MAT 1033C Final Exam Review. BY: Math Connections/Hands-On Math

Transcription

1 MAT 1033C Final Exam Review BY: Math Connections/Hands-On Math

2 Useful Formulas Rational Expressions/Equations #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 Table of Contents Radicals and Rational Exponents/Equations #10, 11, 12, 13, 14, 15, 16, 17, 18, 19 Complex Numbers #20, 21, 22, 23 Quadratic Equations/Applications #24, 25, 26, 27, 28, 29, 30, 31, 32 Graphing #33, 34, 35, 36, 37, 38, 39, 40, 41 Equations of Lines/Slope #42, 43 Inequalities (Solving/Graphing) #44, 50, 51, 52, 53, 54, 55 Systems of Equations/Applications #45, 46 Other Topics to Study Study and Test Taking Tips

3 Helpful Formulas to Remember

4 Slope/Linear Equations m = yy 22 yy 11 xx 22 xx 11 Slope-intercept form: y = mx + b Function Notation: f(x) = mx + b Point-slope form: y y 1 = m(x x 1 ) Parallel Lines: Same Slope Perpendicular Lines: opposite, reciprocal slopes

5 Work Formulas Set-Up: 1 aa + 1 bb + 1 cc = 1 xx (3 people) x = aaaaaa aaaa+aaaa+bbbb

6 Formula: Factor: a 3 + b 3 OR a 3 b 3 Formula: (a b)(a 2 ab b 2 ) M Match O Opposite P Plus a and b are cube roots!

7 x xx 3 = xx a 3 8 = 2 b (a b)(a a ab b b) (x 2)(x x 2x 2 2) + + M Match O Opposite P Plus (x 2)(x 2 + 2x + 4)

8 Problem #1 (Rational Expressions) Factor & Cancel like factors! 9x 4 72x 9x(x 3 8) 9x(x 2)(x 2 + 2x + 4) 3x (x 2 4) 3(x 2)(x + 2) x 2 + 1x 2 (x + 2)(x 1) 4x 3 + 8x x 4x(x 2 + 2x + 4)

9 Problem #1 CONT 9xx(xx 22)(xx ) 3(xx 22)(xx + 22) (xx + 22)(xx 1) 4xx(xx ) 3 = 9(xx 11) = 3(xx 11) 3 4 4

10 Problem #2 (Rational Expressions) Factor & Cancel like factors! x x + 36 (x + 9)(x + 4) x x + 45 (x + 9)(x + 5) x 2 + 5x x(x + 5) x 2 3x 28 (x 7)(x + 4)

11 Problem #2 CONT (xx + 99)(xx + 44) (xx + 99)(xx + 55) xx(xx + 55) (xx 77)(xx + 44) = xx (xx 77) = xx xx 77

12 Problem #3 (Rational Expressions) Division Multiply by the reciprocal. Keep, Change, Flip! Always flip the 2 nd fraction! xx xx xx xx 22 11

13 Problem #3 CONT Factor & Cancel like factors! xx xx xx xx x 2 + 5x 6 (x + 6)(x 1) x 2 + 9x + 18 (x + 6)(x + 3) x 2 + 7x + 12 (x + 4)(x + 3) x 2 1 (x 1)(x + 1)

14 Problem #3 CONT xx xx xx xx (xx + 66)(xx 11) (xx + 66)(xx + 33) (xx + 44)(xx + 33) (xx + 11)(xx 11) (xx + 44) xx + 44 = (xx + 11) = xx + 1

15 Problem #4 (Complex Fractions) Find LCD and multiply each term by LCD! LCD = 12x 9 12x + 3 xx 12x 3 xx 4 12x x = 3333 xx 108xx xx 2 + xx = 36( ) xx( )

16 Problem #5 (Complex Fractions) Find LCD and multiply each term by LCD! LCD = x 2 5 x + 4 xx xx 2 25 xx 2 16 xx x 2 x 2 x 2 x x 2 =

17 Problem #6 (Complex Fractions) Find LCD and multiply each term by LCD! LCD = x 2 1 xx x x 2 x xx 2 xx x 2 x 2 xx + 9 xx 2 xx Sum of Cubes Factor!

18 Formula: Factor: a 3 + b 3 OR a 3 b 3 Formula: (a b)(a 2 ab b 2 ) M Match O Opposite P Plus a and b are cube roots!

19 x xx 3 = xx a = 9 b (a b)(a a ab b b) (x 9)(x x 9x 9 9) + + M Match O Opposite P Plus (x + 9)(x 2 9x + 81)

20 Problem #6 CONT xx + 9 xx = 11(xx+99) (xx+99)(xx ) = 11 xx

21 Problem #7 (Complex Fractions) Find LCD and multiply each term by LCD! NOTE: 11 x = (x 11) 10 xx xx 11 3 xx + 8 xx 11

22 Problem #7 CONT LCD = x(x 11) 10 (xx 1111) 3 xx (xx 1111) x(x 11) x(x 11) 8 (xx 1111) x(x 11) x(x 11) = 10xx + 11xx 33 xx xx

23 Problem #7 CONT = 10xx + 11xx 33 xx xx = xx = xx

24 Problem #8 (Polynomial Division) (x 8) 4x 2 33x + 8 Long Division of polynomials! Steps on next slide!

25 4x -1 x 4x = 4x 2 (x 8) 4x 2 33x + 8 4x x 4x(x 8) = 4x 2 32x Change signs! -1 1x + 8 x = -1x + 1x 8 Remainder = 0 4x - 1-1(x 8) = -1x + 8 Change signs!

26 Problem #9 (Polynomial Division) (3x + 5) 15x x 2 2x 17 Long Division of polynomials! Steps on next slide!

27 5x 2 +2x 4 (3x + 5) 15x x 2 2x 17 15x 3 25x 2 6x 2 2x 17 6x 2 10x 12x x + 20 Remainder = +3 5x 2 + 2x ( ) 3x 5x 2 = 15x3 5x 2 (3x + 5) = 15x x 2 Change signs! 3x +2x = 6x 2 2x(3x + 5) = 6x x Change signs! 3x -4 = -12x -4(3x + 5) = -12x 20 Change signs!

28 Problem #10 (Rational Exponents) Find LCD: LCD for 3, 5, and 2 = xx xx xx

29 Problem #10 CONT xx xx xx xx xx xx Properties to Remember: 1) x m x n = x m + n 2) xxmm xxnn = xxmm nn = xx ( 3333 ) =

30 Problem #11 (Rational Exponents) 1111 = 99xx 33 xx = (33xx ) (33xx33) 1 xx6 Subtract exponents! xx mm = xxmm nn xxnn = = Add exponents! x m x n = x m + n = 99xx

31 Problem #12 (Radicals) 3333kk 66 qq 88 22kk kk 66 = kk 33 qq 88 = qq 44 6k 3 q 4 6k 3 q

32 Problem #13 (Radicals) = = 6666xx 5 yy xx 5 yy 66 33yy 44 3x 2 y xx 44 yy 22 3x 2 y 77xx

33 Problem #14 (Radicals) = = = NOTE: 99 = 3 44 = = = 9 =

34 Problem #15 (Radicals) NOTE: = 3 = yy yy = yy yy =33 33 yy yy Since = = 4 Since = 64 = yy yy

35 Problem #16 (Radicals) 2xx = 10 Isolate radical! xx 1 = 66 2x 1 = x = x =

36 Problem # Checking: 2xx = ? = 10? = 10 x = = = 10

37 Problem #17 (Radicals) xx = (xx 1) Square both sides! 3333 xx = (xx 11)(xx 11) FOIL! 3333 xx = xx x + 1x 31 Make equation equal to zero! 0 = xx 22 xx 3333

38 Problem #17 CONT xx 22 11xx 3333 = 0 (xx 6 )(xx + 5 ) = 0 Solve: x 6 = 0; x + 5 = Solutions: x = 6; x = 5 Factor trinomial!

39 Problem # Checking! x = 6; x = 5 31 xx = xx 1? = = 5 x = 6 Extraneous Solution? 31 ( 55) = = 66

40 Problem #18 (Radicals) Square both 2 2 2xx + 5 = (3 + xx 2) sides! FOIL! 2xx + 5 = (3 + xx 2) (3 + xx 2) 2xx + 5 = 9+33 xx xx 22 + xx xx + 5 = xx 22 + x 2

41 Problem #18 CONT 2xx + 5 = xx 22 + x 2 2xx + 5 = xx 22 + x x 7 7 x 2 2 xx 2 = [66 xx 22] Isolate 66 xx 22 xx 2 xx 2 = 3333(xx 22) FOIL! Distribute! x 2 4x + 4 = 36x 72

42 Problem #18 CONT. x 2 4x + 4 = 36x 72 36x x +72 Make equation equal to zero! x 2 40x + 76 = 0 (x 38)(x 2) = 0 x = {2, 38} Check your solutions! Solve by factoring!

43 Problem # Checking! x = 2; x = 38 2xx + 5 = 3 + xx = NO Extraneous Solutions! 9 = = = = = 3 + 6

44 Problem #19 (Radicals) 2 2 4xx + 5 = ( 2xx 2 3) Square both sides! = ( 22xx 22 3)( ) FOIL! = =

45 Problem #19 CONT = = Isolate ( ) = [ ] Square Both Sides! ( )( ) = 3333( ) FOIL! Distribute!

46 Problem #19 CONT ( )( ) = 3333( ) 4x 2 4x 4x + 4 = 72x 72 4x 2 8x + 4 = 72x x x +772 Make equation equal to 0. 4x 2 80x + 76 = 0 Divide each term by x 2 20x + 19 = 0

47 Problem #19 (By Factoring): x 2 20x + 19 = 0 Solve: (x 19)(x 1) = 0 x = {1, 19} We should check our solutions!

48 Problem #19 (By Quadratic Formula): x 2 20x + 19 = 0 Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Solve by quadratic formula! 1x 2 20x + 19 = 0 So a = 1, b = ( 20), c = 19 xx = ( 2222)± ( 2222)22 44(11)(1111) 22(11)

49 Problem #19 CONT xx = ( 2222)± ( 2222)22 44(11)(1111) 22(11) xx = 2222 ± = ± 1111 xx = = ± x = {10 9, } x = {1, 19} = 1111 ± 99 We should check our solutions!

50 Problem #19 Checking x = {1, 19} 4xx + 5 = 2xx 2 3 No Solution!? 4(11) + 5 = 2(11) = = -3? 4(1111) + 5 = 2(1111) = = 6 3

51 Problem #20 (Complex Numbers) 3 6ii ii =10 4ii Combine Like Terms!

52 Problem #20 (Calculator Tip) ii = 2 nd key + period (next to 0) =10 4ii

53 Problem #21 (Complex Numbers) Combine Like Terms! Distribute minus sign! (7 + 8ii) 1( 9 + ii) 7 + 8ii + 9 1ii =16 + 7ii

54 Problem #21 (Calculator Tip) ii = 2 nd key + period (next to 0) =16 + 7ii

55 Problem #22 (Complex Numbers) Write the binomial twice! (8 + 9ii) (8 + 9ii) Multiply (FOIL)! ii +72ii + 81ii 2

56 Problem #22 CONT ii +72ii + 81ii 2 Remember: ii 2 = ( 1) = ii + 81( 1) = ii 81 = ii

57 Problem #22 (Calculator Tip) ii = 2 nd key + period (next to 0) = ii

58 Problem #23 (Complex Numbers) Rationalize the denominator! To do so, multiply by the conjugate. Conjugates: (a + b)(a b) Conjugate of 8 + 2ii 8 2ii (8 5ii) (8 + 2ii) ( ) ( ) Multiply/FOIL!

59 Problem #23 CONT Numerator: Remember: (8 5ii) (8 2ii) 64 16ii 40ii + 10ii 2 ii 2 = ( 1) = 54 56ii Numerator = 64 56ii + 10( 1) = 64 56ii 10

60 Problem #23 CONT Denominator: Remember: (8 + 2ii) (8 2ii) 64 16ii + 16ii 4ii 2 ii 2 = ( 1) = 68 Denominator = 64 4( 1) = = 68

61 Problem #23 CONT Numerator = 54 56ii Denominator = ii 68 = ii = ii

62 Problem #23 (Calculator Tip) ii = 2 nd key + period (next to 0)

63 Problem #23 Calculator CONT Change to Fraction: MATH KEY; Select 1) FRAC

64 Problem #24 (Quadratic Equation) x 2 4x = 13 x 2 4x + 4 = ) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! = 22; 22 (-2) 2 = +4 Factor trinomial!

65 Problem #24 CONT x 2 4x + 4 = -9 xx 22 = xx; 44 = 22 Match middle sign! (x 2)(x 2) = -9 (xx 22) 22 = ± 99 (x 2) 2 = -9 x 2 = ± 99 x = 2 ± x = {2 3333, }

66 Problem #25 (Quadratic Equation) x 2 + 3x = ) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! = ; ( )2 = = = =

67 Problem #25 CONT x 2 + 3x = Factor trinomial! (x )(x ) = xx 22 = xx; = Match middle sign! (x )2 = (xx )22 = ±

68 Problem #25 CONT x = ± = = = 22 x = ±

69 Problem #25 CONT x = ± x = ± = 33 ±

70 Problem #25 CONT 33 ± xx = ,

71 Problem #25 (Alternative Method) x 2 + 3x = LCD = x x + 9 = ) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! = ; ( )2 = x x + 9 = 45 Factor trinomial!

72 Problem #25 CONT 4x x + 9 = = 2222; 99 = 33 Match middle sign! (2x + 3)(2x + 3) = 45 (2x + 3) 2 = 45 2x + 3 = ± x + 3 = ±33 55 ( ) 22 = ± = =3 55

73 Problem #25 CONT 2x + 3 = ± x = 3 ± x = 33± xx = ,

74 Problem #26 (Quadratic Equation) 11 8x 2 5x = -1 8 x xx Coefficient of x 2 must be 1! = ) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! = ; ( )2 =

75 Problem #26 CONT x x = = = x x =

76 Problem #26 CONT x x = Factor trinomial! xx 22 = xx; = Match middle sign! (x )(x ) = 77 (x )2 =

77 Problem #26 CONT (x )2 = (xx )22 = ± = 1 7 = ii = 1111 x = ± ii

78 Problem #26 CONT x = ± ii x = ± ii = 55 ± ii

79 Problem #26 CONT 55 ± ii xx = 55 ii , 55 + ii

80 Problem #26 (Alternative Method) x 2 5x = -1 8 x xx Coefficient of x 2 must be 1! = ) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! = ; ( )2 =

81 Problem #26 CONT LCD = x xx = x 2 160x + 25 = x 2 160x + 25 = 7 Factor trinomial! = ; 2222 = 55 Match middle sign! (16x 5)(16x 5) = 7 (16x 5) 2 = 7

82 Problem #26 CONT (16x 5) 2 = 7 ( ) 22 = ± 77 16x 5 = ± 77 16x 5 = ±ii x = 5 ±ii = 1 7 = ii 77 x = 55±ii

83 Problem #26 CONT x = 55±ii xx = 55 ii , 55 + ii

84 Problem #27 (Quadratic Equation) Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Solve by quadratic formula! 1x x + 3 = 0 So a = 1, b = 10, c = 3 xx = 1111± (11)(33) 22(11)

85 xx = Problem #27 CONT xx = 1111± (11)(33) 22(11) xx = 1111 ± = = ± xx = ± All outsides numbers are divisible by 2. = 55 ± 2222

86 Problem #27 CONT xx = 55 ± 2222 x = { , }

87 Problem #28 (Quadratic Equation) Make equation = x 2 3x + 1 = 0 Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Solve by quadratic formula! 16x 2 3x + 1 = 0 So a = 16, b = 3, c = 1

88 Problem #28 CONT Solve by quadratic formula! 16x 2 3x + 1 = 0 So a = 16, b = 3, c = 1 xx = ( 33)± xx = 33 ± ( 33)22 44(1111)(11) 22(1111) 5555 = 1 55 = ii 5555 xx = 33 ii , 33 + ii

89 Problem #29 (Square Root Method) (x + 7) 2 = 24 Do opposite operations: Opposite of exponents Roots (xx + 77) 22 = ± 2222 x + 7 = ± = 4 6 = x = 77 ± x = { , }

90 Problem #30 (Quadratic Equation/ Problem Solving) Hitting the ground: Height = 0-4.9t t = 0 Solve by quadratic formula! -4.9t t = 0 So a = (-4.9), b = 42, c = 130

91 Problem #30 CONT Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Solve by quadratic formula! -4.9t t = 0 So a = (-4.9), b = 42, c = 130 xx = 4444± ( 44.99)(111111) 22( 44.99) xx = ,

92 Problem #30 CONT xx = , x {10.986, } x 11.0 seconds Nearest tenth Extraneous Solution; Negative solution does not make sense!

93 Problem #31 (Quadratic Equation/ Problem Solving) Hitting the ground: Height = 0-16t t = t 2 21t 7 = 0

94 Problem #31 CONT Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Solve by quadratic formula! 1t 2 21t 7 = 0 So a = 1, b = (-21), c = (-7) xx = ( 2222)± ( 2222)22 44(11)( 77) 22(11) xx = ,

95 Problem #31 CONT xx = , x { , } Extraneous Solution; Negative solution does not make sense! x 21.3 seconds Nearest tenth

96 Problem #32 (Vertex Problem) Find vertex! Maximum value of parabola = Vertex (h, k). Vertex formula: t = bb 2aa (x-coordinate) h(t) = y-coordinate a = -16, b = 64 h(t) = -16t t h(2) = -16(2) (2) = 64 feet (Maximum height) t = = = 2 The arrow reaches the maximum height after 2 seconds.

97 f(x) = 1x 2 + 2x 3 ; a = 1, b = 2 x = = 2 2 = 11 f(-1) = (-1) 2 + 2(-1) 3 = -4 Since a = 1 (positive) Opens UPWARD! Problem #33 (Graphing Quadratics) Vertex formula: x = bb 2aa (x-coordinate) f(x) = y-coordinate Vertex: (-1, -4)

98 Problem #33 (Sketch Graph): Vertex: (-1, -4) (-1, -4) Touches x-axis 2 times; 2 x-intercepts!!

99 Problem #33 CONT y-intercept: x = 0 f(0) = (0) 3 = 3 y-intercept: (0, 3) y-intercept: x = 0 x-intercept: y = 0 x-intercept: y = 0 x 2 + 2x 3 = 0 Solve by factoring! x 2 + 2x 3 = 0 (x + 3)(x 1) = 0 x = {-3, 1} x-intercepts: (-3, 0), (1, 0)

100 Problem #33 CONT Line of Symmetry: x = -1 (Vertical) x = h of vertex (h, k) (-3, 0) (1, 0) (0, -3) Vertex: (-1, -4) y-intercept: (0, -3) x-intercepts: (-3, 0), (1, 0) Opens UPWARD! (-1, -4)

101 Problem #33 Solution: (-3, 0) (0, -3) (1, 0) (-1, -4)

102 Domain and Range for all parabolas DOMAIN: (-, ) OR All real numbers k UP DOWN k - RANGE: [k, ) RANGE: (-, k]

103 Find Domain and Range for #33 x = - Goes to left forever! y = Goes up forever x = Goes to right forever! RANGE: [-4, ) DOMAIN: (-, ) OR All real numbers y = -4 (-1, -4)

104 Range Visualized: RANGE: [-4, ) (-1, -4) y = -4 y = Goes up forever

105 Problem #34 (Graphing Quadratics) F(x) = 2x 2 4x + 5; a = 2, b = 4 x = = = 11 F(1) = 2(1) 2 4(1) + 5 = 3 Since a = 2 (positive) Opens UPWARD! Vertex formula: x = bb 2aa (x-coordinate) f(x) = y-coordinate Vertex: (1, 3)

106 Problem #34 (Sketch Graph): Vertex: (1, 3) (1, 3) Does not touch x-axis; No x-intercepts!!

107 Problem #34 CONT y-intercept: x = 0 F(0) = 2(0) 2 4(0) + 5 = 5 y-intercept: (0, 5) y-intercept: x = 0 x-intercept: y = 0 x-intercept: y = 0 2x 2 4x + 5 = 0 2x 2 4x + 5 = 0 Solve by quadratic formula! 2x 2 4x + 5 = 0 So a = 2, b = (-4), c = 5

108 Problem #34 CONT Use Quadratic Formula: xx = bb± bb22 44aacc ; Just plug in a, b, c 22aa Quadratic Equation to solve: 2x 2 4x + 5 = 0 So a = 2, b = (-4), c = 5 xx = ( 44)± ( 44)22 44(22)(55) 22(22) xx = 44 ± NO SOLUTION! NO x-intercepts! 2222 = UNDEFINED!

109 Problem #34 CONT (0, 5) (2, 5) (1, 3) Vertex: (1, 3) y-intercept: (0, 5) Another point: (2, 5) x-intercepts: NONE! Opens UPWARD! Line of Symmetry: x = 1 (Vertical) x = h of vertex (h, k)

110 Problem #34 Solution: (0, 5) (1, 3)

111 Find Domain and Range for #34 x = - Goes to left forever! y = Goes up forever x = Goes to right forever! y = 3 (1, 3) RANGE: [3, ) DOMAIN: (-, ) OR All real numbers

112 Range Visualized: RANGE: [3, ) (1, 3) y = 3 y = Goes up forever

113 Problem #35 (Graphing Quadratics) f(x) = 1(x + 2) 2 5 Vertex = ( 2, 5) Vertex: (-2, -5) Vertex form: f(x) = a(x h) 2 + k Vertex = (h, k) x-coordinate: Opposite Sign y-coordinate: Copy! Since a = 1 (positive) Opens UPWARD! Line of Symmetry: x = -2 (Vertical) x = h of vertex (h, k)

114 Problem #35 CONT Vertex: (-2, -5) Opens UPWARD! Line of Symmetry: x = -2 (Vertical) x = h of vertex (h, k) (-2, -5)

115 Problem #35 Solution: (-2, -5)

116 Find Domain and Range for #35 x = - Goes to left forever! y = Goes up forever x = Goes to right forever! RANGE: [-5, ) y = -5 (-2, -5) DOMAIN: (-, ) OR All real numbers

117 Range Visualized: RANGE: [-5, ) (-2, -5) y = -5 y = Goes up forever

118 Problem #36 (Graphing Quadratics) f(x) = -1(x 3) Vertex = (3, 0) Vertex: (-2, -5) Vertex form: f(x) = a(x h) 2 + k Vertex = (h, k) x-coordinate: Opposite Sign y-coordinate: Copy! Since a = -1 (negative) Opens DOWNWARDS! Line of Symmetry: x = 3 (Vertical) x = h of vertex (h, k)

119 Problem #36 CONT Vertex: (3, 0) Opens DOWN! Line of Symmetry: x = 3 (Vertical) x = h of vertex (h, k) (3, 0)

120 Problem #36 Solution (3, 0)

121 Find Domain and Range for #36 y = 0 (3, 0) RANGE: (, 0] x = - Goes to left forever! y = Goes down forever DOMAIN: (-, ) OR All real numbers x = Goes to right forever!

122 Range Visualized: y = Goes down forever (3, 0) y = 0 RANGE: (, 0]

123 Problem #37 (Linear Graphing) y-intercept (x = 0): x + 2y = y = 8 2y = 8 x-intercept (y = 0): x + 2y = 8 x + 2(0) = 8 x = 8 y = 4 4) x (0, y (8, 0)

124 Problem #37 CONT (0, 4) (8, 0)

125 Problem #37 Solution:

126 Problem #38 (Linear Graphing) Make x-y table! x (0, y 3) x = 0: y = y = 3 x = 4: y = y = 6 (4, 6)

127 Problem #38 CONT (4, 6) (0, 3)

128 Solution for #38 (0, 3) (4, 6)

129 Problem #38 (Alternative Method) y = mx + b (Slope-intercept form) y-intercept = (0, b) Slope = m y = xx + 33 y-intercept: (0, 3) Slope = Up 3, Right 4

130 Problem #38 Alternative CONT y-intercept: (0, 3) RIGHT 4 (4, 6) Slope = Up 3, Right 4 UP 3 (0, 3)

131 Solution for #38 (0, 3) (4, 6)

132 Find Domain and Range for #38 y = Goes up forever x = - Goes to left forever! x = Goes to right forever! y = Goes down forever DOMAIN/RANGE: (-, ) OR All real numbers

133 y-intercept (x = 0): -5x + 3y = y = -15 Problem #39 (Linear Graphing) 3y = -15 Find intercepts! x-intercept (y = 0): -5x + 3y = -15-5x + 3(0) = -15-5x = -15 x (0, y = -5 x = 3-5) y (3, 0)

134 Problem #39 CONT (3, 0) (0, -5)

135 Solution for #39 (3, 0) (0, -5)

136 Problem #40 (Linear Graphing) Horizontal line; y = k

137 Solution for #40

138 Problem #41 (Linear Graphing) Vertical line; x = k

139 Solution for #41

140 Problem #42 (Slope-Intercept Form) Write the equation in the form y = mx + b Point-slope form: y y 1 = m(x x 1 ) y ( 7) = -3(x 7) y + 7 = -3(x + 7) y +7 = -3x y = -3x 28

141 Problem #43 (Function Notation) (xx 22, yy 22 ) and (xx 11, yy 11 ) (9, 43) and (1, 11) Slope = yy 22 yy 11 xx 22 xx 11 = = = 44 Point-slope form: y y 1 = m(x x 1 ) Choose (1, 11) for (xx 11, yy 11 ) and m = 4.

142 Problem #43 CONT Choose (1, 11) for (xx 11, yy 11 ) and m = 4. y 11 = 4(x 1) y 11 = 4x y = 4x + 7 f(x) = 4x + 7

143 Problem #44 (Graphing Inequalities) Notes: Notes: <, >: Dotted Line <, >: Solid Line y > mx + b Shade Above Line y < mx + b Shade Below Line Graph: y < 2x + 6; Dotted Line/Below y > x 8; Solid Line/Above

144 Problem #44 CONT Graph: y < 2x + 6; Dotted Line/Below y > x 8; Solid Line/Above For y < 2x + 6: y-intercept = (0, 6) Slope = 2 (Go up 2, over by 1) For y > 1x 8: y-intercept = (0, -8) Slope = 1 (Go up 1, over by 1) (0, 6) (1, 8) (1, -7) (0, -8)

145 Problem #44 FINAL SOLUTION (0, 6) (1, 8) (1, -7) (0, -8)

146 Problem #44 Solution

147 Problem #45 (Systems of Equations) Let x = number of adult tickets Let y = number of senior citizens tickets Set-Up: x + y = 491 tickets $25x + $13y = $8195 Solve for y (Substitution): x + y = 491 xx Solve: 25x + 13(491 x) = x x = 8195 xx y = (491 x)

148 Problem #45 CONT 25x x = 8195 y = 491 x 12x = x = y = = 340 senior citizen tickets x = 151 adult tickets

149 Problem #46 (Systems of Equations) Let x = pounds of trail mix Let y = pounds of cashew Set-Up: x + y = 75 pounds $5x + $15y = ($13)(75) Solve (elimination): x + y = 75 5x + 15y = 975

150 Problem #46 CONT. We will eliminate x -5( x + y = 75) 5x + 15y = 975 x + y = 75 x + 60 = x = 15 pounds of trail mix -5x 5y = x + 15y = y = y = 60 pounds of cashew

151 Problem #47 (Rational Equations) 1 (xx + 44) 7 xx 44 = 4 (xx + 44)(xx 44) Factor: x 2 16 (x + 4)(x 4) Restrictions: Denominator 0 Solve: x x -4 Solve: x 4 0 x 4 Drop all solutions where x = {-4, 4}

152 Problem #47 CONT 1 (xx + 44) 7 xx 44 LCD = (x + 4)(x 4) (xx + 44)(xx 44) (xx + 44)(xx 44) (xx + 44)(xx 44) = 4 (xx + 44)(xx 44) 1(x 4) 7(x + 4) = 4 1x 4 7x 28 = 4 x = 6 No extraneous solutions! 6x 32 = x =

153 Problem #48 (Rational Equations) Factor: x 2 xx xx = 12 xxxx x 0 Restrictions: Denominator 0 Drop all solutions where x = {0}

154 Problem #48 CONT LCD = x 2 (xx) xx = 12 xxxx xxxx xxxx xxxx x 2 + 1x = x 2 + 1x 12 = 0 Make equation = 0 Factor trinomial! (x + 4)(x 3)= 0 x = {-4, 3} No extraneous solutions!

155 Problem #49 (Rational Equations) Factor: x 2 + 3x 4 (x + 4)(x 1) x 2 2x + 1 (x 1)(x 1) (xx + 5) (xx + 44)(xx 11) 5 xx 11 (xx 11) = (xx 5) (xx + 44)(xx 11) Restrictions: Denominator 0 Solve: x x -4 Solve: x 1 0 x 1 Drop all solutions where x = {-4, 1}

156 Problem #49 CONT xx + 44 xx 11 xx 11 LCD = (x + 4)(x 1)(x 1) xx + 44 xx 11 xx 11 (xx + 5) (xx + 44)(xx 11) 5 xx 11 (xx 11) = (xx 5) (xx + 44)(xx 11) xx + 44 xx 11 xx 11 xx + 5 xx 11 5 xx + 44 = (xx 5)(xx 11) FOIL! DISTRIBUTE! FOIL! x 2 1x + 5x 5 5x 20 = x 2 1x 5x + 5 x 2 1x + 5x 5 5x 20 = x 2 1x 5x + 5 x 2 1x 25 = x 2 6x + 5

157 Problem #49 CONT x 2 1x 25 = x 2 6x + 5 xx 22 xx 22 1x 25 = 6x x = 30 5x 25 = x = 6 No extraneous solutions!

158 Problem #50 (Compound Inequalities) Intersection ONLY! x < 3 (Shade left) x > -2 (Shade right) [ ] [ ] [-2, 3]

159 Problem #51 (Compound Inequalities) 11 < x + 6 < < 5x + 12 < < x < 10 x is between 2 and 10! 10 < 5x <

160 Problem #51 (Compound Inequalities) 2 < x < 10; x is between 2 and 10! [ ) [2, 10) Remember: <, >: [] <, >: ()

161 Problem #52 (Compound Inequalities) x + 4 < x < -3-4x < x > -1 Sign Flipped! x < -3 and x > -1 Intersection ONLY! ) ( There is no intersection/overlap! NO SOLUTION!

162 Problem #52 Solution Blank Number Line

163 Problem #53 (Compound Inequalities) ] [ (, 4] [6, ) (, 4] [6, )

164 Problem #54 (Compound Inequalities) Beginning (left) to End (right) ) ) (, 7) )

165 Problem #55 (Compound Inequalities) -5x + 1 > x > x < -2 Sign Flipped! 3x + 3 > x > x > -4 x < -2 or x > -4 Beginning to end!

166 Problem #55 CONT x < -2 or x > -4 Beginning to end! [ ] Beginning (left) to End (right) ( ) (, ) All real numbers!

167 Problem #55 CONT ( ) (, ) All real numbers!

168 Other Topics to Study: Functions, Domain/Range Midpoint and Distance Formula Problems Pythagorean Theorem Work Word Problems Review Factoring! Review Basic Graphing! Review Solving Equations and Simplifying Expressions

169 Helpful Study Tips: 1) The final is cumulative and you should study the course materials beyond this workshop! 2) The final exam is made by your individual instructor. Use any study guide/tips provided by your instructor. 3) Study your lab project worksheets and lab assessment. 4) Study previous class exams and quizzes! 5) Of course, study and review your homework assignments!

170 Helpful Study Tips: 6) Visit the Math Connections for additional support and resources! Study a little each day, DO NOT CRAM!!

171 General Test Taking Tips: 1) Preview the exam and do the problems that are easy and you are familiar with. 2) Pace yourself do not spend too much time on any 1 problem. 3) DO NOT RUSH! 4) Go back and check your answers (if time allows). 5) Follow instructions carefully! 6) Double check your work! Review your exam before you submit it!

172 Now go study and do well on your final exam!