C H A P T E R Polnomial Functions Section. Quadratic Functions and Models............. 9 Section. Polnomial Functions of Higher Degree......... Section. Polnomial and Snthetic Division............ 8 Section. Zeros of Polnomial Functions.............. 0 Section. Mathematical Modeling and Variation..........8 Review Eercises............................ 7 Problem Solving............................. 8 Practice Test.............................. 87
C H A P T E R Polnomial Functions Section. Quadratic Functions and Models You should know the following facts about parabolas. f a b c, a 0, is a quadratic function, and its graph is a parabola. If a > 0, the parabola opens upward and the verte is the point with the minimum -value. If a < 0, the parabola opens downward and the verte is the point with the maimum -value. The verte is ba, f ba. To find the -intercepts (if an), solve a b c 0. The standard form of the equation of a parabola is f a h k where a 0. (a) The verte is h, k. (b) The ais is the vertical line h. Vocabular Check. nonnegative integer; real. quadratic; parabola. ais or ais of smmetr. positive; minimum. negative; maimum. f opens upward and has verte, 0.. f opens upward and has verte, 0. Matches graph (g). Matches graph (c).. f opens upward and has verte 0,.. f opens downward and has verte 0,. Matches graph (b). Matches graph (h).. f opens downward. f opens upward and has verte and has verte,. Matches graph (f).,. Matches graph (a). 7. f opens downward and has 8. f opens downward and has verte, 0. verte,. Matches graph (e). Matches graph (d). 9
Section. Quadratic Functions and Models 97 9. (a) (b) 8 Vertical shrink Vertical shrink and reflection in the -ais (c) (d) Vertical stretch Vertical stretch and reflection in the -ais 0. (a) (b) Vertical translation one unit upward Vertical translation one unit downward (c) (d) 0 8 8 Vertical translation three units upward Vertical translation three units downward
98 Chapter Polnomial Functions. (a) (b) Horizontal translation one unit to the right Horizontal shrink and a vertical translation one unit upward (c) (d) 8 0 8 8 Horizontal stretch and a vertical translation three units downward Horizontal translation three units to the left. (a) (b) 8 0 8 8 0 8 8 (c) Horizontal translation two units to right, vertical shrink each -value is multiplied b, reflection in the -ais, and vertical translation one unit upward Horizontal translation one unit to the right, horizontal stretch (each -value is multiplied b ), and vertical translation three units downward (d) 7 8 8 Horizontal translation two units to left, vertical shrink each -value is multiplied b, reflection in -ais, and vertical translation one unit downward Horizontal translation one unit to left, horizontal shrink each -value is multiplied b, and vertical translation four units upward
Section. Quadratic Functions and Models 99. f. h Verte: 0, Verte: 0, Ais of smmetr: 0 or the -ais Ais of smmetr: 0 Find -intercepts: Find -intercepts: 0 0 0 ± -intercepts:, 0,, 0 ± -intercepts: ±, 0 0 0 0 0. f 0. f Verte: 0, Verte: 0, Ais of smmetr: 0 or the -ais Ais of smmetr: 0 Find -intercepts: 0 8 -intercepts: ±8 ±, 0,, 0 Find -intercepts: 0 ±8 -intercepts: ±8, 0 9 8 9 9 7. f 8. Verte:, Ais of smmetr: Find -intercepts: 0 ± ± 0 8 -intercepts:, 0,, 0 0 8 f Verte:, Ais of smmetr: Find -intercepts: 0 Not possible for real No -intercepts 0 0 0 0 0 0 0 0 0 0 9. h 8 0. g Verte:, 0 Ais of smmetr: -intercept:, 0 0 8 8 Verte:, 0 Ais of smmetr: -intercept:, 0
00 Chapter Polnomial Functions. f. f 9 9 Verte:, Verte:, Ais of smmetr: Ais of smmetr: Find -intercepts: Find -intercepts: 0 0 Not a real number No -intercepts ± ±9 ± -intercepts: ±, 0. f. f Verte:, Verte:, Ais of smmetr: Ais of smmetr: Find -intercepts: Find -intercepts: 0 0 0 0 ± 0 ± ± ± -intercepts: ±, 0 -intercepts:, 0,, 0
Section. Quadratic Functions and Models 0. h. f 0 Verte:, 0 7 8 Ais of smmetr: Verte:, 7 8 Find -intercepts: 0 Not a real number ± No -intercepts Ais of smmetr: Find -intercepts: Not a real number No -intercepts 0 ± 8 0 0 8 8 7. f 8. f 8 9 9 8 8 Verte:, Ais of smmetr: Verte: 9 9, Find -intercepts: Ais of smmetr: 9 0 8 8 0 0 or -intercepts:, 0,, 0 8 0 8 Find -intercepts: 0 9 8 0 0 -intercepts:, 0,, 0 8 0 9. f 0. f 0 Verte:, Ais of smmetr: -intercepts:, 0,, 0 8 7 Verte: 0 0, 0 0 Ais of smmetr: -intercepts:, 0,, 0 80
0 Chapter Polnomial Functions. g 8. Verte:, Ais of smmetr: -intercepts: ±, 0 8 f 0 0 Verte:, 0 0 Ais of smmetr: -intercepts: ±, 0. f 8. Verte:, Ais of smmetr: -intercepts: ±, 0 f Verte: 9, Ais of smmetr: No -intercepts 0 0 0. g. Verte:, Ais of smmetr: -intercepts: ±, 0 8 f 9 7 Verte:, 0 Ais of smmetr: -intercepts: ±, 0 0 0,, 7., 0 is the verte. 8. 0, is the verte. a 0 a f a 0 a Since the graph passes through the point we have: Since the graph passes through, 0, a0 0 a a a. So,. 9., is the verte. 0., is the verte. a f a Since the graph passes through the point, 0, we have: 0 a Since the graph passes through 0,, a0 a a a a a. So,.
Section. Quadratic Functions and Models 0., is the verte.., 0 is the verte. a f a 0 a Since the graph passes through the point, 0, we have: Since the graph passes through,, 0 a a a a. So,.., is the verte.., is the verte. f a f a Since the graph passes through the point 0, 9, we have: 9 a0 Since the graph passes through,, a a a a a f a. So, f.., is the verte.., is the verte. f a Since the graph passes through the point,, we have: a a a f f a Since the graph passes through 0,, a0 a a a. So, f. 7., is the verte. 8., is the verte. f a f a Since the graph passes through the point 7,, we have: a7 Since the graph passes through, 0, 0 a a a 0 a f a. So, f., 9., is the verte. 0. is the verte. f a Since the graph passes through the point we have: 0 a 9 a a 9 f 9, 0, f a Since the graph passes through,, a 8 a 9 8 a 9 8 a. So, f 9 8.
0 Chapter Polnomial Functions., 0 is the verte.., is the verte. f a f a Since the graph passes through the point we have: a 7 a f 7,, Since the graph passes through a 0 00a 9 00a 0 a. 0,, So, f 0.. 0. 9 -intercepts: ±, 0 -intercept:, 0 ± 0 9 0 0. 0. -intercepts:, 0,, 0 0 -intercepts:, 0,, 0 or 0 0 0 0 7. f 8. f 0 -intercepts: 0, 0, (,0 -intercepts: 0, 0,, 0 0 8 0 0 0 ) 0 0 or 0 0 The -intercepts and the solutions of f 0 are the same. 0 The -intercepts and the solutions of f 0 are the same. 9. f 9 8 0. f 8 0 0 -intercepts:, 0,, 0 -intercepts:, 0, 0, 0 0 9 8 8 0 8 0 0 ) 0 0 0 or 0 The -intercepts and the solutions of f 0 are the same. 0 0 0 The -intercepts and the solutions of f 0 are the same.
Section. Quadratic Functions and Models 0. f 7 0 0. -intercepts:, 0,, 0 0 7 0 0 ) or The -intercepts and the solutions of 0 0 f 0 are the same. f -intercepts: 7, 0,, 0 0 0 7 7 0 7 0 The -intercepts and the solutions of 0 9 70 f 0 are the same.. f 0 7. -intercepts:, 0, 7, 0 0 7 0 7 0 7 or 7 0 The -intercepts and the solutions of f 0 are the same. f 7 0 -intercepts:, 0,, 0 0 7 0 0 0 0 The -intercepts and the solutions of 0 8 0 f 0 are the same.. f opens upward. g opens downward Note: f a has -intercepts, 0 and, 0 for all real numbers a 0. f, opens upward g f, opens downward g Note: f a has -intercepts, 0 and, 0 for all real numbers a 0. 7. f 0 0 opens upward 8. 0 g 00 opens downward 0 Note: f a 0 0 a 0 has -intercepts 0, 0 and 0, 0 for all real numbers a 0. f 8, opens upward g f, opens downward g Note: f a 8 has -intercepts, 0 and 8, 0 for all real numbers a 0. 9. f opens upward 70. f g 7 Note: 7 7 opens downward f a has -intercepts, 0 and, 0 for all real numbers a 0. 0, opens upward g f, opens downward g 0 Note: f a has -intercepts, 0 and, 0 for all real numbers a 0.
0 Chapter Polnomial Functions 7. Let the first number and the second number. Then the sum is 0 0. 7. Let first number and second number. Then, S, S. The product is P S. The product is P 0 0. P S P 0 S 0 0 0 0 S S 0 The maimum value of the product occurs at the verte of P and is 0. This happens when. S S S The maimum value of the product occurs at the verte of P and is S. This happens when S. 7. Let the first number and the second number. 7. Let the first number and the second number. Then the sum is Then the sum is.. The product is P P. The product is P P. 7 The maimum value of the product occurs at the verte of P and is 7. This happens when and. Thus, the numbers are and. 7 The maimum value of the product occurs at the verte of P and is 7. This happens when and 7. Thus, the numbers are and 7. 7. (a) (b) A 00 This area is maimum when feet and 00 feet. 0 0 00 00 00 0 0 00 (c) A 0 8 80 0 000 0 00 0 0 0 This area is maimum when feet and 00 feet. CONTINUED
Section. Quadratic Functions and Models 07 7. CONTINUED (d) A 8 0 8 0 (e) The are all identical. feet and feet 8 0 8 8 000 The maimum area occurs at the verte and is 000 square feet. This happens when feet and 00 00 feet. The dimensions are 0 feet b feet. 7. (a) Radius of semicircular ends of track: r Distance around two semicircular parts of track: d r (b) Distance traveled around track in one lap: d 00 00 00 (c) Area of rectangular region: A 00 00 00 00 00 00 0 000 The area is maimum when 0 and 00 0 00. 77. 9 9 The verte occurs at b 9. The maimum height is feet. 9 a 9 9 78. 0 9. (a) The ball height when it is punted is the -intercept. 0 0 9 0.. feet (b) The verte occurs at The maimum height is b a 9 0 f 0 9..., 9 7 feet 0.0 feet. CONTINUED
08 Chapter Polnomial Functions 78. CONTINUED (c) The length of the punt is the positive -intercept. 0 0 9. 9 ± 9.0 0 0.80 or 8. The punt is approimatel 8. ft..8 ±.8 0.0807 79. C 800 0 0. 0. 0 800 80. The verte occurs at b a 0 0. 0. The cost is minimum when 0 fitures. C 00,000 0 0.0 The verte occurs at 0. 0.0 The cost is minimum when units. 8. P 0.000 0 0,000 8. P 0 0 0. The verte occurs at The profit is maimum when b a 0 0,000. 0.000 0,000 units. The verte occurs at b 0 0. a 0. Because is in hundreds of dollars, 0 00 000 dollars is the amount spent on advertising that gives maimum profit. 8. Rp p 00p 8. (a) R0 $,000 thousand R $,7 thousand R0 $,00 thousand (b) The revenue is a maimum at the verte b 00 a R,00 The unit price that will ield a maimum revenue of $,00 thousand is $. R p p 0p (a) R$ $ 0$ $08 R$ $ 0$ $8 R$8 $8 0$8 $ (b) The verte occurs at p b a 0 $. Revenue is maimum when price $. per pet. The maimum revenue is f$. $. 0$. $8.7. 8. C 99.8t.t, 0 t (a) (c) 000 0 0 C0 0 Annuall: Dail: 8879 09,8,090 8,08,90 cigarettes 8879 cigarettes (b) Verte 0, 99 The verte occurs when 99 which is the maimum average annual consumption. The warnings ma not have had an immediate effect, but over time the and other findings about the health risks and the increased cost of cigarettes have had an effect.
Section. Quadratic Functions and Models 09 8. (a) and (c) 90 87. (a) 0 00 (b) (d) 99 (e) Verte occurs at (f) 0.0 9.98 88.8 Minimum occurs at ear 99. 8 b 9.98.8 a.0.08 9.988 88.8 8.88 There will be approimatel,8,000 hairdressers and cosmetologists in 008. (b) 0.00s 0.00s 0.09 0 a, b, c 0,09 s ± 0,09 s s s 0,09 0 ± 80,7 s 7., 9. s s 9 0,000 The maimum speed if power is not to eceed 0 horsepower is 9. miles per hour. 88. (a) and (c) 0 80 0 (b) 0.008 0.7.7 (d) The maimum of the graph is at., or about. mi/h. Algebraicall, the maimum occurs at b a 0.7 0.008. mi/h. 89. True. The equation 0 has no real solution, 90. True. The verte of f is and the verte of so the graph has no -intercepts. is, 7,. g 9. f a b c a b a c a b a b a a b a b a c a b a f b a a b a b b a c b b a a c b b b ac a a c ac b a ac b a So, the verte occurs at b ac b, a a b a, f a b.
0 Chapter Polnomial Functions 9. Conditions (a) and (d) are preferable because profits would be increasing. 9. Yes. A graph of a quadratic equation whose verte is has onl one -intercept. 0, 0 9. If f a b c has two real zeros, then b the Quadratic Formula the are b ± b ac. a The average of the zeros of f is b b ac a b b ac a This is the -coordinate of the verte of the graph. b a b a. 9., and, 9. m 7,, m 7 97. 0 and m 98. The slope of the perpendicular line through 0, is m and the -intercept is b. m For a parallel line, m. So, for 8,, the line is 8 0. For Eercises 99 0, let f, and g 8. 99. f g f g 00. g f 8 8 7 8 7 0. fg 7 f 7 g 7 7 8 7 8 9 08 9 0. f g.. 8. 8 0. f g fg f 8 8 09 0. g f 0 g f 0 g0 g 8 7