u u 1 u (c) Distributive property of multiplication over subtraction

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1 ADDITIONAL ANSWERS 89 Additional Answers Eercises P.. ; All real numbers less than or equal to ; All real numbers greater than or equal to and less than ; All real numbers less than ; All real numbers between and, including and ; All real numbers less than ; All real numbers between and 6, including and The real numbers greater than 4 and less than or equal to 9 4. The real numbers greater than or equal to, or the real numbers which are at least. The real numbers greater than or equal to, or the real numbers which are at least 6. The real numbers between and 7, or the real numbers greater than and less than 7 7. The real numbers greater than 8. The real numbers between and 0 (inclusive), or the real numbers greater than or equal to and less than or equal to or [9, ); Bill s age 4. 0 or [0, ]; cost of an item or [.099,.99]; dollars per gallon of gasoline or (0.0, 0.06); average percent of all salary raises 4. (a) Associative property of multiplication (b) Commutative property of multiplication (c) Addition inverse property (d) Addition identity property (e) Distributive property of multiplication over addition 46. (a) Multiplication inverse property 66. (a) Step Quotient Remainder (b) Multiplication identity property, or distributive 0 property of multiplication over addition, followed by 0 0 the multiplication identity property. Note that we also use the multiplicative commutative property to say that u u u (c) Distributive property of multiplication over subtraction 8 4 (d) Definition of subtraction; associative property of addition; definition of subtraction 6 6 (e) Associative property of multiplication; multiplication 7 9 inverse; multiplication identity

2 896 ADDITIONAL ANSWERS SECTION P. Quick Review P Distance: 7. Distance: or y 6. A B y C C D D A B Eercises P.. (a) First quadrant (b) On the y-ais, between quadrants I and II (c) Second quadrant (d) Third quadrant 4. (a) First quadrant (b) On the -ais, between quadrants II and III (c) Third quadrant (d) Third quadrant [99, 00] by [0, 0] [99, 00] by [0, ] [99, 00] by [0, 0].. 4. [99, 00] by [0, 00] [99, 00] by [0, 0] [99, 00] by [0, 0] 7. The three sides have lengths,, and. Since two sides have the same length, the triangle is isosceles. SECTION P. Eercises P. 9. (a) The figure shows that is a solution of the equation 6 0. (b) The figure shows that is a solution of the equation (a) The figure shows that is not a solution of the equation (b) The figure shows that 4 is a solution of the equation

3 ADDITIONAL ANSWERS Multiply both sides of the first equation by. 60. Divide both sides of the first equation by. 6. False. 6 because 6 lies to the left of on the number line. 64. True. 6 includes the possibility that 6,which is true. 69. (e) If your calculator returns 0 when you enter 4, you can conclude that the value stored in is not a solution of the inequality 4. SECTION P.4 Eploration. The graphs of y m b and y m c. have the same slope but different y-intercepts.. [ 4.7, 4.7] by [.,.] The angle between the two lines appears to be 90. [ 4.7, 4.7] by [.,.] m= [ 4.7, 4.7] by [.,.] m= [ 4.7, 4.7] by [.,.] m=4 [ 4.7, 4.7] by [.,.] m= In each case, the two lines appear to be at right angles to one another. Eercises P [, 0] by [ 0, 60] [, 0] by [ 0, 40] [, ] by [ 0, 80] [, ] by [ 0, 0] 4. (a) y (b) y 7 4. (a) y (b) y 4 4. (a) y (b) y (a) y (b) y 49. m 8 4, so asphalt shingles are acceptable. 0. Americans income was, respectively, 000: 8., 00: 9, 00: 9.4 trillion dollars.

4 898 ADDITIONAL ANSWERS. (d). (b) [99, 00] by [, 0] [99, 00] by [0, 80]. (a) (b) (c) The year 006 is represented by 6. So the value of y for 6 is 67 million, a little larger than the U.S. Census Bureau estimate of 6 million. [0, ] by [000, 7000] [0, ] by [000, 7000] 4. (a) (b) [0, ] by [0, 00] [0, ] by [0, 00] 9. (a) No; perpendicular lines have slopes with opposite signs. (b) No; perpendicular lines have slopes with opposite signs. 60. (a) If b 0, both lines are vertical; otherwise, both have slope m a. If c d, the lines are coincident. b (b) If either a or b equals 0, then one line is horizontal and the other is vertical. Otherwise, their slopes are a b and b a,respectively. In either case, they are perpendicular. 6. False. The slope of a vertical line is undefined. For eample, the vertical line through (, ) and (, 6) would have slope (6 )/( ) /0, which is undefined. 6. True. If b 0, then a 0 and the graph of c a is a vertical line. If b 0, then y a b c a is a line with slope b b and y-intercept c. b 67. (a) (b) (c) (d) a is the -intercept and b is the y-intercept when c. [, ] by [, ] [, ] by [, ] [, ] by [, ] (e) (f) When c, a is the opposite of the -intercept and b is the opposite of the y-intercept. [ 0, 0] by [ 0, 0] [ 0, 0] by [ 0, 0] [ 0, 0] by [ 0, 0] a is half the -intercept and b is half the y-intercept when c.

5 ADDITIONAL ANSWERS (a) (b) If m 0, then the graphs of y m and y m have the same steepness, but one increases from left to right, and the other decreases from left to right. (c) These graphs have the same slope, but different y-intercepts. [ 8, 8] by [, ] These graphs all pass through the origin. They have different slopes. [ 8, 8] by [, ] 69. As in the diagram, we can choose one point to be the origin, and another to be on the -ais. The midpoints of the sides, starting from the origin and working around counterclockwise in the diagram, are then A a,0, B a b c,, C d b, c e, and D d, e. c e The opposite sides are therefore parallel, since the slopes of the four lines connecting those points are: m AB ; mbc ; b d a c e m CD ; mda. 70. y 4 b d a 4 ( ) 7. A has coordinates b, c, while B is b c a,, so the line c containing A and B is the horizontal line y, and the distance from A to B is a b b a. SECTION P. Eploration.. [, 4] by [, 0] [, 4] by [, 0] [, 4] by [, 0] By this method, we have zeros at 0.79 and The answers in parts,, and 4 are the same. 6. On a calculator, evaluating 4 7 when 0.79 gives y and when. gives y 0.064, so the numbers 0.79 and. are approimate zeros. [.0,.6] by [ 0., 0.4] Zooming in and tracing reveals the same zeros, correct to decimal places. Eercises P. [0.6, 0.94] by [ 0.9, 0.]. 4 or. 0. or.. or y 7. 7 or or or or or or 4 0.4

6 900 ADDITIONAL ANSWERS 0. or. or 4.. or or [, ] by [, ] [, ] by [, ] [, ] by [, ] [, ] by [, ].. [, ] by [, ] [, ] by [, ] [, ] by [, ] [, ] by [, ] ; ;.7 [, ] by [, ] [, ] by [, ] 4. (a) y 4 (the one that begins on the -ais) and y (b) y 4 (c) The -coordinates of the intersections in the first picture are the same as the -coordinates where the second graph crosses the -ais. 46. Any number between.4 and. must have the digit 4 in its thousandths position. Such a number would round to or or or (a) There must be distinct real zeros, because b 4ac 0 implies that b 4 a c are distinct real numbers. (b) There must be real zero, because b 4ac 0 implies that b b 4 a c 0, so the root must be. a (c) There must be no real zeros, because b 4ac 0 implies that b 4 a c are not real numbers. 6. True. If is an -intercept, then y 0 when. That is, a b c 0 when. 6. False. Notice that ( ) 8, so could also be. SECTION P.6 Eercises P.6. The graph either lies entirely above the -ais or entirely below the -ais.. (a bi) (a bi) bi, real part is zero 4. (a bi) ( a b i ) (a bi) (a bi) a b, imaginary part is zero. ( a b i ) ( c d i ) ( a c b d ) ( a d b c ) i (ac bd) (ad bc)i and ( a b i ) ( c d i ) (a bi) (c di) (ac bd) (ad bc)i are equal 6. ( a b i ) ( c d i ) ( a c ) ( b d ) i (a c) (b d)i and ( a b i ) ( c d i ) (a bi) (c di) (a c) (b d)i are equal

7 ADDITIONAL ANSWERS 90 SECTION P.7 Eercises P (, ),. (, ), , 4 4, 9., 4,. (,.4] [0.08, ).,,. Reveals the boundaries of the solution set 7. (b) When is in the interval (, ]. CHAPTER P REVIEW EXERCISES. Endpoints 0 and ; bounded (7 zeros between the decimal point and the 9) 4. (a) 9.8 (b), 7. ( 0) (y 0), or y 4 8. ( ) [y ( )] 4, or ( ) (y ) 6 9. Center: (, 4); radius: 8. y. (a) (b) y (c) 07.6, which is very close to 08. [0, ] by [00, ] [0, ] by [00, ] 6. Both graphs look the same, but the graph on the left has slope less than the slope of the one on the right, which is 4. The different horizontal and vertical scales for the two windows make it difficult to judge by looking at the graphs. 9. ( 6, ] 60., SECTION. Eploration. A statistician might look for adverse economic factors in 990, especially those that would affect people near or below the poverty line. 4. Yes. Table. shows that the minimum wage worker had less purchasing power in 990 than in any other year since 90.

8 90 ADDITIONAL ANSWERS Eercises.. Women ( ), Men ( ) 9. (a) and (b) [, ] by [, 9] L (a). sec 4. (a) (b) The line is y (c) About 00. (d) The terrorist attacks on September, 00 caused a major disruption in American air traffic, from which the airline industry was slow to recover. [, ] by [400, 70] [, ] by [400, 70] [ 0, 0] by [ 0, 0].9 [ 0, 0] by [ 0, 0].09 or.86 [ 0, 0] by [ 0, 0]. or 4 [ 0, 0] by [, ] [, ] by [ 0, 0].77 [, ] by [ 0, 0].6 [ 4, 4] by [ 0, 0].47 [, ] by [, 4] 48. (c) A vertical line through the -intercept of y passes through the point of intersection of y and y. (d) At.6806, y y.7. At , y y At.404, y y (b) [0, ] by [0, ]. Let n be any integer. n n n(n ), which is either the product of two odd integers or the product of two even integers. The product of two odd integers is odd. The product of two even integers is a multiple of 4, since each even integer in the product contributes a factor of to the product. Therefore, n n is either odd or a multiple of 4.

9 ADDITIONAL ANSWERS (a) [ 4, 4] by [ 0, 0] 6. (a) Subscribers Monthly Bills (c) The fit is very good: [7, ] by [0, 00] [7, ] by [, ] [7, ] by [0, 00] (e) Subscribers Monthly Bills [4, ] by [0, 00] [4, ] by [0, 60] SECTION. Eploration. From left to right, the tables are (c) constant, (b) decreasing, and (a) increasing.. X moves from to to 0 0 to X Y X moves from to to 0 0 to X Y X moves from to to 0 0 to X Y. positive; negative; 0 4. For lines, Y/ X is the slope. Lines with positive slope are increasing, lines with negative slope are decreasing, and lines with 0 slope are constant. to 0 to Eercises. to 4 to to to Not a function; y has two values for each value of.. Not a function; y has two values for each positive value of [, ] by [, ] [, ] by [ 0, 0] [ 0, 0] by [ 0, 0] [ 0, 0] by [ 0, 0]

10 904 ADDITIONAL ANSWERS [ 0, 0] by [, ] [, ] by [, ] [, ] by [, ] [, ] by [0, 6]. Yes, non-removable. Yes, removable. Yes, non-removable 4. Yes, non-removable [ 0, 0] by [ 0, 0] [, ] by [ 0, 0] [ 0, 0] by [, ] [, ] by [, ]. Local maima at (, 4) and (, ), local minimum at (, ). The function increases on (, ], decreases on [, ], increases on [, ], and decreases on [, ). 6. Local minimum at (, ), (, ) is neither, and (, 7) is a local maimum. The function decreases on (, ], increases on [, ], and decreases on [, ). 7. (, ) and (, ) are neither, (, ) is a local maimum, and (, ) is a local minimum. The function increases on (, ], decreases on [, ], and increases on [, ). 8. (, ) and (, ) are local minima, while (, 6) and (, 4) are local maima. The function decreases on (, ], increases on [, ], decreases on (, ], increases on [, ], and decreases on [, ). 9. Decreasing on (, ]; 0. Decreasing on (, ];. Decreasing on (, ];. Decreasing on (, ]; increasing on [, ) constant on [, ]; constant on [, ]; increasing on [, ) increasing on [, ) increasing on [, ) [ 0, 0] by [, 8] [ 0, 0] by [, 8] [ 0, 0] by [0, 0] [ 7, ] by [, ]. Increasing on (, ]; 4. Increasing on (, 0.49]; decreasing on [, ) decreasing on [ 0.49,.]; increasing on [., ). [ 4, 6] by [, ] [, ] by [, ]

11 ADDITIONAL ANSWERS f has a local minimum of y.7 at Local maimum: y 4.08 at.. It has no maimum. Local minimum: y.08 at.. [, ] by [0, 6] [, ] by [ 0, 0] 4. Local minimum: y 4.09 at Local minimum: y 9.48 at.67. Local maimum: y.9 at 0.8. Local maimum: y 0 when. [, ] by [ 0, 0] [, ] by [ 0, 0] 4. Local maimum: y 9.6 at Local maimum: y 0 at.. Local minimum: y 0 at 0 and y 0 at 4. Local minimum: y. at.. [, ] by [0, 80] [, ] by [ 0, 0] [ 0, 0] by [ 0, 0] [ 0, 0] by [ 0, 0] [ 8, ] by [ 0, 0] [ 0, 0] by [ 0, 0] [ 0, 0] by [ 0, 0] [, ] by [0, ] [ 4, 6] by [, ] [ 6, 4] by [ 0, 0]

12 906 ADDITIONAL ANSWERS 67. (a) f() crosses the horizontal (b) g() crosses the horizontal (c) h() intersects the horizontal asymptote at (0, 0). asymptote at (0, 0). asymptote at (0, 0). [ 0, 0] by [ 0, 0] [ 0, 0] by [, ] [, ] by [, ] 69. (a) The vertical asymptote is 0, and 70. The horizontal asymptotes are determined by the two limits this function is undefined at 0 lim f() and lim f(). These are at most two different (because a denominator can t be zero). numbers. (b) 7. True; this is the definition of the graph of a function. 7. False; f() 0 is a function that is symmetric with respect to the -ais. [ 0, 0] by [ 0, 0] Add the point (0, 0). 77. (a) (b) 0; but the discriminant of is negative ( ), so the graph never crosses the -ais on the interval (0, ). (d) 0; but the discriminant of is negative ( ), so the graph never crosses the -ais on the interval (, 0). [, ] by [, ] 78. (a) increasing (b) y (c) y 0.0 ; y is none of these since it first increases from to.0 and then decreases [ 6, 6] by [, ]

13 ADDITIONAL ANSWERS 907 SECTION. Eercises. 9. Domain: all reals; Range: [, ) 0. Domain: all reals; Range: [0, ). Domain: ( 6, ); Range: all reals. Domain: (, 0) (0, ); Range: (, ) (, ). Domain: all reals; Range: all integers 4. Domain: all reals; Range: [0, ). (a) Increasing on [0, ) (b) Neither (c) Minimum value of 0 at 0 (d) Square root function, shifted 0 units right 6. (a) Increasing on [(k ) /, (k ) /] and decreasing on [(k ) /, (k ) /], where k is an even integer (b) Neither (c) Minimum of 4 at (k ) / and maimum of 6 at (k ) /, where k is an even integer (d) Sine function shifted units up 7. (a) Increasing on (, ) (b) Neither (c) None (d) Logistic function, stretched vertically by a factor of 8. (a) Increasing on (, ) (b) Neither (c) None (d) Eponential function, shifted units up 9. (a) Increasing on [0, ); decreasing on (, 0] (b) Even (c) Minimum of 0 at 0 (d) Absolute value function, shifted 0 units down 40. (a) Increasing on [(k ), k ] and decreasing on [k, (k ) ], where k is an integer (b) Even (c) Minimum of 4 at (k ) and maimum of 4 at k, where k is an integer (d) Cosine function, stretched vertically by a factor of 4 4. (a) Increasing on [, ); decreasing on (, ] (b) Neither (c) Minimum of 0 at (d) Absolute value function, shifted units right 4. (a) Increasing on (, 0]; decreasing on [0, ) (b) Even (c) Maimum of at 0 (d) Absolute value function, reflected across -ais and then shifted units up 4. y 46. y 47. y 48. y No points of discontinuity 0 No points of discontinuity y 0. y. y. y No points of discontinuity No points of discontinuity 0,, 4,,.... (a). (a) [, ] by [, ] f() [, ] by [, ] (b) The fact that ln(e ) shows that the natural logarithm function takes on arbitrarily large values. In particular, it takes on the value L when e L.

14 908 ADDITIONAL ANSWERS 6. (a) y Cost ($) Weight (oz) (b) One possible answer: It is similar because it has discontinuities spaced at regular intervals. It is different because its domain is the set of positive real numbers, and because it is constant on intervals of the form (k, k ] instead of [k, k ), where k is an integer. 7. Domain: all real numbers; Range: all integers; Continuity: There is a discontinuity at each integer value of ; Increasing/decreasing behavior: constant on intervals of the form [k, k ), where k is an integer; Symmetry: none; Boundedness: not bounded; Local etrema: every non-integer is both a local minimum and local maimum; Horizontal asymptotres: none; Vertical asymptotes: none; End behavior: int() as and int() as. 64. (a) Answers will vary. (b) In this window, it appears that. (c) (d) On the interval (0, ),. On the interval (, ),. All three functions equal when. [0, ] by [0,.] 66. Answers will vary. 67. (a) Pepperoni count ought to be proportional to the area of the pizza, which is proportional to the square of the radius. SECTION.4 Eploration f g f g 4 ( ) ( ) 0.6 ln(e ) ln sin cos sin sin cos

15 ADDITIONAL ANSWERS 909 Eercises.4. ( f g)() ; ( f g)() ; ( fg)() ( )( ). There are no restrictions on any of the domains, so all three domains are (, ).. ( f g)() 4; ( f g)() ; ( fg)() ( ) ( ) 7. There are no restrictions on any of the domains, so all three domains are (, ).. ( f g)() sin ; ( f g)() sin ; ( fg)() sin. Domain in each case is [0, ). 4. ( f g)() ; ( f g)() ; ( fg)(). Domain in each case is [, ).. ( f/g)() ; 0 and 0, so the domain is [, 0) (0, ). (g/f )() ; 0, so the domain is (, ). 6. ( f/g)() ; 0 and 4 0, so and 4; the domain is [, ). 4 4 (g/f )() 4 4 ; 0 and 4 0, so and 4; the domain is (, ). 7. ( f/g)() / ; 0, so ; the domain is (, ). ( g/f )() / ; 0 and 0; the domain is [, 0) (0, ]. 8. ( f/g)() / ; 0, so ; the domain is (, ) (, ). ( g/f )() / ; 0, so 0; the domain is (, 0) (0, ) f(g()) ; (, ); g( f()) ; (, ) [0, ] by [0, ] [, ] by [ 0, ] 6. f(g()) ( ) ; (, ) (, ); g( f()) ( ) ( ) 7. f(g()) ; [, ); g( f()) ; (, ] [, ) 8. f(g()) ; [0, ) (, ); g( f ()) ; (, ) (, ) (, ) 9. f(g()) ; [, ]; g( f()) ; 4 [, ] 0. f(g()) ; (, ); g( f()) 9 ; (, ). f(g()) ; (, 0) (0, ); g( f()) ; (, 0) (0, ). f(g()) ; all reals ecept 0 and /( ) g(f()) ; all reals ecept and 0 /( ). One possibility: f() and g() 4. One possibility: f() ( ) and g(). One possibility: f() and g() 6. One possibility: f() / and g() 7. One possibility: f() and g() 8. One possibility: f() e and g() sin 9. One possibility: f() cos and g() 0. One possibility: f() and g() tan. V 4 r 4 (48 0.0t) ; 77,74.6 in.. t.6 sec 7. y and y 8. y and y 9. y and y 40. y and y 4. y and y 4. y and y 4. y and y or y and y 44. y and y ; (, )

16 90 ADDITIONAL ANSWERS 4. False; is not in the domain of (f/g)() if g() False; it is the set of all numbers that belong to both the domain of f and the domain of g... f g D e ln (0, ) ( ) [, ) ( ) (, ] ( ) 0 [ 9.4, 9.4] by [ 6., 6.] (, ) SECTION. Eploration. T starts at 4, at the point ( 8, ). It stops at T, at the point (8, ). 6 points are computed.. The graph is smoother because the plotted points are closer together.. The graph is less smooth because the plotted points are further apart. 4. The smaller the Tstep, the slower the graphing proceeds. This is because the calculator has to compute more X and Y values.. The grapher skips directly from the point (0, ) to the point (0, ), corresponding to the T-values T and T 0. The two points are connected by a straight line, hidden by the Y-ais. 6. With Tmin set at, the grapher begins at the point (, 0), missing the bottom side of the curve entirely. 7. Leave everything else the same, but change Tmin back to 4 and Tma to. Eercises.. (a) ( 6, 0), ( 4, 7), (, 4), (0, ), (, ), 6. (a) (, ), (, 8), (0, ), (, 0), (, ), (, 0), (4, ) (4, ), (6, 8) (b) 4 ; It is a function. (b). ; It is a function. (c) (c) [, ] by [, 6] [, ] by [, ] 7. (a) (9, ), (4, 4), (, ), (0, ), (, ), 8. (a) (0, ), (, ), (, ), (, ) (4, 0), (9, ) (b) y ; It is a function. (b) (y ) ; It is not a function. (c) (c) [, ] by [, ] [, 4] by [ 6, 4]. f (), (, ) 4. f (),(, ). f (), (, ) (, ) 6. f (), (, ) (, ) 7. f (), 0 8. f (), 0

17 ADDITIONAL ANSWERS 9 9. f (), (, ) 0. f (), (, ). f (), (, ). f (), (, ). One-to-one 4. Not one-to-one. One-to-one 6. Not one-to-one y 7. f(g()) [ ] ( ) ; g( f()) [( ) ] () 8. f(g()) 4 [(4 ) ] 4 [ (4) ; g( f()) 4 4 ] ( ) 9. f(g()) [( ) / ] ( ) ; g( f()) [( ) ] / ( ) / 7 0. f(g()) ; g( f()) f(g()) ( )(. f(g()) ) ( ) ; g( f()) ( ) ( ) ; ( ) ( ( ) ) g( f()) ( ) ( ) ( ) ( ). (b) y. This converts euros () to dollars (y) (a) c () 9. This converts Celsius temperature to Fahrenheit temperature. (b).8. This converts Fahrenheit temperature to Kelvin temperature. 9. y e and y ln are inverses. If we restrict the domain of the function y to the interval [0, ), then the restricted function and y are inverses. 6. y and y / are their own inverses. 9. True. All the ordered pairs swap domain and range values. 40. True. This is a parametrization of the line y. 4. (Answers may vary.) (a) If the graph of f is unbroken, its reflection in the line y will be also. (b) Both f and its inverse must be one-to-one in order to be inverse functions. (c) Since f is odd, (, y) is on the graph whenever (, y) is. This implies that ( y, ) is on the graph of f whenever (, y) is. That implies that f is odd. (d) Let y f(). Since the ratio of y to is positive, the ratio of to y is positive. Any ratio of y to on the graph of f is the same as some ratio of to y on the graph of f, hence positive. This implies that f is increasing. 46. (Answers may vary.) (a) f() e has a horizontal asymptote; f () ln does not. (b) f() e has domain all real numbers; f () ln does not. (c) f() e has a graph that is bounded below; f () ln does not. (d) f() has a removable discontinuity at because its graph is the line y with the point (, 0) removed. The inverse function is the line y with the point (0, ) removed. This function has a removable discontinuity, but not at. 47. (a) y 0.7 (b) y 4 ( ). It converts scaled scores to raw scores.

18 9 ADDITIONAL ANSWERS SECTION.6 Eploration. They raise or lower the parabola along the y-ais.. They move the parabola left and right along the -ais.. Yes Eploration. Graph C. Points with positive y-coordinates remain unchanged, while points with negative y-coordinates are reflected across the -ais.. Graph A. Points with positive -coordinates remain unchanged. Since the new function is even, the graph for negative values will be a reflection of the graph for positive values.. Graph F. The graph will be a reflection across the y-ais of graph C. 4. Graph D. The points with negative y-coordinates in graph A are reflected across the y-ais. Eploration. The. and the stretch the graph vertically; the 0. and the 0. shrink the graph vertically.. The. and the shrink the graph horizontally; the 0. and the 0. stretch the graph horizontally. [ 4.7, 4.7] by [.,.] [ 4.7, 4.7] by [.,.] Eercises.6. Vertical translation down units. Vertical translation up. units. Horizontal translation left 4 units 4. Horizontal translation right units. Horizontal translation to the right 00 units 6. Vertical translation down 00 units 7. Horizontal translation to the right unit, and vertical translation up units 8. Horizontal translation to the left 0 units and vertical translation down 79 units 9. Reflection across -ais 0. Horizontal translation right units. Reflection across y-ais. This can be written as y ( ) or y. The first of these can be interpreted as reflection across the y-ais followed by a horizontal translation to the right units. The second may be viewed as a horizontal translation left units followed by a reflection across the y-ais.. Vertically stretch by 4. Horizontally shrink by, or vertically stretch by 8. Horizontally stretch by 0., or vertically shrink by Vertically shrink by Translate right 6 units to get g 8. Translate left 4 units, and reflect across the -ais to get g 9. Translate left 4 units, and reflect across the -ais to get g 0. Vertically stretch by to get g. y. y. y 4. f 0 g 6 7 g 0 h 6 h f 6 g h 0 y h f g 6 0 f. f() 6. f() ( ) 7. f() 8. f() 9. (a) (b) 0. (a) 4 (b) 4. (a) y f() ( 8 ) (b) y f( ) 8( ). (a) (b)

19 ADDITIONAL ANSWERS 9. Let f be an odd function; that is, f( ) f() for all in the domain of f. To reflect the graph of y f() across the y-ais, we make the tranformation y f( ). But f( ) f() for all in the domain of f, so this transformation results in y f(). That is eactly the translation that reflects the graph of f across the -ais, so the two reflections yield the same graph. 4. Let f be an odd function; that is, f( ) f() for all in the domain of f. To reflect the graph of y f() across the y-ais, we make the transformation y f( ). Then, reflecting across the -ais yields y f( ). But f( ) f() for all in the domain of f, so we have y f( ) [ f()] f(); that is, the original function.. y 6. y 7. y 8. y 9. (a) 8 (b) (a) (b) 4. (a) 4 (b) 9 4. (a) (b) 4. Starting with y,translate right units, vertically stretch by, and translate down 4 units. 44. Starting with y, translate left unit, vertically stretch by, and reflect across -ais. 4. Starting with y, horizontally shrink by and translate down 4 units. 46. Starting with y, translate left 4 units, vertically stretch by, and reflect across -ais, and translate up unit.. y. y. y 4. y. Reflections have more effect on points that are farther away from the line of reflection. Translations affect the distance of points from the aes, and hence change the effect of the reflections. 6. The -intercepts are the values at which the function equals zero. The stretching (or shrinking) factors have no effect on the number zero, so those y-coordinates do not change. 7. First vertically stretch by 9, then translate up units. 8. First vertically shrink by 9, then translate down units (a) y (b) Change the y-value by multiplying by the conversion rate from dollars to yen, a number 6 that changes according to international market conditions. This results in a vertical 4 stretch by the conversion rate. Price (dollars) Month

20 94 ADDITIONAL ANSWERS 67. (a) The original graph is on the top; (b) The original graph is on the top; the graph of y f() is on the bottom. the graph of y f( ) is on the bottom. [, ] by [ 0, 0] [, ] by [ 0, 0] [, ] by [ 0, 0] [, ] by [ 0, 0] (c) y (d) y 68. (a) You should get a graph that looks like this: (b) Let cos t and leave y unchanged. [ 4.7, 4.7] by [.,.] [ 4.7, 4.7] by [.,.] (c) Let cos t and y sin t. (d) Let 4 cos t and y sin t. [ 4.7, 4.7] by [.,.] [ 4.7, 4.7] by [.,.]