rational function is an function that can be written in the form f() = n() d(), d() 0 where n() and n() are polnomials. For eample, f() = 1, g() = 2 2 6, h() = , p() = , q() = 123, r() = 0 are all rational functions. If n() and d() are polnomials, then the both have domain R. However, Domain of a Rational Function If f() = n() d() is a rational function, then its domain is the set of all real numbers such that d() 0. 8

9 Eample 1 Find the domain of f() =

10 Vertical and Horizontal Asmptotes Recall that a polnomial function is alwas continuous and smooth. It is also true that if increases or or decreases without bound, then function also increases or decreases without bound. However, this ma not be true for a rational function. Also, a rational function ma not have a intercept. Consider a rational function f() = 3 2. Its domain (, 2] [2, ), or all real numbers ecept for = 2, f() = = = = = = = = = = = = = = = = undefined = = = = = = = = = = = = = = = = 1 10

11 f() = = = = = = = 1003 = 1003 = 13 = 8 12 = = = 3 2 = = = = = = 0 1 = = 2 3 = = 7 8 = = = = = = = = = = 2 The graph of f() gets closer to the line = 2 as gets closer to 2. Line = 2 is a vertical asmptote for f(). = 1 The graph of f() gets closer to the line = 1 as increases or decreases without bound. The line = 1 is a horizontal asmptote for f(). 11

12 Definition (Vertical Asmptote) A vertical line = a is called a vertical asmptote for a function f() if the graph of f() gets closer to the line = a as gets closer to a. Note: the number of vertical asmptotes of a rational function f() = n() d() is at most equal to the degree of d(). Definition (Horizontal Asmptote) A horizontal line = b is called a horizontal asmptote for a function f() if the graph of f() gets closer to the line = b as gets increases or decreases without bound. Note: a rational function has at most one horizontal asmptote. Moreover, the graph of a rational function approaches the horizontal asmptote (when one eists) both as increases and decreases without bound. = 2 = = f() = 8 2 = 8 ( 2)( + 2) 12 f() = + 1 = 2 + 1

13 Eample 2 Given the rational function f() = , (a) Find the domain. (b) Find the and intercepts. (c) Find the equations of all vertical asmptotes. (d) If there is a horizontal asmptote, find its equation. (e) Using the information from parts (a)-(d) and additional points as necessar, sketch a graph of f for

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15 Consider the rational function g() = = 3 =

16 Eample 3 Find the vertical and horizontal asmptotes of the rational function f() =

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18 Applications Eample (Emploee Training) A compan that manufactures computers has established that, on the average, a new emploee can assemble N(t) components per da after t das of on-the-job training, as given b N(t) = 25t + 5 t + 5, t 0 Sketch a graph of N, 0 t 100, including an vertical or horizontal asmptotes. What does N(t) approach as t increases without bound? 18

19 N(t) t 19