CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............ Section. Comple Numbers...................... 7 Section. The Fundamental Theorem of Algebra........... 7 Section. Rational Functions and Asmptotes............. 79 Section.7 Graphs of Rational Functions................ Section. Quadratic Models...................... 9 Review Eercises.............................. 9 Practice Test...............................
CHAPTER Polnomial and Rational Functions Section. Quadratic Functions You should know the following facts about parabolas. f a b c, a, is a quadratic function, and its graph is a parabola. If a >, the parabola opens upward and the verte is the minimum point. If a <, the parabola opens downward and the verte is the maimum point. The verte is ba, fba. To find the -intercepts (if an), solve a b c. The standard form of the equation of a parabola is f a h k where a. (a) The verte is h, k. The ais is the vertical line h. Vocabular Check. nonnegative integer, real. quadratic, parabola. ais. positive, minimum. negative, maimum. f opens upward and has verte. f opens upward and has verte,.,. Matches graph. Matches graph.. 9 c d a b 9 7. f Verte:, -intercepts:,,, (a), vertical shrink, vertical shrink and vertical shift one unit downward, vertical shrink and horizontal shift three units to the left (d), horizontal shift three units to the left, vertical shrink, reflection in -ais, and vertical shift one unit downward 9
Chapter Polnomial and Rational Functions 9. f. f. h Verte:, -intercepts: ±, Verte:, -intercepts: ±, Verte:, -intercepts:, 7. f Verte:, -intercepts: None 7. f Verte:, -intercepts:,,, 9. h. Verte:, -intercept: None f Verte:, -intercepts:,,,. g Verte:, -intercepts: ±,. f Verte: -intercept:, ±,
7., is the verte. 9., is the verte. f a Since the graph passes through the point we have: a a a,, Thus, f. Note that, is on the parabola. Section. Quadratic Functions f a Since the graph passes through the point, 9, we have: 9 a a a., is the verte.. is the verte. f a Since the graph passes through the point,, we have: a a a a f f, f a Since the graph passes through the point,, we have: a a a a f. 7. -intercepts:,,, -intercept:, or 9.. 7. 7 9 or -intercepts:,,, 7 or -intercepts:,,, 7 7 7, 7 -intercepts:,, 7,
Chapter Polnomial and Rational Functions. f, opens upward 7. f, opens upward g, opens downward 7 g 7, opens downward 7 Note: f a has -intercepts Note: f a has -intercepts, and, for all real numbers a., and, for all real numbers a. 9. Let the first number and the second number. Then the sum is. The product is P. P The maimum value of the product occurs at the verte of P and is. This happens when.. Let be the first number and be the second number. Then. The product is P. Completing the square, P 7 7. The maimum value of the product P occurs at the verte of the parabola and equals 7. This happens when and.. (a) Radius of semicircular ends of track: r Distance around two semicircular parts of track: d r Distance traveled around track in one lap: (e) d The area is maimum when and. (d) Area of rectangular region: A The area is maimum when and.
Section. Quadratic Functions. (a) 7. C. When, feet. The verte occurs at b a 9.9. The maimum height is 9. feet. (d) Using a graphing utilit, the zero of occurs at., or. feet from the punter. C 7 7. 7 7. 7 From the table, the minimum cost seems to be at. The minimum cost occurs at the verte. b a.. C 7 is the minimum cost. Graphicall, ou could graph C. in the window,, and find the verte, 7. 9. (a) The verte occurs at R. R R p b a $, thousand $,7 thousand $, thousand $. R $, thousand (d) Answers will var. Ct.t.t, t t corresponds to 9. (a) The maimum consumption per ear of cigarettes per person per ear occurred in 9 t. Answers will var. For, C(). 9,7, 99 cigarettes per,, 99 smoker per ear, cigarettes per smoker per da. True, impossible. The parabola opens downward and the verte is,. Matches and (d).
Chapter Polnomial and Rational Functions 7. For f a is a a b c a b 9. For a f a a b >, c a b is a maimum when b In this case, the a. minimum when b In this case, the minimum a. maimum value is c b Hence, a. value is c b Hence, a. 7 b b b b ±. b b b b ±. 7. Model (a) is preferable. a > means the parabola opens upward and profits are increasing for t to the right of the verte, t b a. 7. Then and..,. 7. 9, Thus,, and, are the points of intersection. 77. Answers will var. (Make a Decision.) Section. Polnomial Functions of Higher Degree You should know the following basic principles about polnomials. f a n n a n n... a a a, a n, is a polnomial function of degree n. If f is of odd degree and (a) a n >, then. f as.. f as. a n <, then. f as.. f as. If f is of even degree and (a) a n >, then. f as.. f as. a n <, then. f as.. f as. CONTINUED
Section. Polnomial Functions of Higher Degree Section. CONTINUED The following are equivalent for a polnomial function. (a) a is a zero of a function. a is a solution of the polnomial equation f. a is a factor of the polnomial. (d) a, is an -intercept of the graph of f. A polnomial of degree n has at most n distinct zeros. If f is a polnomial function such that a < b and fa fb, then f takes on ever value between f a and f b in the interval a, b. If ou can find a value where a polnomial is positive and another value where it is negative, then there is at least one real zero between the values. Vocabular Check. continuous. Leading Coefficient Test. n, n, relative etrema. solution, a, -intercept. touches, crosses. Intermediate Value. f is a line with -intercept,.. f is a parabola with -intercepts Matches graph (f)., and, and opens downward. Matches graph.. f has intercepts, and ±,. Matches graph (e). 7. f has intercepts, and,. Matches graph (g). 9. (a) f Horizontal shift two units to the right f Vertical shift two units downward f (d) f Reflection in the -ais and a vertical shrink Horizontal shift two units to the right and a vertical shift two units downward
Chapter Polnomial and Rational Functions. f 9 ; g. f ; g f g g f. f 7. g 7 Degree: Degree: Leading coefficient: Leading coefficient: The degree is even and the leading coefficient is positive. The graph rises to the left and right. The degree is even and the leading coefficient is negative. The graph falls to the left and right. 9. Degree: (odd) Leading coefficient: > Falls to the left and rises to the right. ht t t Degree: Leading coefficient: The degree is even and the leading coefficient is negative. The graph falls to the left and right.. f. ht t t 9 7. f f ht t f ± ht (multiplicit ), 9. f t t t t. tt t, (multiplicit ) f ±.,. ± 7. (a). (a) 7.7,. t ± f gt t t t t ± ± t ±
Section. Polnomial Functions of Higher Degree 7 7. (a) 9. (a),.,..,. f f, ± ±. (a). (a),,, f or (multiplicit ) ±,. f 7. f 9 9 Zeros: ±., ±. Relative maimum:, Relative minimums:.,.,.,. Zeros:.7 Relative maimum:.,. Relative minimum:.,.7 9. f. f Note: f a a has Note: f a has zeros,, zeros and for all nonzero real numbers a. and for all nonzero real numbers a.
Chapter Polnomial and Rational Functions. f 9 9 Note: f a 9 has zeros,,, and for all nonzero real numbers a.. f Note: f a has zeros and for all nonzero real numbers a. 7. f 9. 7 Note: f a has zeros,, and for all nonzero real numbers a. f Note: f a has zeros,, and for all nonzero real numbers a.. f. Note: f a has zeros,,, for all nonzero real numbers a. f Note: f a, a <, has zeros,,, rises to the left, and falls to the right.. 7. 7 7 7 = + = + For eample, f. 9. (a) The degree of f is odd and the leading coefficient is. The graph falls to the left and rises to the right. f 9 9 Zeros:,, and (d) 7. (a) The degree of f is odd and the leading coefficient is. The graph falls to the left and rises to the right. f Zeros:, and (d)
Section. Polnomial Functions of Higher Degree 9 7. (a) The degree of f is even and the leading coefficient is. The graph falls to the left and falls to the right. f 9 Zeros: ±, ±: ±,, ±, and (d) 7. (a) The degree is odd and the leading coefficient is. The graph falls to the left and rises to the right. 9 7 9 Zeros:, :,,, and (d) 9 77. gt t t (a) Falls to left and falls to right; < and (d) gt t t t t,,,,,, ; zeros 7 79. f (a) 7.9.9.9...7..9979....7 The function has three zeros. The are in the intervals,,, and,. Zeros:.79,.7,...7....7.7...9..777..9..9...7.........7..
Chapter Polnomial and Rational Functions. g (a)..7.7..9.9.7. The function has two zeros. The are in the intervals, and,. Zeros:.,.779..7.......79.7.77.7.9..79...7..77.. 7. 9 9 f No smmetr Two -intercepts gt t t Smmetric about the -ais Two -intercepts f Smmetric to origin Three -intercepts 9. g 9 Three -intercepts No smmetr 9. (a) Volume length width height Because the bo is made from a square, length width. Thus: Volume length height Domain: < < < < > > Height, Length and Width Volume, V 7 7 7 7 7 Maimum volume for (d) 7 when V is maimum.
Section. Polnomial Functions of Higher Degree 9. The point of diminishing returns (where the graph changes from curving upward to curving downward) occurs when. The point is, which corresponds to spending $,, on advertising to obtain a revenue of $ million. 9. 97. For, t, and $7. thousand $. thousand. Answers will var. The model is a good fit. 99. True. f has onl one zero,.. False. The graph touches at, but does not cross the -ais there.. True. The eponent of is odd.. The zeros are,,, and the graph rises to the right. Matches. 7. The zeros are,,,, and the graph rises to 9. the right. Matches (a). g f g f 7 9. g f f.. g.. g f g f g 7. and and or and or and or 7. or or
Chapter Polnomial and Rational Functions Section. Real Zeros of Polnomial Functions You should know the following basic techniques and principles of polnomial division. The Division Algorithm (Long Division of Polnomials) Snthetic Division fk is equal to the remainder of f divided b k. fk if and onl if k is a factor of f. The Rational Zero Test The Upper and Lower Bound Rule Vocabular Check. fis the dividend, d is the divisor, q() is the quotient, and r is the remainder.. improper, proper. snthetic division. Rational Zero. Descartes s Rule, Signs. Remainder Theorem 7. upper bound, lower bound. )., ),. ) 7 7, 7. ) 7 7 7 7 7
Section. Real Zeros of Polnomial Functions 9. ). ) 9 9 9 9. ). 7 7 7 7, 7. 7 7 9. 9 7 9 7 7 9 9,.. 7,,. 7. 9 9 9 9 9
Chapter Polnomial and Rational Functions 9. f, k. f f f f. f f. f 7 (a) 7 f 7. h 7 (a) 7 9 h 7 7 f h 7 f 7 7 7 h (d) 7 (d) 7 f h 9. 7 Zeros:,, 7. 7 Zeros:,, 7 7
Section. Real Zeros of Polnomial Functions. (a) Remaining factors:, f () ( (d) Real zeros:, (e), 7 7. (a) Remaining factors:, f (d) Real zeros:,,, (e) 7 7 7. (a) Remaining factors:, 7 f 7 (d) Real zeros:,, (e) 9 9 7 9. f p factor of q factor of Possible rational zeros: ±, ± f Rational zeros: ±, 9. f 7 9 p factor of q factor of Possible rational zeros: ±, ±, ±, ±9, ±, ±, ± ± ± ± 9 ±,,,,, Using snthetic division,,, and are zeros. f Rational zeros:,,, ±
Chapter Polnomial and Rational Functions. z z z Possible rational zeros: ±, ±, ± z z z z z z The onl real zeros are and. You can verif this b graphing the function fz z z z.. 7 7. Using a graphing utilit and snthetic division,,, and are rational zeros. Hence,,,. Using a graphing utilit and snthetic division,,,, are rational zeros. Hence,,,,. 9. 7. Using a graphing utilit and snthetic division,,,, and, are rational zeros. Hence,,,,,. ht t t 7t (a) Zeros:,.7,. 7 ht t t t t is a zero. t t t. h 7 (a) h 7 From the calculator we have,, and ±.. 7 h The eact roots are,,, ±.. f 7. variations in sign, or positive real zeros f variations in sign negative real zeros g variations in sign or positive real zeros g variation in sign negative real zero
Section. Real Zeros of Polnomial Functions 7 9. f 7. f (a) f has variation in sign positive real zero. (a) f has variations in sign or positive real zeros. f has variations in sign or negative real zeros. f has variation in sign negative real zero. Possible rational zeros: ±, ±, ± Possible rational zeros: ±, ±, ±, ±, ± 7 (d) Real zeros:,, (d) Real zeros:,,, 7. f 7 7. f (a) f has variations in sign or positive real zeros. f 7 has variation in sign negative real zero. Possible rational zeros: ± ± ±,,, ± ± ±, ± ± ± ± ±,,,,, ±,, is an upper bound. is a lower bound. Real zeros:.97,.7 (d) Real zeros:,, 77. f 79. P 9 is an upper bound. 7 is a lower bound. Real zeros:, 7 7 9 9 The rational zeros are and ±. ±
Chapter Polnomial and Rational Functions. f. f Rational zeros: Irrational zeros: Matches (d). The rational zeros are and ±.. f Rational zeros:, ± Irrational zeros: Matches. 7. 9 9. Using the graph and snthetic division, is a zero: is a zero of the cubic, so. For the quadratic term, use the Quadratic Formula. 9 ± ± The real zeros are,, ±. 7 Using the graph and snthetic division, and are zeros: Using the Quadratic Formula: ± ± 7 The real zeros are,, ± 7. 9. (a) Pt.t.t.t. (d) For, t and:..... The model fits the data well...7.7..9 7.79 Hence, the population will be about 7. million, which seems reasonable.
Section. Real Zeros of Polnomial Functions 9 9. (a) Combined length and width: Volume, l w h Dimensions with maimum volume:, 7 Using the Quadratic Formula, or The value of, 7 7 ±. is not possible because it is negative. 9. False, 7 is a zero of f. 97. The zeros are,, and. The graph falls to the right. a a < Since f, a. 99. f. k k k k k Hence, k k 7. k k. (a),.,, In general, n n n...,. 9, 7. ± ± ±,
7 Chapter Polnomial and Rational Functions 9. f. [Answer not unique] f 7 [Answer not unique] Section. Comple Numbers You should know how to work with comple numbers. Operations on comple numbers (a) Addition: a bi c di a c b di Subtraction: a bi c di a c b di Multiplication: a bic di ac bd ad bci (d) Division: The comple conjugate of a bi is a bi: a bia bi a b The additive inverse of a bi is a bi. The multiplicative inverse of a bi is a bi a b. a a i for a >. a bi a bi c di c di c di c di ac bd bc ad c d c d i Vocabular Check. (a) ii iii i.,. comple, a bi. real, imaginar. Mandelbrot Set. a bi 9 i. a b i i. a 9 a a i b b b 7. i 9. i i i i. 7 7..9.9 i.i. i 7 i 7 i i 7. 7 i i 7 i
Section. Comple Numbers 7 9. i 7i i 7i i. i i i...i..i. 7.i 9 i 9 7 i. ii 7. i i i 9. i i i i i. i i i i i i i i. i i i. i i i i i i i i 7. i is the comple conjugate of i. 9. i is the comple conjugate of i. i i 9 i i. i is the comple conjugate of i.. i is the comple conjugate of i. ii i i 9. i i i i i i i i 7. i i i i i i 9. i i i i i i i i i i. i i i i i 9 i 9 i 9 i 9i 9 i
7 Chapter Polnomial and Rational Functions. i i i i i i i i i i. i i i i i i i i i i 9i 9 i 9i i i i 7i i 97i 99 97 99 99 i 7. i i i i i 9. i i i 7 i i i 7i. i i i i i i i i. i i i i i 9i i i 9 i i i i i i 9i i i 9 i The three numbers are cube roots of.. i 7. i 9. 7. i 7. i 7. Imaginar ais Imaginar ais Imaginar ais 7 7 Real ais Real ais Real ais
Section. The Fundamental Theorem of Algebra 7 77. The comple number is in the Mandelbrot Set since for c i, 79. z i i, the corresponding Mandelbrot z i sequence is i, i, i,,77 97, 9 i,,,,7,,9,97,9,7,7 i which is bounded. Or in decimal form.i,..i,.7.i,.7.i,.9.777i,..9i. 7 i, z z z i i i z i i i i i i i i i..97i. False. A real number a i a is equal to its conjugate.. False. For eample, i i, which is not an imaginar number.. True. Let z a b i and z a b i. Then z z a b ia b i a a b b a b b a i a a b b a b b a i a b ia b i a b i a b i z z. 7. 9. Section. The Fundamental Theorem of Algebra You should know that if f is a polnomial of degree n >, then f has at least one zero in the comple number sstem. (Fundamental Theorem of Algebra) You should know that if a bi is a comple zero of a polnomial f, with real coefficients, then a bi is also a comple zero of f. You should know the difference between a factor that is irreducible over the rationals such as 7 and a factor that is irreducible over the reals such as 9.
7 Chapter Polnomial and Rational Functions Vocabular Check. Fundamental Theorem, Algebra. Linear Factorization Theorem. irreducible, reals. comple conjugate. f. The three zeros are, and. f 9 i i Zeros: 9, ±i. f Zeros:, ±i The onl real zero of f is. This corresponds to the -intercept of, on the graph. 7. f Zeros: ±i, ±i f has no real zeros and the graph of f has no -intercepts. 9. h h has no rational zeros. B the Quadratic Formula, the zeros are ± ±. h. f f has no rational zeros. B the Quadratic Formula, the zeros are ± ±. f. f. i i The zeros of f are ±i. f 9 9 i i Zeros: ±, ± i 7. f z z z 9. f 9 z ± ± 9 i i i i The zeros of f are ±i and ±i. ± i f z z i z i
Section. The Fundamental Theorem of Algebra 7. f Using snthetic division, The zeros are, f i, i. is a zero: i i. ft t t t Possible rational zeros: B the Quadratic Formula, the zeros of t t are t ± ±, ±, ±, ± ± i. The zeros of ft are t and t ± i. ft t t it i t t it i. f 9 Possible rational zeros: ±, ± ± ±, ±, ±, ±, ±,, B the Quadratic Formula, the zeros of are those of : 9 ± ±i Zeros:, ±i f i i i i 7. g Possible rational zeros: ±, ±, ±, ±, ± g i i The zeros of g are,, and ±i. 9. (a) f. B the Quadratic Formula, ± 7 ±. The zeros are 7 and 7. f 7 7 7 7 -intercepts: 7, and 7, (d)
7 Chapter Polnomial and Rational Functions. (a) f (d) The zeros are -intercept:,, ±i. f i i. (a) (d) f Use the Quadratic Formula to find the zeros of. ± The zeros are, i, and i. f i i -intercept:, ± i.. (a) f 7. 9 The zeros are ±i, ±i. f 9 i i i i No -intercepts (d) f ) i i Note that f () a, where a is an nonzero real number, has zeros, ±i. 9. f ) i i). Because i is a zero, so is i. 7 Note that f () a, where a is an nonzero real number, has zeros,, ± i. f i i 7 Note that f a 7, where a is an nonzero real number, has zeros,, ± i.. (a) f a i i f a f a a f i i
Section. The Fundamental Theorem of Algebra 77. (a) f a i i 7. a a 9 f a 9 a f i i f 9 f 7 (a) f 7 f 7 7 f 7 7 i i 9. f (a) f f f i i. f 7 Since i is a zero, so is i. i i i i i i i i i i 7 7 The zero of is. The zeros of f are and ±i. Alternate Solution Since ±i are zeros of f, i i is a factor of f. B long division we have: ) 7 7 7 Thus, f and the zeros of f are ±i and.. g 7 7. Since i is a zero, so is i. i i 7 i i i i The zero of is. The zeros of f are, ± i. i i i i 7 7
7 Chapter Polnomial and Rational Functions. h. Since i is a zero, so is i. i i The zero of is The zeros of h are., ± i. 7. h 9. Since i is a zero, so is i. i i The zero of is The zeros of h are,. ± i. i i i i i i i i i i i i i i i i 9 9 9. f (a) The root feature ields the real roots and, and the comple roots ±.i. B snthetic division: The comple roots of are ± ± i.. h 9. t t, t (a) The root feature ields the real root.7, and the comple roots. ±.i. B snthetic division: 9 9 The comple roots of are t t t ± 79i Since the roots are imaginar, the ball never will reach a height of feet. You can verif this graphicall b observing that t t and do not intersect. ± ± i.
Section. Rational Functions and Asmptotes 79. False, a third degree polnomial must have at least one real zero. 7. f k (a) f has two real zeros each of multiplicit for k : f. f has two real zeros and two comple zeros if k <. 9. f 7 7 9 9 7. f Verte: 7, 7 Intercepts: f,,,,, f Intercepts:,,,,, 9 Verte:, 9 Section. Rational Functions and Asmptotes You should know the following basic facts about rational functions. (a) A function of the form f PQ, Q, where P and Q are polnomials, is called a rational function. The domain of a rational function is the set of all real numbers ecept those which make the denominator zero. If f PQ is in reduced form, and a is a value such that Qa, then the line a is a vertical asmptote of the graph of f. f or f as a. (d) The line b is a horizontal asmptote of the graph of f if f b as or. (e) Let f P where P and Q have no common Q a n n a n n... a a b m m b m m... b b factors.. If n < m, then the -ais is a horizontal asmptote.. If then a n n m, is a horizontal asmptote. b m. If n > m, then there are no horizontal asmptotes.
Chapter Polnomial and Rational Functions Vocabular Check. rational functions. vertical asmptote. horizontal asmptote. f (a) Domain: all. f (a) Domain: all f f f f.... 9.9..9 7..99..99 97..999..999 997. f f f f...7...9..77..99..97..99..997 f approaches from the left of and from the right of. f approaches from both the left and the right of.. f (a) Domain: all ± f..9.79.99 7..999 9. f.. 7.9.... f... f... f approaches from the left of, and from the right of. f approaches from the left of, and from the right of.
Section. Rational Functions and Asmptotes 7. f 9. f. f Vertical asmptote: Vertical asmptote: Vertical asmptote: Horizontal asmptote: Horizontal asmptote: Horizontal asmptote: Matches graph (a). Matches graph. Matches graph.. f. (a) Vertical asmptote: Horizontal asmptote: Holes: none f, (a) Vertical asmptote: Horizontal asmptote: Hole at :, 7. f 9. f, (a) Vertical asmptote: (a) Domain: all real numbers Vertical asmptote: none Horizontal asmptote: 7 Horizontal asmptote: Hole at :, 7. f. f, g (a) Domain: all real numbers ecept (a) Domain of f: all real numbers ecept Vertical asmptote: Domain of g: all real numbers Horizontal asmptote: to the right (to the left) 9 9 f, f has no vertical asmptotes. Hole at (d) 7 f 7 Undef. 9 g 7 9 (e) f and g differ at, where f is undefined.
Chapter Polnomial and Rational Functions. f, (a) Domain of f: all real numbers ecept, Domain of g: all real numbers ecept f, f has a vertical asmptote at. The graph has a hole at. g (d) f Undef. Undef. g (e) f and g differ at, where f is undefined. Undef. 7. f 9. f (a) As ±, f (a) As ±, f. As, f but is less than. As, f but is greater than. As, f but is greater than. As, f but is less than.. g The zeros of g are the zeros of the numerator: ±. f 7 The zero of f corresponds to the zero of the numerator and is 7.. g Zeros:, 7. f 7, Zero: is not in the domain. 9. C p p, p < (a) C. million dollars C 7 million dollars C7 7 7 million dollars 7 (d) (e) C as. No, it would not be possible to remove % of the pollutants.. (a) Use data,,. 9.7, 9..7.,,,.7,, 9.,..7.. 7.. (Answers will var.) No, the function is negative for 7.
Section.7 Graphs of Rational Functions. N (a) t.t, t N deer N deer N deer The herd is limited b the horizontal asmptote: N deer.. False. A rational function can have at most n vertical 7. There are vertical asmptotes at ±, and zeros asmptotes, where n is the degree of the denominator. at ±. Matches. 9. f. f.. 7 7 7. ) 9 9. 9 9 9 ) 9 9 9 9 Section.7 Graphs of Rational Functions You should be able to graph f p q. (a) Find the - and -intercepts. Find an vertical or horizontal asmptotes. Plot additional points. (d) If the degree of the numerator is one more than the degree of the denominator, use long division to find the slant asmptote. Vocabular Check. slant, asmptote. vertical
Chapter Polnomial and Rational Functions. g. g. g 7. g 7 f g f g f g g f f g f g f f g g Vertical shift one unit upward Reflection in the -ais Vertical shift two units downward Horizontal shift two units to the right 9. f -intercept:, Vertical asmptote: (, ) Horizontal asmptote:. C -intercept:,. f t t t t-intercept: t t, -intercept:, Vertical asmptote: Vertical asmptote: Horizontal asmptote: t Horizontal asmptote: (, ) t (, ) (, ) C 7. f Intercept:, Vertical asmptotes:, Horizontal asmptote: -ais smmetr (, )
Section.7 Graphs of Rational Functions 7. f Intercept:, Vertical asmptotes: Horizontal asmptote: Origin smmetr and (, ) 9. g Intercept:, Vertical asmptotes: and (, ) Horizontal asmptote: 7. f Intercept:, Vertical asmptotes:, Horizontal asmptote: (, ) 9 9. f Intercept:, Vertical asmptote:, There is a hole at. Horizontal asmptote: Undef.
Chapter Polnomial and Rational Functions. f The graph is a line, with a hole at., 7. f Vertical asmptote: Horizontal asmptote: 7 Domain: or,, 9. ft t t. ht t Vertical asmptote: t Domain: all real numbers OR, Horizontal asmptote: Horizontal asmptote: 7 7 Domain: t or,,. f. f 9 Domain: all real numbers ecept, Vertical asmptotes: Horizontal asmptote:, Domain: all real numbers ecept, OR,, Vertical asmptote: Horizontal asmptote: 9 9 7. h 9. g. f There are two horizontal asmptotes, ±. There are two horizontal asmptotes, ±. One vertical asmptote: The graph crosses its horizontal asmptote,.
Section.7 Graphs of Rational Functions 7. f. h Vertical asmptote: Intercept:, Slant asmptote: Vertical asmptote: Origin smmetr Slant asmptote: = = + (, ) 7. g 9. f Intercept:, Intercepts:.9,,, Vertical asmptotes: ± Slant asmptote: Slant asmptote: Origin smmetr 7 (, ) (, ) = + =. (a) -intercept:,. (a) -intercepts: ±, ±
Chapter Polnomial and Rational Functions. 7. Domain: all real numbers ecept Domain: all real numbers ecept Vertical asmptote: or,, Slant asmptote: Vertical asmptote: Slant asmptote: 9. f Vertical asmptotes:, Horizontal asmptote: No slant asmptotes, no holes. f. f,, Vertical asmptote: Horizontal asmptote: Long division gives f 7. No slant asmptotes Hole at,, 7 Vertical asmptote: No horizontal asmptote Slant asmptote: 7 Hole at,,. (a) 7 -intercept:,
Section.7 Graphs of Rational Functions 9 7. 9. 7. (a) (a) (a) 9 9 -intercept:, -intercept:,,,, -intercepts:.,,., ± 9 ±.,. 7. 7. (a) (a) 7 7 No -intercepts -intercepts:.7,,., No real zeros ± ±.7,.
9 Chapter Polnomial and Rational Functions 77. (a)..7 C 79. (a) A and Domain:..7 C Thus, 9. C C and 9 Thus, A. Domain: Since the margins on the left and right are each inch, >, or,. 9 As the tank fills, the rate that the concentration is increasing slows down. It approaches the horizontal asmptote C.7. When the tank is full 9, the concentration is C.7. The area is minimum when.7 in. and.7 in.. C, The minimum occurs when... C t t t, t (a) The horizontal asmptote is the t-ais, or C. This indicates that the chemical eventuall dissipates. The maimum occurs when t.. Graph C together with.. The graphs intersect at t. and t.. C <. when t <. hours and when t >. hours.. (a).t t corresponds to 99 Using the data,, t A, we obtain:.t..t. A 7. False, ou will have to lift our pencil to cross the vertical asmptote.
Section. Quadratic Models 9 9. h Since h is not reduced and is a factor of both the numerator and the denominator, is not a horizontal asmptote. There is a hole in the graph at., 9. a has a slant asmptote and a vertical asmptote. a a a a Hence,. 9. 9. 7 97. 7 99. Domain: all Range: Domain: all Range:. Answers will var. (Make a Decision) Section. Quadratic Models You should know how to Construct and classif scatter plots. Fit a quadratic model to data. Choose an appropriate model given a set of data. Vocabular Check. linear. quadratic. A quadratic model is better.. A linear model is better.. Neither linear nor quadratic
9 Chapter Polnomial and Rational Functions 7. (a) Linear model is better. (d) (e).., linear.7.7.9, quadratic 7 9.....9...... Model....7..9..... 9. (a) Quadratic model is better. (d) (e). 77. 7 7 7 Model 7 9 9 9 7. (a).., linear.7.9.7,.999 for linear model.999 for quadratic model Quadratic fits better. quadratic. (a).9., linear..9., quadratic.999 for the linear model.9997 for the quadratic model The quadratic model is slightl better.. (a) 7. (a) P.t.9t.7.t.7t 7. (d) The model s minimum is H. at t 7.. This corresponds to Jul. (d) According to the model, >, when t, or. (e) Answers will var.
Section. Quadratic Models 9 9. (a) (d).7t.999t., quadratic model,.999 (e) The cubic model is a better fit. (f) Year 7 t t 7 t A * 7.7..9 Cubic 9.9..9 Quadratic. 7. 9.. True. The model is above all data points.. (a) f g f g f g 7 7. (a) f g f g f g 9. f is one-to-one. f. f is one-to-one.,, f,. Imaginar ais. Imaginar ais Real ais Real ais
9 Chapter Polnomial and Rational Functions Review Eercises for Chapter. 9 d a b c 9. f Verte:, -intercept:, No -intercepts (a) is a vertical stretch. is a vertical stretch and reflection in the -ais. is a vertical shift two units upward. (d) the left. is a horizontal shift five units to. f Verte: -intercept:,, -intercepts: ± Use the Quadratic Formula. ±, 7. f() 7 Verte:, 7 Intercepts:,, ± 7, 7 7 9. Verte:, f a Point:, a a Thus, f.. Verte:, f a Point:, a a Thus, f.
Review Eercises for Chapter 9. (a) A since, Since the figure is in the first quadrant and and must be positive, the domain of A is < <. Area 7. 9 The maimum area of occurs at the verte when and. (d) The dimensions that will produce a maimum area seem to be and. A (e) The answers are the same.. The maimum area of occurs when and. A Total amount of fencing Area enclosed Because, A ( ) b The verte is at. Thus feet, a 9 and the dimensions are 7 feet b 7. feet. 9. 7.
9 Chapter Polnomial and Rational Functions 7. (a) (d) f() = ( + ) = f() = + = f() = = f() = ( + ) = 9. f ; g. f g f 9. The degree is even and the leading coefficient is negative. The graph falls to the left and right. f The degree is even and the leading coefficient is positive. The graph rises to the left and right.. (a) 7. (a) t t tt tt t Zeros: t, ± Zeros:,, Zeros:,, ; the same Zeros: t, ±.7, the same 9. (a) Zeros:, Zeros:,, the same. f. f 7 7. (a) Degree is even and leading coefficient is >. Rises to the left and rises to the right. 7 and (d) Zeros: ±,
Review Eercises for Chapter 97 7. f (a) f <, f > zero in, f >, f < zero in, f <, f > zero in, Zeros:.7,.,. 9. f (a) f >, f < zero in, f <, f > zero in, Zeros: ±.7... ) 7. ) Thus,. Thus,, ±. 9. ) Thus,, ±.
9 Chapter Polnomial and Rational Functions. ).. 9. 7 Hence,. 9 9 9 9. 7 Thus, 7 7,. 7. 9. (a) 7 f Thus,. f. f (a) 7 f 7 (d) Zeros:,, 7 is a factor. 7 7 Remaining factors:, 7. f 7 (a) is a factor. 7 9 Remaining factors:, f (d) Zeros:,,, is a factor.
Review Eercises for Chapter 99. f 7. Possible rational zeros: ±, ±, ±, ±, ±, ± Zeros:,, f Real zero: 9. f 7 Possible rational zeros: ±, ±, ±, ±, ±9, ±, ±, ±, ± 9, ±, ±, ± Use a graphing utilit to see that and are probabl zeros. 7 9 7 Thus, the zeros of f are,,, and. 7. g 9 has two variations in sign 7. or positive real zeros. g 9 has one variation in sign negative real zero. All entries positive; is upper bound. 7 Alternating signs; is lower bound. 7. i 77. i 7i 7i 79. 7 i i 7 i i 7i. i i i i i. i i. i i 7i 9 i i 9i 7. i i i i i i 9. 7i 7i 9 i 9 9 i 9 9. i i i i i i i i 9. i i i i i i i i i 7 7 i 9. i
Chapter Polnomial and Rational Functions 97. i 99. i. Imaginar ais Imaginar ais Imaginar ais Real ais 7 Real ais Real ais. f Zeros:,,. f 7. h 7 7 is a zero. is a zero. f is a zero. Appling the Quadratic Formula on, Zeros: ±9, h ± i. i, i i i B the Quadratic Formula, applied to, ± ± i. Zeros:,, ± i f i i 9. f i i Zeros:,,, ±i
Review Eercises for Chapter. f (a) B the Quadratic Formula, for, ± ± i. Zeros:, i, i f i i -intercept:,. (a) f 9 9 f -intercepts:,,,,, 7. f (a) 9 (d) Zeros: ±i, ±i i i i i No -intercepts 7. Since i is a zero, so is i. 9. f i i 7 f i i 9 7. f 9 (a) f 9 For the quadratic, ± f 9 f i i ±.. Zeros: i, i i i is a factor. f Zeros: ±i,. (a) Domain: all Horizontal asmptote: Vertical asmptote: 7. f 9. f 7 7 (a) Domain: all, (a) Domain: all 7 Horizontal asmptote: Vertical asmptotes:, Horizontal asmptote: Vertical asmptote: 7
Chapter Polnomial and Rational Functions. f. f, (a) Domain: all ± ± Horizontal asmptote: Vertical asmptotes: ± ± (a) Domain: all, Vertical asmptote: There is a hole at. Horizontal asmptote:. f (a) Domain: all real numbers No vertical asmptotes Horizontal asmptotes:, 7. C (a) When p p, p < When When p, p, p 7, C 7 C C 7 7 No. As p, C tends to infinit.. million. million. million. 9. f, Vertical asmptotes: Horizontal asmptote: No slant asmptotes Hole at :,. f 7 9. f,, Vertical asmptote:, Horizontal asmptote: Vertical asmptote: No slant asmptotes No horizontal asmptotes Hole at :, 9 Slant asmptote: Hole at :,
Review Eercises for Chapter. f Intercepts: Vertical asmptote: Horizontal asmptote: (, ) (, ),,, 7. f Intercept:, Origin smmetr Horizontal asmptote: (, ) 9. f Intercept:, -ais smmetr Horizontal asmptote: ± ± ± 9 (, ). f Intercept:, Horizontal asmptote: Vertical asmptote: (, ). f Intercept:, Origin smmetr Slant asmptote: (, ) =. f Intercept: Vertical asmptote: Slant asmptote:, 7 (, ) = +
Chapter Polnomial and Rational Functions 7. t N (a) fish N,.t, t N, fish N 7, fish The limit is,,. fish, the horizontal asmptote. 9. Quadratic model. Linear model. (a).t 7.t ;.9 Yes, the model is a good fit. (d) From the model, when t., or. (e) Answers will var.. False. The degree of the numerator is two more than the degree of the denominator. 9. Not ever rational function has a vertical asmptote. For eample,. 7. False. i i, a real number 7. The error is i. In fact, i ii i.
Practice Test for Chapter Chapter Practice Test. Sketch the graph of f b hand and identif the verte and the intercepts.. Find the number of units that produce a minimum cost C if C. 9,.. Find the quadratic function that has a maimum at, 7 and passes through the point,.. Find two quadratic functions that have -intercepts, and,.. Use the leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polnomial function f 7.. Find all the real zeros of f. Verif our answer with a graphing utilit. 7. Find a polnomial function with,, and as zeros.. Sketch f b hand. 9. Divide 7 b. Divide b. using long division.. Use snthetic division to divide b.. Use snthetic division to find f when f 7.. Find the real zeros of f 9.. Find the real zeros of f 9 9.. List all possible rational zeros of the function f.. Find the rational zeros of the polnomial f 9.. Write in standard form. i 7. Write f as a product of linear factors. i 9. Write i in standard form.. Find a polnomial with real coefficients that has, i, and i as zeros.. Use snthetic division to show that i is a zero of f 9.. Find a mathematical model for the statement, z. Sketch the graph of f and label all varies directl as the square of and inversel as intercepts and asmptotes. the square root of.. Sketch the graph of f and label all intercepts and asmptotes.. Find all the asmptotes of f 9.. Find all the asmptotes of f 7. 7. Sketch the graph of f.