Chapter 1: Packing your Suitcase

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1 Chapter : Packing your Suitcase Lesson.. -. a. Independent variable = distance from end of tube to the wall. Dependent variable = width of field of view. e. The equation depends on the length and diameter of the tubes used. The students should have a slope between 0. and 0.4 with a y-intercept around 3.5 cm if they use a paper towel core Answers will depend on the students tube and models. Answers will depend on the students tube and models. Review and Preview a. parabola y = x b. cubic y = x 3 c. hyperbola, inverse variation, d. exponential y = x reciprocal function y = x e. absolute value y = x f. square root y = x CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

2 -6. Examples of non-functions are a circle, x + y = r, and a sleeping parabola, x = y. Other answers are acceptable. -7. a. slope = 7!8 7!4 = 9 4 point! slope form " y! 8 = 9 (x! 4) 4 slope! intercept form " y! 8 = 9 4 x! 9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!y = 9 4 x! b. slope = 7!8 7!4 = 9 4 point! slope form " y! 0 = 9 (x! (!)) 4 slope! intercept form " y! 0 = 9 4 x + 7!or!y = 9 4 x a. =! 3 =!! b. 5 =!!!! = 3 6 =!!!!! = 64 4 =!!! c. They are half as large each time. Divided by, or multiplied by ½ is also acceptable. d. 0 =! =,!" =! =, " =! =,!"3 = 4! = 8,!"4 = 8! = 6 "n = n"! = n"+ = n -9. a.!4 " =!!!!Check :! 6 " 4 = 4 b.! "! =!3 Check :! " 4 = 8 c. 0! "3 = "3!!!!!Check :!! 8 = 8-0. a. x(x + 5) b. 3xy 3 (xy! 3) c. 7x 3 y(! xy) -. a. (x) 3 = 3! x 3 = 8x 3 b. (3x! ) = (3x! )(3x! ) = 9x! 6x! 6x + 4 = 9x! x + 4 c. (3x) 4 = 3 4! x 4 = 8x 4 d. (3x)!3 = (3x) 3 = 3 3 x 3 = 7x 3 -. a. a b b. c b c. a c CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

3 Lesson a. b. y =.5 x looks like the others, but would graph to the right of y = x. c. 0 < b < Example: -5. This is the graph of y = x shifted up five units. -6. y = x! 4-7. This is the graph of y = x 3 shifted left three units. -8. y = x! -9. Parent graph: y = x Shifted right four units: y = x!4 Shifted down three units: y = x!4! 3 CPM Educational Program 0 Chapter : Page 3 Pre-Calculus with Trigonometry

4 -0. a. x y = x y = x A vertical stretch occurs with each of the y-values doubling. b. A vertical stretch with stretch factor. -. a. y = x 3 b. See graph at right. c. Stretched vertically by, some may prefer to call this a compression. -. y =!x Review and Preview a. x + b. x! c. x + 4 d. 5x -4. a. f (4) =! 4 " 3 = 3 " 3 = 9 b. f (!5) = "(!5)! 3 = 50! 3 = 47 c. f (3b) =! (3b) " 3 =! 9b " 3 = 8b " 3 d. f (a + ) =!(a + ) " 3 = a + 4a + " 3 = a + 4a " CPM Educational Program 0 Chapter : Page 4 Pre-Calculus with Trigonometry

5 -5. (3x + ) = (3x + )(3x + ) = 9x + 6x + 6x + 4 The middle terms must be included. = 9x + x + 4! 9x a. a b! a c = a (b+c) b. a!b " a c = a (!b+c) = a (c!b) c. Cannot be simplified. d. a! a b = a! a b = a (+b) e. a 0! a b = a (0+b) = a b f. a (b+c)! a c = a (b+c+c) = a (b+3c) -7. a. 3x y!(7x " 4) b. (x + ) 3 + x + 5 [ ] = (x + )(x + 8) [ ] = (3x! 7) [ 6x! 4 + x! ] = (3x! 7)(7x! 6) c. (3x! 7) (3x! 7) + (x! ) d. (x + y)(m + x + y) -8. a. (5a! ) = 5 " a!" = 5a!4 b. (m! n! ) 3 = m!"3 n!"3 = m!3 n!6 c. (x! ) (x 0 ) = " x!" " = 8x! -9. a.!x = x b.!5"4 =!0 c. ( 3 )! = ( 3 ) = 3 d. ( 3 )! = ( 3 ) = 3 = a. cos 6! = x = x 8 x = 6.8 c. = x = x = x x = 384 = 9.60 b. cos 70! = 8 x x! 0.34 = 8 x = 3.39 d. sin 4! = x = x x = 7.87 Lesson a. It multiplies the input by two and then adds. b. We hope that they will think that 3 would come out the top and if that is true, then the machine must undo itself working backwards in a sense. CPM Educational Program 0 Chapter : Page 5 Pre-Calculus with Trigonometry

6 c. Subtract one and then divide by two. -3. a. Subtract 6, then multiply by. b. f! (x) = (x! 6) -33. a. f (x) + g(x) = 3x! 5 + x + = x + 3x! 3 b. f (x)g(x) = (3x! 5)(x + ) = 3x 3! 5x + 6x! 0 c. f (g(x)) = 3(x + )! 5 = 3x + 6! 5 = 3x + d. g( f (x)) = (3x! 5) + = 9x! 30x = 9x! 30x a. f (x) = x 3! 4x f (x) = x(x! 4) = x(x + )(x! ) 0 = x(x + )(x! ) Either x = 0, x + = 0, x! = 0 x =!, 0, b. c. Shifted left two units d. e. g(x) = (x + ) 3! 4(x + ) -35. a. hours! 3 miles hour = 6 miles b. c. d. miles hr! hr = miles Review and Preview a. 3x 3 + 7! (x! ) = 3x 3 + 7! x + = 3x 3! x + 8 b. 3x +7 x!, x " ± c. (3x 3 + 7)(x! ) = 3x 5! 3x 3 + 7x! 7 CPM Educational Program 0 Chapter : Page 6 Pre-Calculus with Trigonometry

7 -37. a. y = 3x 3! 5 x = 3y 3! 5 x + 5 = 3y 3 x+5 3 = y 3 3 x+5 = y " 3 3 x+5 = f 3! (x) b. y = (x + 4) / x = (y + 4) / x = y + 4 x!4 = y " x!4 = g! (x) c. y = x! x = y! x + = y x + 4 = y x + 4 = y " x + 4 = h! (x) -38. a. g(h(4)) = g!6 " b. h(g(-)) = h! (-) + 4 c. g(h(!)) = g " 4! ( ) = g(6) =! = 6 = 4 ( ) = h ( ) =! ( ) " = " = " ( ) = g(0) = " = d. Part (c) does not have the same output as input. e. A function has one output (y) for every input (x), but y = ± x + 4 has two outputs for every input a. sin x = 7 sin! (sin(x)) = sin! x = 35.7! -40. a. cubic function y = x 3 ( ) flipped over y-axis y =!x 3 shifted down 3 units y =!x 3! 3 shifted right units y =! ( x! ) 3! 3 b. cos y = 5 5 ( ) cos! (cos(y)) = cos! y = 70.5! b. exponential function y = x shifted down 3 units y = x! 3 shifted right units y = x!! 3-4. a. g(x! ) = x! + = x! b. f (g(8)) = f ( 8 + ) = f (3) = 3 + (3) = 5 c. g( f (8)) = g(8 + (8)) = g(80) = 80 + = 9 d. f ( f ()) = f ( + ()) = f (3) = 3 + (3) = 5 e. f (x + ) = (x + ) + (x + ) = x + x + + x + = x + 4x + 3 f. g( f (x)) = x + x + = (x + ) = x + CPM Educational Program 0 Chapter : Page 7 Pre-Calculus with Trigonometry

8 -4. h( j(x)) = 3(ax + b)! = 3ax + 3b! j(h(x)) = a(3x! ) + b = 3ax! a + b 3b! =!a + b a + b = a + b = Lesson a. Shifts right three units and up two units. b. f (x + 4)! c. The point is on the x-axis. It does not change since 0 = 0. d. It still does not move Shifted left two units: g(x) = f (x + ) Shifted down one unit: g(x) = f (x + )! -46. a. It is stretched then shifted down 3. b. It is shifted down 3 and then stretched. c. k(x) = f (x)! 3, m(x) = ( f (x)! 3) = f (x)! 6. These two functions are not equivalent a. They are the same. b.! x! 4 c. Yes, replace x with x in the inequalities and solve. d. 0! x + 3!! 3! 3! 3! 3 " x "! 0! x "! + + +! x! 4 CPM Educational Program 0 Chapter : Page 8 Pre-Calculus with Trigonometry

9 Review and Preview a. f (x + ) + Shifted left two units and up one unit. b. f (x) + Vertical stretch by a factor of two, up two. c.! f (x + 4) Flipped over x-axis, and shifted left four units. -50.!A = 80! " 90! " 38! = 5! cos 38! = 5 c = 5 c c = 5 = 9.04 cm sin 38! = b 9.04 b = #9.04 =.7 cm -5. a. 50(.5) + 75(0.5) = =.5 miles b. two rectangles c. 50(.5) + 75(0.5) = =.5 miles d. miles hour! hours = miles -5. a. f (g(!)) = f ((!)! ) = f (3) = (3) + 5 = ( ( )) = g f + b. g f h() c. y = x + 5 x = y + 5 x! 5 = y x!5 = y x! 5 ( ) = f! (x) ( ( )) = g f () ( ) = g ( () + 5) = g(9) = 9! = 8! = 80 CPM Educational Program 0 Chapter : Page 9 Pre-Calculus with Trigonometry

10 ( ) = f " x + d. f g ( h(x) ) # ( )! $ = f (x + ) = (x + ) + 5 = x = x + 7 % CPM Educational Program 0 Chapter : Page 0 Pre-Calculus with Trigonometry

11 -53. The graph does not give a full line. The line starts at (!, 3) and then follows the line y = x + 7. Since h(x) is defined for only values of x! ", the composite function is only defined for x! " a. Opposite =!5 4 Reciprocal = 5!4 = 5 4 c. Opposite =!6 b. Opposite =!3!5 Reciprocal = 3 5 Reciprocal =! 6 Reciprocal = 7 d. Opposite =! 7 e. Opposite =! 9 Reciprocal = 9 ( ) ( )! = ( 9 ) f. Opposite =! 7 3 Reciprocal = 7 3 ( )!5 = (! 7 3 )!5 ( ) x m x n = xm! x "n = x m+("n) = x m"n Lesson a. (7, ) b. m = 5! 7! = 3 6 = e. = y! x! (x! ) = y! y! = (x! ) c. (x, ) d. m = y! x! f. (x, y ) g. y! y, x! x h. y!y x!x -58. Yes, the point (0, 0) is on the line because f (0) = 3! 0 " 0 = y = (x! ) y! 5 = (x! ) 3 This is the same as point-slope form. original function! y = mx right shift h units! y = m(x " h) shifted up k units! y = m(x " h) + k CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

12 -60. The point-slope form requires a point on the line and the slope of the line. bthe slopeintercept form requires the slope and the y-intercept. -6. a. y = 3 (x! 0)! 3 5 b. y! 7.3 =.85(x! 6.) c. m =!8 5!4 = 3 Point-slope form: y! 8 = 3 (x! 4) or y! = 3 (x! 5) d. m = 9.78! = 4.45 Point-slope form: y! 6.4 = 4.45(x! 5.) or y! 6.4 = 4.45(x! 4.3) -6. The negative reciprocal. slope =! The negative reciprocal. The product of a slope and the perpendicular slope should be When the slope is zero a. m = 3 y! 7 = 3(x + ) b. m = 3 4! m = m! = " 4 3 y " 0 = " 4 (x " ) a. AB = (5! 3) + (! 3) = = 5 = 5 b. midpoint of AB = 3+5 (, 3+ ) = 8, 5 ( ) = 9, 7.5 ( ) Review and Preview a. Parent Graph: y = x Shifted left units: y = x+ Shifted down 3 units: y = x+! 3 b. Parent Graph: y = x Shifted right units: y = (x! ) Shifted up unit: y =!(x! ) + CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

13 -68. a. 50mph! 3hours = 50 miles b. It is a rectangle. c. height = 50 mph, base = 3 hours, ( )! (3hrs) = 50 miles 50 miles hr -69. a. 7 5 = b. 8 ( ) x = (3 3 ) 5 = = 3!5 35 ( ) x =!3 3 ( ) x =!3x c. 6 x! ( 3 ) ( "x) = ( 4 ) x! f (x + ) = x++ x+! = x+ x! = " (x + ) = (x! ) x + 4 = x! x + 4 =! x =!5 ( ) "x ( ) = 4x! "5 ( ) "x ( ) = 4x! "0+5x = 4x"0+5x = 9x"0-7. a. ( 3 ) (x+3) = 5 3x+9 = 5! 3x + 9 = 5 3x = "4 x = " 4 3 b. (3 3 ) x = ( 3 ) (x!) 3 6x = (3! ) (x!) 3 6x = 3!x+ " 6x =!x + 8x = x = 4 c. ( 5 ) (x!3) 3 = 5 (5!3 ) (x!3) = 5! " 5!6x+9 = 5!!6x + 9 =!!6x =! x = 6-7. a. 3x! 6! x! 4 = x + 7 x! 0 = x + 7!x! 0 = 7!x = 37 " x =!37 b. (x + 5)(x! ) = 0 Either x + 5 = 0 or x! = 0 x =!5, CPM Educational Program 0 Chapter : Page 3 Pre-Calculus with Trigonometry

14 c. x! 7x + = 0 (x! 4)(x! 3) = 0 Either x! 4 = 0 or x! 3 = 0 x = 3, 4 d. x 3 + x! 6x = 0 x(x + x! 6) = 0 x(x + 3)(x! ) = 0 x = 0, x + 3 = 0!or x! = 0 x =!3, 0, CPM Educational Program 0 Chapter : Page 4 Pre-Calculus with Trigonometry

15 -73. a. g( f (x)) = x + x + b. f (g(x)) = (x + ) + x = x + x + + x = x + 3x + c. (x + ) + (x + ) = x + x + + x + = x + 3x = 0 x(x + 3) = 0 x = 0 or x =!3-74. sin!p = 8 6 = ( ) = sin " sin " sin!p!p = 30! ( )!R = 80! " 90! " 30! = 60! sin 60! = r 6 3 #6 = r r = 8 3 $ 3.86 cm Lesson a. The coefficients a, b and c. b.!b + D = R and!b! D = S a a c. R and S; the Quadratic Formula has two solutions because of the ± in the formula Sierpinski s Triangle a. By choosing a random integer: 0,, or. b. T = 0 chooses A, T = chooses B. CPM Educational Program 0 Chapter : Page 5 Pre-Calculus with Trigonometry

16 c. :(X+T)/, :(Y+T)/ Y, and Y/ Y. Review and Preview The program will crash since the program tries to take the square root of a negative number a. Flip over y-axis:! f (x) Shifted up 3 units:! f (x) + 3 b. Shifted up unit: f (x) + Shifted left unit: f (x + ) + Doubled: (f (x + ) + ) = f (x + ) + h(x) = f (x + ) Slope of line m = 9!(!3) 8! = 6 = Midpoint of line = 8+ (, 9+(!3) ) = 0, 6 Slope of perpendicular line =! Equation of line y! 3 =! x! 5 ( ) ( ) = 5, 3 ( ) -8. a. p = 6, q = b. not in p q form c. p = x + 3y, q =! r d. not in p q form -83. a. 6!(6 ) "3! = 6! 6 "6 = 6 "4 b. c. d. (5 )! 5 "3 (5 3 ) " = 54! 5 "3 5 "6 = 5 5 "6 = 5"("6) = 5 7 3!9 97 8!9 94 = 3 8!997"94 = 3 8!93 3!9-97 3!994 = 8!9"94 8!9 97 = 3 8!994"97 = 3 8!9"3-84. a. Example: x =, y = 3! ( + 3) " " 3 b. Example: p =, q = 3! + 3 " " 5 Solution continues on next page. CPM Educational Program 0 Chapter : Page 6 Pre-Calculus with Trigonometry

17 -84. Solution continued from previous page. c. Example: w = 4! 3" 4 # $ 3"4 3 6 $ 48 e. Example: x = 5! 3" 5 # 6 5 3" 3 # # 7776 d. Example: a =, b = 3! ( " + 3 " ) " # + 3 ( + 3 ) " # 5 ( ) " # # a. Impossible, different bases. b. Impossible, bases are being added. c. +3 = 5 d. d.!3 = 6 e.!3 =! f. Impossible, bases are being subtracted y = 5 x! 3 x = 5 y! 3 x + 3 = 5 y (x+3) 5 = y (x+3) 5 = f! (x) -87. a. x(x + 8) b. 6x(x + 8) -88. Circumference of circle =! " =! Length of AB! = 4! " = " -89. Area = base! height 30 =!! h 30 = 6! h 5 inches = h sin 36! = 5 KL 5 KL = = 8.5 inches CPM Educational Program 0 Chapter : Page 7 Pre-Calculus with Trigonometry

18 Lesson a. K = ah b. sin C = h b! h = b sin C c. K = ah! K = ab sin C -9. Area = (6)(4) sin 76! =! =.644 cm -9. K = bh sin A = h c! h = c sin A K = bc sin A -93. SA of one side = (0)(0) sin 40! = 50! = 3.39 ft Total surface area of sides = 4(3.39) = ft -94. a. sin A = h b! h = b sin A b. sin B = h a! h = a sin B c. h = b sin A, h = a sin B b sin A = a sin B sin A a e. sin A a = sin B b = sin B b = sin C c d. h = b sin C, h = c sin B b sin C = c sin B sin B b = sin C c -95. sin P p = sin Q q = sin R r -96. a.!nat = 80! " 00! " 38! = 4! b. sin 4! sin 00! = 00 y CPM Educational Program 0 Chapter : Page 8 Pre-Calculus with Trigonometry

19 sin 4! = 00 sin 38! x x = x = x = ft x = ft -97. Cannot in (a) and (b) because you will get two unknowns in any form of the equation. Cannot in (d) for the same reason, and also because the triangle is not determined. Note that (c) is the only diagram in which you re given exactly one side. Review and Preview a. b.!g = 80! " 64! " 38! = 78! c. Area of!dog = 8 sin 78! = OG sin 64! 8 # = OG # = OG 7.35 in = OG 8 sin 78! = 8"5.036"sin 64! == 40.88" = 8.0 sq.in. DG sin 38! 8! = DG! = DG in = DG -99. f (x + ) = (x + ) + (x + ) = x + 4x x = x + 6x = (x + )(x + 4) Either x + = 0 or x + 4 = 0! x = " or x = "4-00. x 6 = x = 90 CPM Educational Program 0 Chapter : Page 9 Pre-Calculus with Trigonometry

20 y n = x y = n x CPM Educational Program 0 Chapter : Page 0 Pre-Calculus with Trigonometry

21 -0. ( ) = 4 = 6 b. ( 5 /3 ) 4 = 5 4 = 65 a. 64 /3 c. 8 3/4 = 8 /4 ( ) 3 = 3 3 = 7 7 d.! ( ) /3 " 8 # $ % = ( 3 ) % = ( 3 ) = a. y = x! x = 3 y! x 3 = y! x 3 + = y (x3 + ) = f! (x) b. y = (x! 3) + x = (y! 3) + (x! ) = y! 3 (x! ) + 3 = y (x! ) + 3 = g! (x) c. y = x 3/ x = y 3/ x = y3/ x ( ) /3 = y ( ) /3 = h! (x) x -04. a. Distance for one revolution = Circumference of circle =!"! r =!"! = " feet b. 0 =!"! r 0 " = r # r = 5 =.59 feet " -05. a. n + n = d n = d n = d n = d c. (n) = n + k 4n = n + k 3n = k 3n = k n 3 = k b.! 90! = 45! d. y = 60! (equilateral triangle) CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

22 Lesson a. e = b! d b. h + e = c! h = c " e c. h + d = a! h = a " d d. a! d = c! e e. c = a! d + (b! d) c = a! d + b! bd + d c = a + b! bd g. c = a + b! b(a cos C) -07. a. c = 0 + 6! (0)(6) cos 74! c = ! c = c = = 4.6 ft Yes, 50 feet of fencing is enough. sin 74 b.! 4.6 = sin B = sin B = sin B sin! = sin! (sin B) 58.! = "B c. Area = (0)(6) sin 74! = 6930! = (area of home) 666 (area of lot) = Yes, the area of the home would be more than 3 a! d + e = c f. cos C = d a! d = a cos C sin 74! 4.6 = sin C = sin C = sin C sin! = sin! (sin C) of the lot size. 47.9! = "C -08. It is not possible in (c) or (d) because you will get two unknowns in any form of the equation. You can solve (c) with the Law of Sines. The triangle for (d) is not determined cos C = cos90! = 0!c!= a!+ b! ab 0 c!= a!+ b ( ) CPM Educational Program 0 Chapter : Page Pre-Calculus with Trigonometry

23 -0. c = 0 + 4! (0)(4) cos 60 c = 96! 80 " c = 56 c = 56 = BE = ! (.8)(3.5) cos 43! BE = 0.09! BE = BE =.399 km -. c = 0 + 0! (0)(0) cos 30! c = 500! c = c =.393 cm -3. x = 8 + 4! (8)(4) cos X! x = ! 35 cos X! x = 548! 35 cos X! x + 35 cos X! = 548 If X = 0! x + 35 = 548 x = 96 If X = 80! x! 35 = 548 x = 4900 x = 96 = 4 x = 4900 = 70 4 < x < 70 inches Review and Preview a. 6 = 5 + 8! " 5 " 8 cos A 36 = 89! 80 cos A = cos A = cos A b. A = 48.5! = #A 6 sin 48.5! = 5 sin B 6 sin B = 5! sin B = = 0.64 "B = 38.6! 8! 6! (sin 38.6) = 8!6!0.64 = = 4.98 square meters "C = 80! # 38.6! # 48.5! = 9.87! CPM Educational Program 0 Chapter : Page 3 Pre-Calculus with Trigonometry

24 -5. a. x + + x! 8 = x + x! 7 b. (x - 8) x + ( ) = x + x! 4x! 8 = x! x! 8 c. ( x + )! 8 = x +! 8 = x! 6 d. (x + ) + = x 4 + x + + = x 4 + x a.! 360! = 80! b.! " = " meters c. AB! = 6! " = " 3 meters -7. ( ) "3! 5 3 = 5 ( 3 )! 3 "3 ( 3 3 ) " = 34! 3"3 a. 5! 5 b. c. d. -8. a. 8 /3 + ( "6 )+3 = 5 " 3 "6 = 3 3 "6 = 3" "6 5!4 98 8!4 95 = 5 8!498"95 = 5 8!43 5!4-98 8!4 "95 = 5!4"98" ( "95 ) = 5 8 8!4"3 ( ) = 3 7 ( ) = = 4 b. ( 00 / ) 3 = 0 3 = 000 ( ) = 5 = 5 c. 5 /3-9. 3x! 7y = 4!7y =!3x + 4 y = 3 7 x! 6 perpendicular slope =! 7 3 equation of line " y + 8 =! 7 ( 3 x + 3) -0. a. x + 3y + 6 = 6x! 30 3y = 4x! 36 y = 4 3 x! b. 6x + = 6y!!!!!y! 0 y = 6 6 x + 6 y = x + 6!!x! " 6 -. a. (x! 3y)(x + 3y) b. x 3 (4! x 4 ) = x 3 ( + x )(! x ) CPM Educational Program 0 Chapter : Page 4 Pre-Calculus with Trigonometry

25 -. 3! x " 0!x "!3 x # 3 Lesson d. π radius lengths = circumference -4. Length of AB! = unit. -5. C =! " =! -6. a. 360 b. π radians c.! = 6.83, nearest whole number = a. Degrees in half a circle: 80! b. π radians = 80 Approximate radians in half a circle: 3 Exact radians in half a circle:!! c. 3 = 80! = 60 3! d. 00! "! 80! = 00! " 80! = 0" 9-9. a. 80!! = 57.96! b.! 80! = 0.07 c. Very different. A radian is much larger, almost 60 times as large. CPM Educational Program 0 Chapter : Page 5 Pre-Calculus with Trigonometry

26 -30. a. 80! "! 80! = " b.!36! # " 80! =!36# =! # 80 5 c.!!! " 80! =!!! 80! =! a. 3! " 80!! = 540 = 70! b.! 7" 6 # 80! " =! 60 6 =!0! c.! 80! ( )! = 360 " " Review and Preview Radians per minute = 500! " = 000" Radians per second = 000" = 00" = 50" G R Y 6!6 : x 6!6 : 3!6 96 x = 96 x = 96 teaspoons x = 96 6 = 536 ounces x = x + b = 7 x = 7! b = gallons x = ± 7! b, b " a. 9: x + 6x + 9 = (x + 3)(x + 3) = (x + 3) CPM Educational Program 0 Chapter : Page 6 Pre-Calculus with Trigonometry

27 b. 8: x! 8x + 6 = (x! 4)(x! 4) = (x! 4) CPM Educational Program 0 Chapter : Page 7 Pre-Calculus with Trigonometry

28 -37. a. d = (!4! (!)) + (! (!6)) d = (!) + 8 = = 68 b. m =!(!6)!4!(!) = 8 =!4! point slope form! y + = "4(x + 4) point slope form! y + 6 = "4(x + ) slope intercept form! y + = "4x " y = "4x " a. tan A = 4 7 tan! (tan A) = tan! 4 7 "A = 9.74! ( ) b. tan B = 7 4 tan! (tan B) = tan! 7 4 "B = 60.6! ( ) -39. a. (a! 3)(a! 3! ) = (a! 3)(a! 4) b. 5x(x! 3) + 4(x! 3) = (x! 3)(5x + 4) -40. Slopes: a.! b. c. d. 3 e.! Parallel Lines (same slope)! a and e, b and c Lesson a. Radian measure for a: b.! 4, 3! 4, 5! 4, 7! 4 Half circle:! " = " Quarter circle: 4! " = " Three fourths of a circle: 3 4! " = 3"! c. 6,! 6 =! 3, 3! 6 =!, 4! 6 =! 3, 5! 6 d. 7! 6, 8! 6 = 4! 3, 9! 6 = 3!, 0! 6 = 5! 3,! 6 CPM Educational Program 0 Chapter : Page 8 Pre-Calculus with Trigonometry

29 -4. a. In the center of each quadrant. b. y-axis c. Closest to the x-axis a.! " 3 + 6" 3 = 4" 3 b.! 5" 4 + 8" 4 = 3" 4 c.! " 6 + " 6 = " a. 0! 3 = 9! +! c.! 5" 6 =! =! + 4! 3 3 = 4! 3 b. 7! 6! +! = = 4! +! =! 4 4" +" =!4"! " 6 6 =! " " or a. The speed does not change. The ratio of the distance and time is constant, or for a set time interval, an object will travel a set distance. b. Faster on the inside. The CD must go around more times on an inside track to cover the same distance as a point on the outside of the CD. c. 00(! " 5.5) # 6597 cm d. Distance around innermost track =! " 6597!" = rotations -46. a. 580 feet = 580! inches = inches in 90 rev = inches in one revolution C = = "d d =.36 inch diameter b. 90!"! 6 = inches = 640 feet 640 =.63 miles 580 c. Linda could get a speeding ticket = x 6 x = 40!6 = 46.5 mph.36 Review and Preview a. 0!! " 80! = 0" 80 = " 3 c. 80!! " 80! = 80" 80 = 4" 9 b.!5! " # 80! =!5# 80 =! 5# 4 CPM Educational Program 0 Chapter : Page 9 Pre-Calculus with Trigonometry

30 -48. a. tan x = 0 = 5 3 ( ) tan - (tan x) = tan! 5 3 x = 59.0! c. x = ! " 6.5 " 7.cos9! x = ! 9.3(!0.4848) x = = 37.4 x = 37.4 =.7 b. 0 sin 35! = x sin 80! 0! = x! = 7.7 = x d. 60 =!5! x!sin 8! = x = 7.04cm = x a. y = x + b. y 6 f - (x) x = 3 y + x 3 = y + x 3! = y x 3! = f! (x) 4 f(x) x ( ) = f! ( ) = f ( 3! ) = f 6 c. f! ( f (6)) = f! f f! () Composing f and f! ( ) = 3! = 6 3 ( ) = ax + bx + c = d(x! ex + e ) + f ax + bx + c = dx! dex + (de + f ) a = d, b =!de, c = de + f -5. a. Parent Graph! y = x 3 Transformation Flip over y-axis! y = "x 3 Shifted right two units! y = " ( x " ) 3 3 = 8 = in either order returns the original number. Shifted down three units! y = " ( x " ) 3 " 3 b. Parent graph! y = x Transformation Shifted left one unit! y = x + Shifted down two units! y = x + " CPM Educational Program 0 Chapter : Page 30 Pre-Calculus with Trigonometry

31 -5. f (g(x)) = (x! 3)! (x! 3) a. f (g(x)) = x! 6x + 9! x f (g(x)) = x! 8x + 0 x! 8x + 0 = 8 x! 8x + = 0 (x! )(x! 6) = 0 x! = 0 or x! 6 = 0 x = or x = 6 30!!!!! b. sin 30! = PQ c. cos 30! = QR 3! = 6 3 = QR! = 6 = PQ d. sin P = 6 3 = a. b. k(x! ) + 3!k(x)! c. k(x) + CPM Educational Program 0 Chapter : Page 3 Pre-Calculus with Trigonometry

32 Closure Problems CL -55. a. f (x)! g(x) = x! x! 3x + = x! 4x + b. g( f (x)) = 3(x! x)! = 6x! 3x! c. g(x+) f (x+) = 3(x+)! (x+)!(x+) = 3x+6! (x = 3x+5 +4 x+4)!x! x = 3x+5 +8x+8!x! x +7x+6 CL -56. a. b. y! 4 =! (x! ) CL -57. a. Third angle = 80!! 50!! 45! = 85! b. Law of Sines sin 85! sin 50! = 7 a 0.43 = a a = = sin 85! sin 45! = 7 b 0.43 = b b = = c. ASA CL -58. a. c = 4 + 7! (4)(7) cos 50! b. Law of Cosines c = ! 36 c = 9 c = 9 = c. SAS CL -59. a. 5x 4 = 5! 5! x! x = 5x b. c. x 3 + (x ) / = x 3 + x (xy ) 3 (x y 3 ) / = x3 y 6 xy 3/ = x3! y 6!3/ = x y 9/ CL -60. CPM Educational Program 0 Chapter : Page 3 Pre-Calculus with Trigonometry

33 a. 3! " = 6" b. 6! 6" = 78" in/sec # 0.4 ft / sec CPM Educational Program 0 Chapter : Page 33 Pre-Calculus with Trigonometry

34 CL -6. a. sin 45! = 4 hyp. hyp = 4 / = 4 Isosceles triangle leg = 4 b. sin 45! = hyp. 6 hyp =! 6 = 6 = 6 = 3 Isosceles triangle leg = 3 c. sin 30! = hyp.!!!!!!!!cos 30! = leg 4 hyp = = 4!!!!!!!!leg = 3! 4 = 3 CL -6. f (x) = x! 3 + Parent graph: y = Stretched: y = x Shifted right three units: y = x! 3 Shifted up one unit: y = x! 3 + x Inverse: x = y! 3 + x! = y! 3 x! = y! 3 x! ( ) = y! 3 ( ) + 3 = 4 (x! ) + 3 = y x! f! (x) = 4 (x! ) + 3 CL -63. a. b. CL -64. a. Total distance = 30! + 40!+ 50! = = 5 miles b = = 35.7 mph CPM Educational Program 0 Chapter : Page 34 Pre-Calculus with Trigonometry

( ) b.! = 7" 3 has coordinates 1

( ) b.! = 7 3 has coordinates 1 Chapter 4: Circular Functions Lesson 4.. 4-. a.! b.! c. i. 0!! " radians 80! = " 6 radians 4-. a. and b. ii. iii. 45!! " radians 80! = " 4 radians 60!! " radians 80! = " radians 4-. Possible patterns that