You analyzed parent functions and their families of graphs. (Lesson 1-5)

Transcription

1 You analyzed parent functions and their families of graphs. (Lesson 1-5) Graph and analyze power functions. Graph and analyze radical functions, and solve radical equations.

2 power function monomial function radical function extraneous solution

3

4 Analyze Monomial Functions A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

5 Analyze Monomial Functions B. Graph and analyze f (x) = x 5. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

6 Describe where the graph of the function f (x) = 2x 4 is increasing or decreasing. A. increasing: (, ) B. decreasing: (, 0), increasing: (0, ) C. decreasing: (, ) D. increasing: (, 0), decreasing: (0, )

7 Functions with Negative Exponents A. Graph and analyze f (x) = 2x 4. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

8 Functions with Negative Exponents B. Graph and analyze f (x) = 2x 3. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

9 Describe the end behavior of the graph of f (x) = 3x 5. A. B. C. D.

10 Rational Exponents A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

11 Rational Exponents B. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

12 Describe the continuity of the function A. continuous for all real numbers B. continuous on C. continuous on (0, ] D. continuous on [0, ) and.

13 Power Regression A. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Create a scatter plot of the data.

14 Power Regression B. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Determine a power function to model the data.

15 Power Regression C. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Use the data to predict the mass of an African Golden cat with a length of 77 centimeters.

16 AIR The table shows the amount of air f(r) in cubic inches needed to fill a ball with a radius of r inches. Determine a power function to model the data. A. f (r) = 5.9r 2.6 B. f (r) = 0.6r 0.3 C. f (r) = 19.8(1.8)r D. f (r) = 5.2r 2.9

17

18 Graph Radical Functions A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

19 Graph Radical Functions B. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.

20 Find the intercepts of the graph of A. x-intercept: B. x-intercepts: C. x-intercept: D. x-intercepts:, y-intercept:, y-intercept:, y-intercept:, y-intercept 4.

21 Solve Radical Equations A. Solve.

22 Solve Radical Equations B. Solve.

23 Solve Radical Equations C. Solve.

24 Solve A. 0, 5 B. 11, 11 C. 11 D. 0, 11.

25 You analyzed graphs of functions. (Lesson 1-2) Graph polynomial functions. Model real-world data with polynomial functions.

26 polynomial function polynomial function of degree n leading coefficient leading-term test quartic function turning point quadratic form repeated zero multiplicity

27 A. Graph f (x) = (x 3) 5. Graph Transformations of Monomial Functions

28 B. Graph f (x) = x 6 1. Graph Transformations of Monomial Functions

29 Graph f (x) = 2 + x 3. A. C. B. D.

30

31 Apply the Leading Term Test A. Describe the end behavior of the graph of f (x) = 3x x + x + x 1 using limits. Explain your reasoning using the leading term test.

32 Apply the Leading Term Test B. Describe the end behavior of the graph of f (x) = 3x 2 + 2x 5 x 3 using limits. Explain your reasoning using the leading term test.

33 Apply the Leading Term Test C. Describe the end behavior of the graph of f (x) = 2x 5 1 using limits. Explain your reasoning using the leading term test.

34 Describe the end behavior of the graph of g (x) = 3x 5 + 6x 3 2 using limits. Explain your reasoning using the leading term test. A. Because the degree is odd and the leading coefficient negative,. B. Because the degree is odd and the leading coefficient negative,. C. Because the degree is odd and the leading coefficient negative,. D. Because the degree is odd and the leading coefficient negative,.

35

36 Zeros of a Polynomial Function State the number of possible real zeros and turning points of f (x) = x 3 + 5x 2 + 4x. Then determine all of the real zeros by factoring.

37 State the number of possible real zeros and turning points of f (x) = x 4 13x Then determine all of the real zeros by factoring. A. 4 possible real zeros, 3 turning points; zeros 2, 2, 3, 3 B. 4 possible real zeros, 2 turning points; zeros 4, 9 C. 3 possible real zeros, 2 turning points; zeros 2, 3 D. 4 possible real zeros, 4 turning points; zeros 2, 3

38

39 Zeros of a Polynomial Function in Quadratic Form State the number of possible real zeros and turning points for h (x) = x 4 4x Then determine all of the real zeros by factoring.

40 State the number of possible real zeros and turning points of g (x) = x 5 5x 3 6x. Then determine all of the real zeros by factoring. A. 3 possible real zeros, 2 turning points; real zeros 0, 1, 6 B. 5 possible real zeros, 4 turning points; real zeros 0, C. 3 possible real zeros, 3 turning points; real zeros 0, 1,. D. 5 possible real zeros, 4 turning points; real zeros 0, 1, 1,

41 Polynomial Function with Repeated Zeros State the number of possible real zeros and turning points of h (x) = x 4 + 5x 3 + 6x 2. Then determine all of the real zeros by factoring.

42 State the number of possible real zeros and turning points of g (x) = x 4 4x 3 + 4x 2. Then determine all of the real zeros by factoring. A. 4 possible real zeros, 3 turning points; real zeros 0, 2 B. 4 possible real zeros, 3 turning points; real zeros 0, 2, 2 C. 2 possible real zeros, 1 turning point; real zeros 2, 2 D. 4 possible real zeros, 3 turning points; real zeros 0, 2

43

44 Graph a Polynomial Function A. For f (x) = x(3x + 1)(x 2) 2, apply the leadingterm test.

45 Graph a Polynomial Function B. For f (x) = x(3x + 1)(x 2) 2, determine the zeros and state the multiplicity of any repeated zeros.

46 Graph a Polynomial Function C. For f (x) = x(3x + 1)(x 2) 2, find a few additional points.

47 Graph a Polynomial Function D. For f (x) = x(3x + 1)(x 2) 2, graph the function.

48 Determine the zeros and state the multiplicity of any repeated zeros for f (x) = 3x(x + 2) 2 (2x 1) 3. A. 0, 2 (multiplicity 2), (multiplicity 3) B. 2 (multiplicity 2), (multiplicity 3) C. 4 (multiplicity 2), (multiplicity 3) D. 2 (multiplicity 2), (multiplicity 3)

49 A. POPULATION The table to the right shows a town s population over an 8-year period. Year 1 refers to the year 2001, year 2 refers to the year 2002, and so on. Create a scatter plot of the data, and determine the type of polynomial function that could be used to represent the data. Model Data Using Polynomial Functions

50 B. POPULATION The table below shows a town s population over an 8-year period. Year 1 refers to the year 2001, year 2 refers to the year 2002, and so on. Write a polynomial function to model the data set. Round each coefficient to the nearest thousandth, and state the correlation coefficient. Model Data Using Polynomial Functions

51 C. POPULATION The table below shows a town s population over an 8-year period. Year 1 refers to the year 2001, year 2 refers to the year 2002, and so on. Use the model to estimate the population of the town in the year Model Data Using Polynomial Functions

52 D. POPULATION The table below shows a town s population over an 8-year period. Year 1 refers to the year 2001, year 2 refers to the year 2002, and so on. Use the model to determine the approximate year in which the population reaches 10,712. Model Data Using Polynomial Functions

53 BIOLOGY The number of fruit flies that hatched after day x is given in the table. Write a polynomial function to model the data set. Round each coefficient to the nearest thousandth, and state the correlation coefficient. Use the model to estimate the number of fruit flies hatched after 8 days. A. y = x x + 5.3; r = 0.84; 190 B. y = x x x ; r 2 = 0.99; 40,922 C. y = 10x x x x + 182; r 2 = 1; 82,966 D. y = x ; r = 0.829; 346

54 You factored quadratic expressions to solve equations. (Lesson 0 3) Divide polynomials using long division and synthetic division. Use the Remainder and Factor Theorems.

55 synthetic division depressed polynomial synthetic substitution

56 Use Long Division to Factor Polynomials Factor 6x x 2 104x + 60 completely using long division if (2x 5) is a factor.

57 Factor 6x 3 + x 2 117x completely using long division if (3x 4) is a factor. A. (3x 4)(x 5)(2x + 7) B. (3x 4)(x + 5)(2x 7) 2 C. (3x 4)(2x + 3x 35) D. (3x 4)(2x + 5)(x 7)

58

59 Long Division with Nonzero Remainder Divide 6x 3 5x 2 + 9x + 6 by 2x 1.

60 4 3 Divide 4x 2x + 8x 10 by x + 1. A. 3 2 B. 4x + 2x + 2x + 10 C. D.

61 Division by Polynomial of Degree 2 or Higher Divide x 3 x 2 14x + 4 by x 2 5x + 6.

62 Divide 2x 4 + 9x 3 + x 2 x + 26 by x 2 + 6x + 9. A. B. C. D.

63

64 Synthetic Division A. Find (2x 5 4x 4 3x 3 6x 2 5x 8) (x 3) using synthetic division.

65 Synthetic Division B. Find (8x x 3 + 5x 2 + 3x + 3) (4x + 1) using synthetic division.

66 Find (6x 4 2x 3 + 8x 2 9x 3) (x 1) using synthetic division. A. B. C. 6x 3 8x D. 6x 3 + 4x x + 3

67

68 Use the Remainder Theorem REAL ESTATE Suppose 800 units of beachfront property have tenants paying $600 per week. Research indicates that for each $10 decrease in rent, 15 more units would be rented. The weekly revenue from the rentals is given by R (x) = 150x x + 480,000, where x is the number of $10 decreases the property manager is willing to take. Use the Remainder Theorem to find the revenue from the properties if the property manager decreases the rent by $50.

69 REAL ESTATE Use the equation for R(x) from Example 5 and the Remainder Theorem to find the revenue from the properties if the property manager decreases the rent by $100. A. $380,000 B. $450,000 C. $475,000 D. $479,900

70

71 Use the Factor Theorem A. Use the Factor Theorem to determine if (x 5) and (x + 5) are factors of f (x) = x 3 18x x Use the binomials that are factors to write a factored form of f (x).

72 Use the Factor Theorem B. Use the Factor Theorem to determine if (x 5) and (x + 2) are factors of f (x) = x 3 2x 2 13x 10. Use the binomials that are factors to write a factored form of f (x).

73 Use the Factor Theorem to determine if the binomials (x + 2) and (x 3) are factors of f (x) = 4x 3 9x 2 19x Use the binomials that are factors to write a factored form of f (x). A. yes, yes; f(x) = (x + 2)(x 3)( 14x + 5) B. yes, yes; f(x) = (x + 2)(x 3)(4x 5) C. yes, no; f(x) = (x + 2)(4x 2 17x 15) D. no, yes; f(x) = (x 3)(4x 2 + 3x + 10)

74

75 You learned that a polynomial function of degree n can have at most n real zeros. (Lesson 2-1) Find real zeros of polynomial functions. Find complex zeros of polynomial functions.

76 Rational Zero Theorem lower bound upper bound Descartes Rule of Signs Fundamental Theorem of Algebra Linear Factorization Theorem Conjugate Root Theorem complex conjugates irreducible over the reals

77

78 Leading Coefficient Equal to 1 A. List all possible rational zeros of f (x) = x 3 3x 2 2x + 4. Then determine which, if any, are zeros.

79 Leading Coefficient Equal to 1 B. List all possible rational zeros of f (x) = x 3 2x 1. Then determine which, if any, are zeros.

80 List all possible rational zeros of f (x) = x 4 12x 2 15x 4. Then determine which, if any, are zeros. A. B. C. D.

81 Leading Coefficient not Equal to 1 List all possible rational zeros of f (x) = 2x 3 5x 2 28x Then determine which, if any, are zeros.

82 List all possible rational zeros of f (x) = 4x 3 20x 2 + x 5. Then determine which, if any, are zeros. A. B. C. D.

83 Solve a Polynomial Equation WATER LEVEL The water level in a bucket sitting on a patio can be modeled by f (x) = x 3 + 4x 2 2x + 7, where f (x) is the height of the water in millimeters and x is the time in days. On what day(s) will the water reach a height of 10 millimeters?

84 PHYSICS The path of a ball is given by the function f (x) = 4.9x x + 40, where x is the time in seconds and f (x) is the height above the ground in meters. After how many seconds will the ball reach a height of 25 meters? A. 4 seconds, 10 seconds B. 4 seconds C. 5 seconds, seconds D. 5 seconds

85

86 Use the Upper and Lower Bound Tests Determine an interval in which all real zeros of f (x) = x 4 4x 3 11x 2 4x 12 must lie. Explain your reasoning using the upper and lower bound tests. Then find all the real zeros.

87 Determine an interval in which all real zeros of f (x) = 2x 4 5x 3 13x x 10 must lie. Then find all the real zeros. A. [0, 4]; 1, 2 B. [ 1, 2]; 1, C. [ 3, 5]; 1, D. [ 2, 1]; 1,

88

89 Use Descartes Rule of Signs Describe the possible real zeros of f (x) = x 4 3x 3 5x 2 + 2x + 7.

90 Describe the possible real zeros of g (x) = x 3 + 8x 2 7x + 9. A. 3 or 1 positive real zeros, 1 negative real zero B. 3 or 1 positive real zeros, 0 negative real zeros C. 2 or 0 positive real zeros, 0 negative real zeros D. 2 or 0 positive real zeros, 1 negative real zero

91

92

93 Find a Polynomial Function Given Its Zeros Write a polynomial function of least degree with real coefficients in standard form that has 1, 2, and 2 i as zeros.

94 Write a polynomial function of least degree with real coefficients in standard form that has 2 (multiplicity 2), 0, and 3i as zeros A. f (x) = x + 4x + 13x + 36x + 36x B. f (x) = x + 4x + 9x + 18x C. f (x) = x + 2x 3ix 6xi D. f (x) = x + 4x 5x 36x 36

95 Write a polynomial function of least degree with real coefficients in standard form that has 2 (multiplicity 2), 0, and 3i as zeros A. f (x) = x + 4x + 13x + 36x + 36x B. f (x) = x + 4x + 9x + 18x C. f (x) = x + 2x 3ix 6xi D. f (x) = x + 4x 5x 36x 36

96

97 Factor and Find the Zeros of a Polynomial Function A. Consider k (x) = x 5 + x 4 13x 3 23x 2 14x 24. Write k (x) as the product of linear and irreducible quadratic factors.

98 Factor and Find the Zeros of a Polynomial Function B. Consider k (x) = x 5 + x 4 13x 3 23x 2 14x 24. Write k (x) as the product of linear factors.

99 Factor and Find the Zeros of a Polynomial Function C. Consider k (x) = x 5 + x 4 13x 3 23x 2 14x 24. List all the zeros of k (x).

100 Write k (x) = x 4 4x 3 + 4x 2 + 4x 5 as the product of linear factors. A. (x + 1)(x 1)(x + (2 i))(x + (2 + i)) B. (x + 1)(x 1)(x 2 4x + 5) C. (x + 1)(x 1)(x (2 + i))(x (2 i)) D. (x + 1)(x 1)(x + 5)

101 Find the Zeros of a Polynomial When One is Known Find all complex zeros of p (x) = x 4 6x x 2 50x 58 given that x = 2 + 5i is a zero of p. Then write the linear factorization of p (x).

102 Find all complex zeros of h(x) = x 4 + x 3 3x 2 + 9x 108 given that x = 3i is a zero of h. A. 3i, 3i B. 3i, 4, 3 C. 3i, 3i, 4, 3 D. 3i, 3i, 4, 3

103 You identified points of discontinuity and end behavior of graphs of functions using limits. (Lesson 1-3) Analyze and graph rational functions. Solve rational equations.

104 rational function asymptote vertical asymptote horizontal asymptote oblique asymptote holes

105

106 Find Vertical and Horizontal Asymptotes A. Find the domain of and the equations of the vertical or horizontal asymptotes, if any.

107 Find Vertical and Horizontal Asymptotes B. Find the domain of and the equations of the vertical or horizontal asymptotes, if any.

108 Find the domain of and the equations of the vertical or horizontal asymptotes, if any. A. D = {x x 4, x }; vertical asymptote at x = 4; horizontal asymptote at y = 10 B. D = {x x 5, x }; vertical asymptote at x = 5; horizontal asymptote at y = 4 C. D = {x x 4, x }; vertical asymptote at x = 4; horizontal asymptote at y = 5 D. D = {x x 4, 4, x }; vertical asymptote at x = 4; horizontal asymptote at y = 2

109

110 Graph Rational Functions: n < m and n > m A. For, determine any vertical and horizontal asymptotes and intercepts. Then graph the function and state its domain.

111 Graph Rational Functions: n < m and n > m B. For, determine any vertical and horizontal asymptotes and intercepts. Then graph the function and state its domain.

112 Determine any vertical and horizontal asymptotes and intercepts for. A. vertical asymptotes x = 4 and x = 3; horizontal asymptote y = 0; y-intercept: B. vertical asymptotes x = 4 and x = 3; horizontal asymptote y = 1; intercept: 0 C. vertical asymptotes x = 4 and x = 3; horizontal asymptote y = 0; intercept: 0 D. vertical asymptotes x = 4 and x = 3; horizontal asymptote y = 1; y-intercept:

113 Graph a Rational Function: n = m Determine any vertical and horizontal asymptotes and intercepts for function, and state its domain.. Then graph the

114 Determine any vertical and horizontal asymptotes and intercepts for. A. vertical asymptote x = 2; horizontal asymptote y = 6; x-intercept: 0.833; y-intercept: 2.5 B. vertical asymptote x = 2; horizontal asymptote y = 6; x-intercept: 2.5; y-intercept: C. vertical asymptote x = 6; horizontal asymptote y = 2; x-intercepts: 3 and 0; y-intercept: 0 D. vertical asymptote x = 6, horizontal asymptote y = 2; x-intercept: 2.5; y-intercept: 0.833

115

116 Graph a Rational Function: n = m + 1 Determine any asymptotes and intercepts for. Then graph the function, and state its domain.

117 Determine any asymptotes and intercepts for. A. vertical asymptote at x = 2; oblique asymptote at y = x; x-intercepts: 2.5 and 0.5; y-intercept: 0.5 B. vertical asymptote at x = 2; oblique asymptote at y = x 5; x-intercepts at ; y-intercept: 0.5 C. vertical asymptote at x = 2; oblique asymptote at y = x 5; x-intercepts: ; y-intercept: 0 D. vertical asymptote at x = 2; oblique asymptote at y = x 2 5x + 11; x-intercepts: 0 and 3; y-intercept: 0

118 Graph a Rational Function with Common Factors Determine any vertical and horizontal asymptotes, holes, and intercepts for graph the function and state its domain.. Then

119 Determine the vertical and horizontal asymptotes and holes of the graph of. A. vertical asymptote at x = 2, horizontal asymptote at y = 2; no holes B. vertical asymptotes at x = 5 and x = 2; horizontal asymptote at y = 1; hole at ( 5, 3) C. vertical asymptotes at x = 5 and x = 2; horizontal asymptote at y = 1; hole at ( 5, 0) D. vertical asymptote at x = 2; horizontal asymptote at y = 1; hole at ( 5, 3)

120 Solve. Solve a Rational Equation

121 Solve. A. 22 B. 2 C. 2 D. 8

122 Solve. Solve a Rational Equation with Extraneous Solutions

123 Solve. A. 2, 1 B. 1 C. 2 D. 2, 5

124 Solve a Rational Equation WATER CURRENT The rate of the water current in a river is 4 miles per hour. In 2 hours, a boat travels 6 miles with the current to one end of the river and 6 miles back. If r is the rate of the boat in still water, r + 4 is its rate with the current, r 4 is its rate against the current, and, find r.

125 ELECTRONICS Suppose the current I, in amps, in an electric circuit is given by the formula, where t is time in seconds. At what time is the current 2 amps? A. 1.7 or 8.3 seconds B. 2 or 7 seconds C. 4.7 seconds D. 12 seconds

126 You solved polynomial and rational equations. (Lessons 2-3 and 2-4) Solve polynomial inequalities. Solve rational inequalities.

127 polynomial inequality sign chart rational inequality

128 Solve Solve a Polynomial Inequality

129 2 Solve x 9x + 10 < 46. A. ( 3, 12) B. C. D. ( 12, 3)

130 Solve a Polynomial Inequality Using End Behavior Solve x 3 22x > 3x 2 24.

131 Solve 2x x 3x + 4. A. (, 4] B. (, 4) C. [ 4, 1] or D. ( 4, 1) or

132 A. Solve x 2 + 2x + 3 < 0. Polynomial Inequalities with Unusual Solution Sets

133 B. Solve x 2 + 2x Polynomial Inequalities with Unusual Solution Sets

134 Polynomial Inequalities with Unusual Solution Sets C. Solve x x + 36 > 0.

135 Polynomial Inequalities with Unusual Solution Sets D. Solve x x

136 Solve x 2 + 6x + 9 > 0. A. no solution B. (, ) C. x = 3 D. (, 3) ( 3, )

137 Solve. Solve a Rational Inequality

138 Solve. A. (, 3) [11, ) B. [, 3] [11, ) C. (3, 11] D. [3, 11]

139 Solve a Rational Inequality CARPENTRY A carpenter is making tables. The tables have rectangular tops with a perimeter of 20 feet and an area of at least 24 square feet. Write and solve an inequality that can be used to determine the possible lengths of the tables.

140 GARDENING A gardener is marking off rectangular garden plots. The perimeter of each plot is 36 feet and the area is at least 80 square feet. Write and solve an inequality that can be used to find the possible lengths of each plot. A. l(36 l) 80; 0 ft < l 8 ft or l 10 ft B. l(18 l) 80; 8 ft l 10 ft C. I 2 36l 80; 0 ft < l 8 ft or 10 ft l 36 ft D. l(18 l) 80; 4 ft l 5 ft