3 Polynomial and Rational Functions

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1 3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental Theorem of Algebra 3.6 Rational Functions Homework Problem Sets 3.1 (5, 7, 9, 13, 17, 19, 35, 41, 47-51, 53, 57, 63, and page ) 3.2 (1-4, 7, 9, 11, 13, 19, 23, 29, 33, 39, 43, 53, 57, 85, 88) 3.3 (5, 7, 13, 17, 21, 25, 27, 29, 33, 37, 41, 47, 55, 59, 65, 68, 71-74) 3.4 (11, 13, 15, 27, 35, 45, 47, 55, 59, 91, 93) 3.5 (7, 9, 23, 25, 29, 31, 35, 41, 43, 45, 47, 49, 61, 67) 3.6 (27, 29, 43, 49, 51, 55, 61, 63, 65, 69, 73) mathminer.org

2 3.1 Quadratic Functions and Models Quadratic Function A polnomial function of the form f() = a 2 +b+c, where a 0, is called a quadratic function. Its -shaped graph is called a parabola. verte ais of smmetr -intercepts -intercept The -intercepts are The -intercept is (0, f(0)). ( b± ) b 2 4ac 2a, 0. The ais of smmetr is = b 2a. The verte is ( b 2a, f ( b 2a)). It opens up if a > 0 and down if a < 0. Eample 1 Show that the -coordinate of the verte of the graph of = a 2 + b + c is given b = b 2a. We can also write the equation of a parabola in a form that utilizes our function transformation techniques. Verte Form of a Quadratic Function The equation of a parabola with verte (h, k) is given b f() = a( h) 2 + k. If a > 0, the parabola opens up and the point (h, k) is a minimum. If a < 0, the parabola opens down and the point (h, k) is a maimum. 2

3 Eample 2 Sketch a graph of the parabola given b f() = ( + 4) 2 2. Eample 3 Sketch a graph of the parabola given b g() = 1 2 ( 2) Eample 4 Find the equation of the quadratic function shown in the graph

4 What if a quadratic function is in the form = a 2 + b + c? How can we convert it to verte form? Eample 5 Write the quadratic function f() = in verte form. Identif the maimum or minimum value and sketch its graph. Eample 6 Write the quadratic function f() = in verte form. Identif the maimum or minimum value and sketch its graph. 4

5 Eample 7 The height of a projectile launched straight up in the air is given b the equation h = 1 2 gt2 + v 0 t + h 0, where h is the height, t is the time after launch, g is the acceleration due to gravit, v 0 is the initial velocit, and h 0 is the initial height of the projectile. Suppose that a water rocket is launched from ground level with an initial velocit of 64 ft/s. (Note that g = 32 ft/s 2 or g = 9.8 m/s 2 ) a. When will the object hit the ground? b. What if the rocket is launched from the top of a 100 ft high platform? c. At what time does the rocket in part (b) reach its maimum height? What is that height? 5

6 3.2 Polnomial Functions and Their Graphs In this section we are going to etend what we know about linear and quadratic functions to more general polnomial functions. Power Function A function of the form f() = k n, where k and n are real numbers, is called a power function. Eample 1 Use our graphing calculator to compare the graphs of the given functions. What do ou notice? a. = 3, = 5, = 7, = 3, = 5, and = 7 b. = 2, = 4, = 6, = 2, = 4, and = 6 We call the long-term behavior of a power function its end-behavior. This describes the behavior of the function as grows infinitel large positive or negative ( ± ). So what are the possibilities for the end-behavior of the function f() = k n? Even Power Odd Power Wh are there no other possibilities for the end behavior of a power function? 6

7 Polnomial Function A polnomial function is a function whose terms are of the form a n, where a is a real number and n is a nonnegative integer. The degree of a polnomial is the value of the largest eponent. In general, a polnomial function of degree n has the form f() = a n n + a n 1 n a a 1 + a 0. The constant a n is called the leading coefficient and a 0 is the constant term. Properties of Polnomial Functions Domain is (, ). The graph has no breaks, jumps, holes, or asmptotes. We sa it is continuous. The graph has no corners or cusps. We sa it is smooth. The graph has at most n 1 turning points. These are called local etrema. The graph has at most n -intercepts. As ±, the graph of f() = a n n + a n 1 n a a 1 + a 0 has the same end behavior as the power function f() = a n n. Eample 2 Which of the following are polnomials? For those that are, give the degree and lead coefficient. a. f() = b. f() = c. f() = i 2 + (2 + 3i) 17 d. f() = e. f() =

8 Eample 3 Describe the end-behavior of each polnomial function. a. P () = b. P () = c. P () = 5 2 ( 3)( + 1) 2 Real Zeros of Polnomials If P is a polnomial and c is a real number, then the following statements are equivalent: c is a zero of P = c is a solution of the equation P () = 0 c is a factor of P () c is an -intercept of the graph of P Eample 4 Find the -intercepts of the polnomial function P () = 5 2 ( 3)( + 1) 2. Eample 5 Find the -intercepts of the polnomial function P () =

9 What can the graph of a polnomial function do at its -intercepts? To answer this, we are going to solve polnomial inequalities, as seen earlier in the course. Eample 6 Use the graph of the equation = 2 ( 3)( + 1) to solve the following inequalities. a. 2 ( 3)( + 1) < 0 15 b. 2 ( 3)( + 1) > c. 2 ( 3)( + 1) 0 d. 2 ( 3)( + 1) Note the connection between the solutions to the inequalities and the location of the graph of the polnomial function. Now let s put all these ideas together. Eample 7 Sketch the graph of P () = 2 6 using -intercepts and inequalities. 9

10 Eample 8 Sketch each polnomial function using -intercepts and inequalities. a. f() = b. g() = ( + 2) 2 ( 3) 2 c. h() = ( + 1)( 4) 2 ( + 3)( 1) 2 10

11 Can ou make a conjecture concerning the behavior of the graphs at the -intercepts and the connection to the corresponding terms in the formula? Multiplicit If ( r) k is a factor of a polnomial function p() = a n n + a n 1 n a a 1 + a 0, we sa that r is a root of multiplicit k. If p() is completel factored, then the sum of the multiplicities of all the factors will equal the degree n of p(). If a root has even multiplicit, the graph of the polnomial will have a turning point on the -ais; if a root has odd multiplicit, the graph of the polnomial will cross through the -ais. Eample 9 What is the lowest possible degree of the polnomial function whose graph is shown? What is the sign of the leading coefficient? Give a possible polnomial function that could have the graph shown. a. 30 b Can ou make a conjecture concerning the behavior of the polnomial where the graphs above are tangent to the -ais and the connection to the corresponding terms in the formula ou gave? 11

12 Eample 10 Find a formula for the polnomial function whose graph is shown. Note that it passes through the point ( 1, 2) Eample 11 Find a formula for the polnomial function whose graph is shown. Note that it passes through the point (0, 4)

13 3.3 Dividing Polnomials Recall the long division process for numbers. Evaluate 53 4 : We use the same process to divide polnomials. Division Algorithm P () = dividend D() = divisor Q() = quotient R() = remainder P () R() = Q() + D() D() or P () = D()Q() + R() Eample 1 Perform long division on each of the following. a b

14 Snthetic Division: This can be used if the divisor is of the form r. Eample 2 Perform snthetic division on each of the following. a b c

15 The following division theorems are equivalent, but each version is useful. Remainder Theorem If the polnomial P () is divided b c, then the remainder is P (c). Factor Theorem The term c is a factor of P () if and onl if P (c) = 0. Eample 3 Is 1 a factor of f() = ? Eample 4 Suppose P is a polnomial function. a. If the onl zeros of P are 4 and 1, what are P (4) and P ( 1)? b. If the multiplicit of 4 is 2 and the multiplicit of 1 is 3, what are the factors of P? c. Write a formula for P and sketch its graph. 15

16 Eample 5 Given that 8 is a root of f() = , find the other roots. 16

17 3.4 Real Zeros of Polnomials In the last eample of the previous section, it was eas to come up with the solutions because we were given one root to start with. What if we didn t have that information? The following theorem will give us a list of potential rational roots of a polnomial. If the polnomial has an rational roots, the will be on this list. Rational Root Theorem Consider the polnomial P () = a n n + a n 1 n a a 1 + a 0, where n 1, a 0 0, and a n 0, and suppose that all the coefficients are integers. Suppose p is a factor of a 0 and q is a factor of a n and that p and q have no common factors other than ±1. Then all rational roots of P () are of the form p q. Eample 1 List all possible rational roots of the polnomial P () = Eample 2 Consider the polnomial function P () = a. List all possible rational roots of the polnomial. b. Are an in our list actuall roots? What theorem can we use to check? 17

18 Eample 3 Find all zeros of the polnomial P () = and sketch its graph. Eample 4 Find all zeros of the polnomial P () = and sketch its graph. 18

19 3.5 Comple Zeros and the Fundamental Theorem of Algebra The Fundamental Theorem of Algebra can be stated a number of different was. The first one given is generall how the theorem is presented, but the others are equivalent statements. Fundamental Theorem of Algebra Ever non-constant polnomial with comple coefficients has at least one comple zero. This theorem, together with the factor theorem, implies that a polnomial can be factored as a product of linear factors. Complete Factorization Theorem Ever polnomial of degree n 1 can be completel factored as a product of linear factors over the comple number sstem. This also implies that ever polnomial of degree n 0 has eactl n zeros, provided a zero of multiplicit k is counted k times. Also, ever polnomial can be factored as a product of linear and irreducible quadratic factors over the real number sstem. Eample 1 According to the Fundamental Theorem of Algebra, which of the following have at least one root? a. f() = b. f() = c. f() = i 2 + (2 + 3i) 17 d. f() = e. f() = Eample 2 Identif all zeros of P () = and write the complete factorization of P. What do ou notice about the comple zeros in the above? 19

20 The following theorem tells us about comple roots of polnomials. Conjugate Root Theorem If a + bi is a root of the polnomial P (), with real coefficients, then so is its comple conjugate a bi. Eample 3 Given that 4 i is a root of = 0, find the other roots. 20

21 3.6 Rational Functions In this section we are going to etend what we know about polnomial functions to rational functions. Rational Function A function of the form r() = P (), where P and Q are polnomials with Q() 0, is called a rational function. Q() Eample 1 What is the domain of the rational function f() = ? Now look at the graph of this function on our 6 graphing calculator. What do ou see occurring at = 2 and = 3? Eample 2 Use our calculator to sketch a graph of f() = 2 3. Identif each of the following. + 4 a. Domain b. Vertical asmptote c. Horizontal asmptote Note that as rational functions approach their vertical asmptotes, ±. In the above, the function approaches as gets close to 4 on the left, and approaches as gets close to 4 on the right. Is there a wa to algebraicall find the horizontal asmptotes of a rational function? 21

22 Graphing rational functions is similar to graphing polnomial functions; i.e. we will use zeros and inequalities. The difference here is that we will include locations of horizontal asmptotes (zeros of the denominator) as critical values. Eample 3 Identif the -intercepts and asmptotes of each rational function and sketch its graph. a. r() = 1 ( 2) 2 b. r() =

23 Eample 4 Identif the -intercepts and asmptotes of each rational function and sketch its graph. a. r() = b. r() =

24 Eample 5 Can a rational function intersect its horizontal asmptote? Sketch a graph of f() = Eample 6 Here is an eample that illustrates the difference between a hole or an asmptote in a graph. Sketch a graph of g() =

25 Eample 7 Here is an eample that has an oblique asmptote. Graph h() = Note that a rational function onl has an oblique asmptote if the degree of the numerator is eactl 1 larger than that of the denominator. 25

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define