5. 2. The solution set is 7 6 i, 7 x. Since b = 20, add

Transcription

1 Chapter : Quadratic Equations and Functions Chapter Review Eercises The solution set is 8, The solution set is 5,5. Rationalize the denominator. 6 The solution set is ,. The solution set is, i The solution set is 76 i, 7 6 i. b. Since b =, add b 9 Since b =, add Since b 6 6. b, add Apply the square root property The solution set is, Copyright Pearson Education, Inc.

2 Introductory and Intermediate Algebra for College Students E Chapter Review Since b 7, add b 7 9. Apply the square root property The solution set is. Since b, add b Apply the square root property. 6 The solution set is.. A P r 96 5 r 96 r 5 Apply the square root property. r 96 5 t r.66 r.8 The solutions are.8 =.8 and +.8 =.8. Disregard.8 since we cannot have a negative interest rate. The interest rate is.8 or 8%.. W t t 588 t 96 t Apply the square root property. t 96 t 96 t The solutions are and. Disregard, because we cannot have a negative time measurement. The fetus will weigh 588 grams after weeks. Copyright Pearson Education, Inc. 67

3 Chapter : Quadratic Equations and Functions Midpoint,,, Use the Pythagorean Theorem. 9, 5 9, 8, 8, The solutions are 6 5 meters. Disregard 6 5 meters, because we can t have a negative length measurement. Therefore, the building is 6 5 meters, or approimately. meters high.. 9 d d Midpoint,, 5, a b c The solution set is 5. 9 a b c i i i The solution set is i. 68 Copyright Pearson Education, Inc.

4 Introductory and Intermediate Algebra for College Students E Chapter Review a b c 6 The solution set is. a b c Find the discriminant. b ac Since the discriminant is negative, there are two imaginary solutions which are comple conjugates. 9 9 a 9 b c Find the discriminant. b ac Since the discriminant is greater than zero and a perfect square, there are two real rational solutions. a b c Find the discriminant. b ac 6 Since the discriminant is greater than zero but not a perfect square, there are two real irrational solutions. Apply the zero-product principle. and The solution set is, Use the quadratic formula. a b c 9 ( ) or The solution set is 5,. 5 Use the quadratic formula. a 5 b c 5 5 ( ) The solution set is. 6 6 Apply the square root principle. 6 The solution set is,. Copyright Pearson Education, Inc. 69

5 Chapter : Quadratic Equations and Functions Apply the square root principle. 8 The solution set is. Use the quadratic formula. a b c 6 i The solution set is i Use the quadratic formula. a b 8 c The solution set is 5.. Because the solution set is,, we have 5 or Apply the zero-product principle in reverse Because the solution set is i i 9, 9, we have 9i or 9i 9i 9i. Apply the zero-product principle in reverse. 9i9i 9i9i8i 8 8. Because the solution set is,, we have or. Apply the zero-product principle in reverse Copyright Pearson Education, Inc.

6 Introductory and Intermediate Algebra for College Students E Chapter Review. a. 5. a. g ( ).5. 7 g(5).5(5).(5) 7 6 On wet pavement, a motorcycle traveling at 5 miles per hour will require a stopping distance of 6 feet. This answer overestimates the stopping distance shown in the graph by foot. b. f( ) b b ac a (.8) (.8) (.5)( 68) (.5).6 or On dry pavement, a stopping distances of 67 feet will be required for a motorcycle traveling miles per hour. g(5).5(5).(5) 7 6 This value is shown in the graph by the point (5, 6). f 7. Since a is negative, the parabola opens downward. The verte of the parabola is hk,, and the ais of symmetry is. Replace f with to find intercepts. Apply the square root property. or The intercepts are and. Set = and solve for y to obtain the y intercept. y y y 6. b. f ().5().8() This value is shown in the graph by the point (, 67). 6t t Apply the Pythagorean Theorem. a 6 b c 9, 6 9 9, or or 8.8 or. Disregard. because we cannot have a negative time measurement. The solution is approimately 8.8. The ball will hit the ground in about 8.8 seconds. Ais of symmetry:. f 8. Since a is positive, the parabola opens upward. The verte of the parabola is hk,, and the ais of symmetry is. Replace f with to find intercepts. Apply the square root property. or The intercepts are and. Set and solve for y to obtain the y intercept. Copyright Pearson Education, Inc. 6

7 Chapter : Quadratic Equations and Functions y y y 6 y The intercepts are and. Set = and solve for y to obtain the y intercept. y y y Ais of symmetry:. f 9. Since a is negative, the parabola opens downward. The coordinate of the verte of the b parabola is and the y a coordinate of the verte of the parabola is b f f a. The verte is (, ). Replace f intercepts. Apply the zero-product principle. or with to find Ais of symmetry:. f 6. Since a is positive, the parabola opens upward. The coordinate of the verte of the parabola is b and the a y coordinate of the verte of the parabola is b f f a The verte is, 8. intercepts. 6 Replace f Apply the zero-product principle. or with to find The intercepts are and. Set = and solve for y to obtain the y intercept. y 6 6 Copyright Pearson Education, Inc.

8 Introductory and Intermediate Algebra for College Students E Chapter Review y 6 y 6 6 Ais of symmetry:. f.. Since a. is negative, the function opens downward and has a maimum at b 5. a.. When 5 inches of rain falls, the maimum growth will occur. The maimum growth is f A maimum yearly growth of.5 inches occurs when 5 inches of rain falls per year. s t 6t t. Since a 6 is negative, the function opens downward and has a maimum at b.5. a 6 At.5 seconds, the rocket reaches its maimum height. The maimum height is s The rocket reaches a maimum height of 5 feet in.5 seconds. f Since a is positive, the function opens upward and b 5.5 has a minimum at 7.. a.5 At 7. hours, the death rate reaches its minimum. The minimum death rate is f f 7..5(7.) 5.5(7.) 66 6 U.S. men who average 7. hours of sleep have a death rate of about 6 per,.. Maimize the area using A lw. A A Since a is negative, the function opens downward and has a maimum at b 5. a The maimum area is achieved when the width is 5 yards. The maimum area is A ,. The area is maimized at 5, square yards when the width is 5 yards and the length is 5 5 yards. 5. Let = one of the numbers. Let + = the other number. We need to minimize the function P The minimum is at. b 7. a The other number is 7 7. The numbers which minimize the product are 7 and 7. The minimum product is Copyright Pearson Education, Inc. 6

9 Chapter : Quadratic Equations and Functions 6. Let u. 8. Let u u 6u8 u u u or u u u Replace u by. or The solution set is,,,. 7. Let u u 7u8 u8u Apply the zero-product principle. u8 or u u 8 u Replace u by. 8 or Disregard 8 because the square root of cannot be a negative number. 5 5 u u5 5 u u u5 or u u 5 u Replace u by. First, consider u = or 5 Net, consider u =. The solution set is 5,,. We must check, because both sides of the equation were raised to an even power. Check: The solution set is. 6 Copyright Pearson Education, Inc.

10 Introductory and Intermediate Algebra for College Students E Chapter Review 9. Let u u u u u u8 or u7 u 8 u 7 Replace u by. 5. Let 8 or The solution set is,. 8 7 u. u u uu u or u u u Replace u by. or 6 7 The solution set is 7, Let u. u u u5u u5 or u u 5 u Replace u by. 5 or 5 6 Disregard 5 because the fourth root of cannot be a negative number. We must check 6, because both sides of the equation were raised to an even power. Check The solution checks. The solution set is 6. Copyright Pearson Education, Inc. 65

11 Chapter : Quadratic Equations and Functions 5. 5 Solve the related quadratic equation. 5 or The boundary points are and. Interval Test Value Test Conclusion, 5 false, does not belong to the solution set.,, 5 true 5 false, belongs to the solution set., does not belong to the solution set. The solution set is, Solve the related quadratic equation. 9 or The boundary points are and. Interval Test Value Test Conclusion, true, belongs to the solution set. 5, 9 false, 9 true The solution set is,,., does not belong to the solution set., belongs to the solution set. 66 Copyright Pearson Education, Inc.

12 Introductory and Intermediate Algebra for College Students E Chapter Review 5. Solve the related polynomial equation. or or The boundary points are,, and. Interval Test Value Test Conclusion, False, does not belong to the solution set., True, belongs to the solution set..5, False, does not belong to the solution set., True, belongs to the solution set.. The solution set is,, Find the values of that make the numerator and denominator zero. 6 6 The boundary points are and 6. Interval Test Value Test Conclusion 6, true, belongs to the solution set. 6, 6 false, 6 does not belong to the solution set , 7 true 7 6, belongs to the solution set.. The solution set is, 6, Copyright Pearson Education, Inc. 67

13 Chapter : Quadratic Equations and Functions Epress the inequality so that one side is zero Find the values of that make the numerator and denominator zero. and The boundary points are and. Eclude from the solution set, since this would make the denominator zero. Interval Test Value Test Conclusion, 5 5, true, belongs to the solution set. 5 5, 5 5, does not belong to the solution set. 8 5, false, , true The solution set is,,., belongs to the solution set. 68 Copyright Pearson Education, Inc.

14 Introductory and Intermediate Algebra for College Students E Chapter Review st 6t 8t 57. To find when the height is more than feet above the ground, solve the inequality 6t 8t. Solve the related quadratic equation. 6t 8t 6t 8t t t t t t or t t t The boundary points are and. Interval Test Value Test Conclusion,.5,.5, , false , true, false, does not belong to the solution set., belongs to the solution set., does not belong to the solution set. The solution set is,. This means that the ball will be more than feet above the graph between and seconds. 5 5 H 8 8 The heart rate is beats per minute immediately following the workout. 58. a. b Apply the zero-product principle. or Copyright Pearson Education, Inc. 69

15 Chapter : Quadratic Equations and Functions The boundary points are and. Interval Test Value Test Conclusion, , true 8, belongs to the solution set., , false 8, does not belong to the solution set., , true 8, does not belong to the solution set. The solution set is,,. This means that the heart rate eceeds beats per minute between and minutes after the workout and more than minutes after the workout. Between and minutes provides a more realistic answer since it is unlikely that the heart rate will begin to climb again without further eertion. Model breakdown occurs for the interval,. Chapter Test Rationalize the denominators. 5 The solution set is The solution set is 5. 6 Copyright Pearson Education, Inc.

16 Introductory and Intermediate Algebra for College Students E Chapter Test.. 6 Since 6 b b, add Since b = 5, add b d 5 6 midpoint, 7 8, 7, Since 6 b 6 9. b, add Apply the square root property. The solution set is. 6. Use the Pythagorean Theorem The solutions are 5 feet. Disregard 5 feet because we can t have a negative length measurement. The width of the pond is 5 feet a b c Find the discriminant. b ac 6 Since the discriminant is greater than zero but not a perfect square, there are two real irrational solutions. 8 8 a b c 8 Find the discriminant. b ac Since the discriminant is negative, there are two imaginary solutions which are comple conjugates Apply the zero-product principle. and 5 5 The solution set is 5,. Copyright Pearson Education, Inc. 6

17 Chapter : Quadratic Equations and Functions. 85 Solve using the quadratic formula. a b 8 c The solution set is Apply the square root principle. 5 5i The solution set is 5 i. 65 a b 6 c i 6 i i The solution set is. i 5. Because the solution set is, 7, we have or 7 7 Apply the zero-product principle in reverse Because the solution set is i i,, we have i or i i i Apply the zero-product principle in reverse. i i i 7. a. is 8 years after. f( ).7 66 f (8).7(8) 6(8) According to the function, in there were 8 Bicycle Friendly communities. This overestimates the number shown in the graph by. b. f( ) b b ac a (6) (6) (.7)( 8) (.7) (.7) or 9 7 According to the function, there will be 86 Bicycle Friendly communities years after, or. 6 Copyright Pearson Education, Inc.

18 Introductory and Intermediate Algebra for College Students E Chapter Test f 8. Since a is positive, the parabola opens upward. The verte of the parabola is hk,, and the ais of symmetry is. Replace f with to find intercepts. The intercepts are and. Set = and solve for y to obtain the y intercept. y This will be result in comple solutions. As a result, there are no intercepts. Set = and solve for y to obtain the y intercept. y 5 Ais of symmetry:. st 6t 6t 5. Since a 6 is negative, the function opens downward and has a maimum at b 6 6. a 6 Ais of symmetry:. f 9. Since a is positive, the parabola opens upward. The coordinate of the verte of the parabola is b and the a y coordinate of the verte of the parabola is b f f a. The verte is,. intercepts. Replace f Apply the zero-product principle. or with to find. The ball reaches its maimum height in two seconds. The maimum height is s The baseball reaches a maimum height of 69 feet after seconds. 6t 6t 5 Solve using the quadratic formula. a 6 b 6 c or. Disregard. since we cannot have a negative time measurement. The solution is. and we conclude that the baseball hits the ground in approimately. seconds. Copyright Pearson Education, Inc. 6

19 Chapter : Quadratic Equations and Functions f 6 6. Since a is negative, the function opens downward and has a maimum at b 6 6. a f Profit is maimized when computers are manufactured. This produces a profit of $69 hundreds or $6,9.. Let u u u u u u or u u u Replace u by 5. 5 or 5 The solution set is,.. Let u. 6 6 u u6 9 u u u9 or u u 9 u Replace u by. or 9 The solution set is,,,. 6 Copyright Pearson Education, Inc.

20 Introductory and Intermediate Algebra for College Students E Chapter Test 5. Let u /. / / 9 8 / / 9 8 u 9u8 8 u u u8 or u u 8 u Replace u / by. / 8 or / 8 5 The solution set is, Solve the related quadratic equation. or The boundary points are and. Interval Test Value Test Conclusion,,, 5 The solution set is,. 8, false, true 5 5 8, false, does not belong to the solution set., belongs to the solution set., does not belong to the solution set. Copyright Pearson Education, Inc. 65

21 Chapter : Quadratic Equations and Functions 7. Epress the inequality so that one side is zero. 9 Find the values of that make the numerator and denominator zero. and The boundary points are and. Eclude from the solution set, since this would make the denominator zero. Interval Test Value Test Conclusion,, true, belongs to the solution set.,, 9, false, true 8, does not belong to the solution set., belongs to the solution set.. The solution set is,, 66 Copyright Pearson Education, Inc.

22 Introductory and Intermediate Algebra for College Students E Chapter Test Cumulative Review Eercises (Chapters ) The solution set is.. y 7 y 9 Multiply the second equation by and add the result to the first equation. y 7 y Back substitute into the second equation. 5y 9 y y The solution set is 5,.. yz 9 yz 6 5yz 5 Multiply the second equation by and add to the first equation to eliminate z. yz 9 69yz 8 78 y 9 Multiply the second equation by and add to the third equation. yz 6 5yz 5 y We now have a system of two equations in two variables. 78y 9 y Multiply the second equation by 8 and add to the second equation. 78y 9 8y 8 Back-substitute for to find y. y () y y y y Back-substitute for and for y to find z. yz 9 z 9 z 9 z 6 z The solution set is,, The solution set is,. 5. and 8 6 For a value to be in the solution set, it must be both less than and less than or equal to. Now only values that are less than or equal to meet both conditions. Therefore, the solution set is,. Copyright Pearson Education, Inc. 67

23 Chapter : Quadratic Equations and Functions 6. 8 or 7 For a value to be in the solution set, it must satisfy either of the conditions. Now, all numbers that are greater than or equal to are also greater than. Therefore, the solution set is, The solution set is,. 8. or The solution set is, Disregard since it causes a result of in the denominator of a fraction. Thus, the equation has no solution. The solution set is. 68 Copyright Pearson Education, Inc.

24 Introductory and Intermediate Algebra for College Students E Cumulative Review. 6 9 Since we square both sides of the equation, we must check to make sure is not etraneous: 6 9 True Thus, the solution set is {} Apply the quadratic formula: a b c The solution set is. Copyright Pearson Education, Inc. 69

25 Chapter : Quadratic Equations and Functions Let u. u 5u6 uu u or u u u Substitute back in for u. or 7 8 The solution set is 8, Solve the related quadratic equation. 6 or The boundary points are and. Interval Test Value Test Conclusion, 6, does not belong to the solution set. 9, False 6,, belongs to the solution set. 6, True, 6, does not belong to the solution set., False The solution set is,. 6 Copyright Pearson Education, Inc.

26 Introductory and Intermediate Algebra for College Students E Cumulative Review. y 6 y 6 6 y y This is a true statement. This means that the point,,, will fall in the shaded half-plane. The slope is m and the y-intercept is b. f 7. Since a is negative, the parabola opens downward. The verte of the parabola is hk,,. Replace f with to find intercepts. f This is a linear function with slope 5. intercept b. m and y- or or The intercepts are and. Set to obtain the y intercept. 6. y 6 First, graph the equation y 6 as a dashed line. y 6 y 6 6 y y The slope is m and the y-intercept is b. Net, use the origin as a test point. 6 6 f 986 The y-intercept is y 66y 6y 6y Copyright Pearson Education, Inc. 6

27 Chapter : Quadratic Equations and Functions y y y 7 y y. 8 9yy 7 y5y 8 9yy 7 y5y 5y6y y yy.. 9 y6y or 6 5y y 5 y 5 y 5y i 7 i 5 5ii9i 5 6i 9 5 6i 9 6i y y y yy y y 9y. f g f g 5 5 f g Copyright Pearson Education, Inc.

28 Introductory and Intermediate Algebra for College Students E Cumulative Review. f f 5 g g. The domain of f g is,, f 5. ( ah) ( ah) 5 a a5 f a h f a h a ahh ah5a a5 h ah h h h ah h 6. R I R r I Rr R IR Ir R Ir R IR Ir I R Ir Ir R or R I I 7. y 9 y 9 The line whose equation we want to find has a slope of m, the same as that of the line above. Using this slope with the point through which the line passes,, 5, we first find the point-slope equation and then put it in slopeintercept form. y y m y5 ( ) y5 y5 6 y The slope-intercept equation of the line through, 5 and parallel to y 9 is y. Copyright Pearson Education, Inc. 6

29 Chapter : Quadratic Equations and Functions 8. Let = the computer s original price The original price of the computer was $6. 9. Let = the width of the rectangle. Let + = the length of the rectangle. 5 5 or Disregard because the width of a rectangle cannot be negative. If, then (). Thus, the length of the rectangle is yards and the width is yards.. Let = the amount invested at %. Let = the amount invested at % Thus, $6 was invested at % and $ was invested at %.. Because I varies inversely as R, we have the following for a constant k: k I R Use the fact that I = 5 when R = to find k: k 5 k 5 Thus, the equation relating I and R is If R =, then I. I. R A current of amperes is required when the resistance is ohms. 6 Copyright Pearson Education, Inc.