Answer Key
Interval Notation
1
4. , 5. (2,5) 6. [12, 3] 7. (17, 24]
2
8. [125, 400) 9. (, 0.40) U (0.40, ) 10. (, 2) U (2, 2) U (2, )
11. (, 4) U[3, ) 12. (, 7] U (10,12) 13(a) [5, 7] (b) (5, 7) (c) 3 x 8 (d)
(,9]
(e) (4, ) (f) (4, 7] (g) (,1) (h) [3, ) 14.(a) 3 x 9
(b) 4 x 5
(c) x 5 (d) x 6 15. (a) [0, 6) (b) [7, ) (c) [5,8]
(d) (7,10) (e) (5,8] (f) (2,9) (g){5} (h) (i) [2,10) (j) (4,10)
16. -3
Y N
Y
Y
17. (a) [3, ) ; (0, 9) ; [2, 6] ; [1, 4] U [7,8]
N N
N
Y
Y Y
Y
Y
7
N N
Y
Y
1. (, 10]
2. (,3]
3. (6, )
(c) 3 x 4; [3, 4]
(b)
Interval Notation Worksheet
1. (,3] ;
4. x 5 ;
7. 5 x 1; (5, 1)
5. x 1; (,1]
8. (, ) ;
6. (,1) U[5, ) ;
9. x 1 or x 2 ; (, 1] U (2, )
11. (7, ) ;
10. 1 x 4 ;
Lesson 1.1
1. Commutative for Addition
3. Identity for Addition
5. Inverse for Multiplication
7. Distributive
9. Associative for Addition
12.
5a(b c)
Given
5(ab ac)
Distributive
5ab 5ac
3. 2 x 6; [2, 6]
2. x 4 ;
Distributive
12. 2 x 2 ;
2. Commutative for Multiplication
4. Distributive for Multiplication over Additon
6. Associative for Multiplication
8. Commutative for Addition
10. Inverse for Multiplication 11. Identity for Addition
(3 b)3b
1
3 3b
b
1
3 3b
b
1
3 3 b
b
3 (3 1)
33
=9
Given
Def. of Division
Associative for Multiplication
Commutative for Multiplication
Inverse for Multiplication
Identity for Multiplication
Multiplication
Answer Key
14. 3a ( 4 2a )
3a (2 2a)
3a (2 2a)
Given
15. 52
Def Sqr. Rt.
Def of Subtraction
Commutative for Add.
Associative for Add.
Addition
Identity for Mult
3a (2a 2)
(3a 2a ) 2
1a 2
a2
1
1
in 28 in 80 inches
4
4
16. 132 oz + 12 oz = 144 oz. 17. 1
2
miles
3
18. 0.25 miles 60 sec 60 min
3.12 sec 1 min 1 hr
900
miles
288.46
3.12
hour
5 days 8 rounds 3980 ft 1mile
30.152 miles 20. 5 pieces
1 week 1 day 1 round 5280 ft
21. Total = 6 min 38 sec; Average = 99.5 sec 22. 1800 mph 23. 5.682 mph
3
3
123 in
lbs
24. 31.25 drops per minute 25. yes; .001 kg3 1000 g 1 pound 2.54 cm
62.372 3
w
19. 1 week
cm
1 kg
454 grams
1 in
1 ft
ft
26a. 21,195,743 burgers 26b. 87 years + present age 27. Answers vary 28. Answers vary
29. Commutative for Mult. 30. Associative for Mult. 31. Inverse for Mult.
32. Identity for Mult. and Distributive 33. Distributive 34. always; associative prop
35. always; associative prop. 36. sometimes; If a, b, and c are all 0 it works; else it doesn’t
37. sometimes; if a, b and c are all 1 it works 38. always; distributive prop.
39. sometimes; if a, b and c are 1 it works 40. 4.032 liters of first solution and 12.019 liters of
second solution
Lesson 1.2
1. x 4 2. 8a 3 ( x y ) 2 3. (6)8 4. 16 5. 1 6. 4 7. 43 8. 3 / 23
9. 17 /12
2
2
10. 2,000,000 11. 20 4x 12. x 15 x 6
13. 2 x 15 y
14. x 2 xy y 2
1
15. x3 x 2 y y x 2 16. x x 6 17. 2 xy 2 x y 1 18. x 3 y 2 x 3 y 2 y 2 2 y
2
2
20. if p > 6, 340 + 35(p – 6) = 35p +130; 10 people = $480; 4 people = $340
19. 55x
21. if m > 300, 19.99 + .10(m – 300) + .25d ; Cost for m = 375 and d = 54 is $40.99
22a. 51x + 12y
22b. 50.4x + 15y
22c. purchase = $930; proceeds = $1029
2
2
2
23. 4ax +7x b (bx ax + 3) = 3x + 9x 8
3
b
24. 3ax 8bx2 (10x2 + 2ax ) + 3x3 4ax = 12x3 18x2 18x;
25. (2ax + b)2 (abx2 6a) = 6x2 + 40x + 37
26. (7 + 3a) (2b ÷ 4)2 =74
27. 4a (ab + b) a2 = 15
28. ab 5b ((2b)2 + 4a) = 21
Lesson 1.3
1. x 21
8. x
11
3
2. x
20
3
9. (, )
3. x
10. x 4
5
8
4. no solution
11. x
6
5
5. x 1
12. x 0
6. x 5
13. x 1
7. x 2
14. x 5
Answer Key
d r
7
2 S 2an
19. t
20. d
a 2
9
n(n 1)
21. $106.21 22. 72 feet 23. 58 mph 24. 2.5 hrs 25. 6 hours 26. 9 minutes
600
27.
85.714 min 28. 22 min and 38.491 sec 29. 8.7 feet 30. 25 31. 40,140
7
32. 34 and 56 33. 43, 45, 47, 49 34. 34, 36, 38, 40 35. 4 hours 36. 300 and 600 mph
37. 3.5 and 15.5 38. 18, 18, 29 39. 20 miles 40. 76 yds by 236 yds 41. 3 pm
42. $3000 43. 119
15. (, ) 16. x 5
17. x 1
18. x
Section 1.4
6 3x
3
3
1
16 6 x
14 12 x
1. y
4. y
5. y
2. y x
3. y
x2
2x
4
2
2
15
20 x 35
9
2A
S
3V
6. y
b2 9. r
10. R
r
7. F C 32 8. b1
s
3x 5
5
h
h
x
z 5x
2 xy
xy
12. y
14. y
11. z
13. z
15. peanut=$2.00;
xy 4 x 3 y
x 1
xy y x
3 xz 4
caramel = $1.50 16. P 2b 1.5c ; P = profit; b = # of peanut; c = # of caramel
2
3V 2 r 3
20. 8 ft
17. 130 peanut bars, 260 caramel bars
18. V r 2 h r 3 19. h
3 r 2
3
V
22. R 2 2
2 r
Literal Equations
C2
1. A(C )
4
s2 3
2. A( s)
4
3. S (V ) 4
3
9V 2
16 2
36 V
3
2
4. V (h)
h3
48
S Ph
3 (5 h)
6. V ( x) x5 x5 3 x5 (1 3) 7. B
2
25
2
R1
2A b2 h
S 2 r
9
3V h3
10. F C 32 11. R2
8. b1
12. r
9. h
RR1 1
2 r
5
h
3 h 2
3
5. V (h) 15
S
Lad
y b
T
15. r t 1 16. m
R 14. n
17. y m( x x1 ) y1
s
x
d
P
v f v0 at0
dr
1 dA dr
in
3 2
3 3
20. A( s)
;
18. t f
19.
2
s 21. V ( s)
s
a
dt 2 r dt dt
min
4
24
1
22. V ( s) s 3 23. 24 ft 25. 15 ft 26. 20 ft 28. 24 ft 29. 7 ft 31. b(t ) 7 2t
4
13. r
32. h(b) 625 b 2
Section 1.5
1. y 8 x 18
33. h(t ) 625 (7 2t ) 2
2. y 7 x 89
3. y 2 x 28
34. no; h(t) is not a linear function
4. y 0.6 x 4
5. y 0.5 x 0.75
Answer Key
6. y 0.7 x 7.2 7. 24 = 7.50 +0.75s; 22 songs 8. 14.25 mph 9. $525,300 10. 5 painters
11. 6 feet 12. 4.75 inches 13. minimum amt = $1.59; total = $7.95 14. 6 quarters
15. 6 inches 16. 2 feet and 3 inches
Section 1.6
1
1. , 2. (5, 3]
2
7. (,3) (3.5, )
3. (,3] [7, )
8. [2,3]
32
9. ,
7
4. (, 0)
10.
5. (, 4)
11.
6. (,3.5]
12. (, )
14
5
x 2
13. x x or x 2 14. x x 3.1 or x 4 15. x 40 x 100 16. x
8
33
17. 1,108, 451 P 1, 211,537 18. $10 shirts = 0, 1, 2 and $7 shirts = 6, 5, 4
19. 5 frames and 10 candles or 6 frames and 9 candles or 7 frames and 8 candles
2
2
20. x 50 , y 85 , z 63 21. 5 paint kits and 10 brush kits or 6 paint kits and 9 brush
3
3
kits or 7 paint kits and 8 brush kits or 8 paint kits and 7 brush kits or 9 paint kits and 6 brush kits
2
92
or 10 paint kits and 5 brush kits or 11 paint kits and 4 brush kits.
22. x
3
3
63
21 105
35
24. 20 stamps at 22 cents 25. 6, 12, 10
23.
x ;
y
8
2 8
2
26. Daphne = 4, Jeanine = 4, Greg = 8, Suzanne = 1 27. x 98 28. n 37
Section 1.7
1. 1 or -5 2. 0 or 6 3. 2/5 or 8/5 4. x 5 or x 1 5. -1 or 9 6. 7. 7/8 or 25/8
9. 9/4 10. 20 x 4 11. 8/5 or 28 12. x 6 or x 1 13. 10 / 9 x 6
8.
16. -1.5 or -0.75 15. x a b 16. x ab
14. x 1.7 15. 2 x 3
c b
cb
c b
cb
17. x 2a or x 0 18.
19.
x
x
a
a
a
a
c b
cb
c b
c b
20.
x or x
21.
x or x
22. d 0.365 0.01
a
a
a
a
23. d 238,850 13,150 24. w 9.75 0.375 25. d 8 4
26. c 3.85 0.35; t 1.2 0.1; w 5.05 0.45
27a. 238.095 x 263.158
27b. x 90 0.05 x
27c. 85.714 x 94.737
Answer Key
Functions the Beginning
1. D:{0, 1, 2, 3, 4}, R: {1, 2, 3, 4}, is a function, is not one-to-one
2. D :{2, 1, 0,1}, R :{3, 1,1,3,5} , not a function
3. D : (, ), R : (, 0] , is a function, not one-to-one
4. D :[0, ), R : (, ) , is not function
5. D :{2, 1, 0,1, 2}, R :{2, 1,1} ,not a function
6. D :{2, 1, 0, 2, 4}, R :{1} , is a function, is not one-to-one
7. D :{0,1, 2,5}, R :{3, 0, 2, 4} , not a function
8. D :[7,8], R :[4,5] ,is a function, not a one-to-one
9. function 10. function 11. not a function 12. function 13. function 14. not a function
15. not a function 16. function 17. not a function 18. not a function 19. function
20. function 21. ( ,3) (3, ) 22. (, 1) (1, ) 23. (, 3) (3,3) (3, )
24. (, ) 25. (, 4) (4, 7) (7, ) 26. (, 10) (10, 2) (2, ) 27. [5, )
31. (2.5, ) 32. (, 4) (6, )
28. (, )
29. [7, ) 30. (, 4) (4,1.5]
7 2 2
33. , , 34. (, 2) (7, ) 35. [4.5, )
3 3 3
Real World Functions –
Student answer will vary.
Function Value and Functions Notation
x 10
1. 6 2. 2 x 4 3. 2 4 4.
5. 24
2
9. 9 10. 2 11. 1 12. 0.5 13. 9.5 14.
19.
20.
18. 6 x 3h
22.
6. 3n 2 18n 22 7. 3n 2 18n 22
8. –9
5 15. 0.5 16. h 2 6 h 6 17. 2
always true 21. not always true
always true 23. not always true
Function Behaviors and Characteristics
1a. 6 b. doesn’t exist c. positive d. negative e. –2 f. (–3, 2) g. (–3, 4]
h. (2,3) (4, 6] i. 1 and 1 j. 2 k. 5.5 l. once m. once n. 4 o. none p. 6
q. (3, 1) and [1, 4] 2a. 0; 3 b. 5.1 c. 2.9 and 2.6 d. 12.1 and 1.6
e. (2.9, 0.3) and (2.6,) f. (, 2.9) and (0.3,2.6) g. (4, 1) and (2,3) h. 4, 1, 2,3
i. 12.1 j. 3a. [0, 6) and (16,18]; function is positive b. (6, 16); function is negative
c. at 6 and at 16 di. 60 m dii. 105 m diii. 15 m div. 180 m e. – 30 f. (6, 10); (16, 18)
g. (2, 6); (14, 16) 4a. (, 6) b. (,5] c. 2 d. g (2) 0.5, g (0) 5, g (2) 1, g (4) 1,
g (6) doesn't exist e. g (2) f. (, 0), (2, 6) g. (0, 2) h. 2 i. 5 j. (, 2.25), [2,3)
Answer Key
Characteristics of Functions
1. a) (, ) b) (, 4] c) -2; 2 d) 4 e) (-2,2) f) ( , 2);(2, ) g) (, 0) h) (0, ) i) none
2. a) (, ) b) [0, ) c) -3; 3 d) 3 e) ( , 3);( 3,3);(3, ) f) none g) (3, 0);(3, )
h) (, 3);(0,3) i) none
3. a) [8,8] b) [6, 2] c) -2; 4 d) 2 e) (-2,4) f) ( 8, 2);(4,8] g) (8, 0) h) (3, 6)
i) (0,3); (6,8)
4. a) [3,3) b) (4, 2] c) -3; 1/3 d) about ½ e) (-3, 1/3) f) (1/ 3,3) g) ( 3, 1) h) (1,3)
i) none
5. a) [7,3] b) [4, 4] c) -3; 5/3 d) 5/2 e) (-3, 5/3) f) [ 7,3);(5 / 3,3] g) (7, 1)
h) (1,3) i) none
6. a) [7,5] b) [5, 4] c) -7; 2 d) 4 e) (-7, 2) f) (2,5] g) ( 7, 4);(1, 0)
h) (0,5 / 2);(4,5) i) (-4, -1); (5/2, 4)
8. a) [7,5] b) [3, 6] c) 5.25, 4.25 d) 6 e) (5.25, 4.25) f) [7, 5.25);(4.25,5]
g) (7, 3) h) (3,5) i) (-3, 3)
9. a) [3,3] b) [2, 2] c)-3; 0; 3 d) 0 e) (-3, 0) f) (0, 3) g) ( 3, 1.5);(1.5,3)
h) (1.5,1.5) i) none
Even and Odd Functions
1. Fig. 1 and Fig. 2 2. Fig. 9 3. Even 4. Odd 5. Neither 6. Neither
7. Odd 8. Even 9. Odd
10. Neither 11. Even 12. Odd 13. Odd
14. Answer vary. Some possible answer are:
Polynomials have all even or all odd exponents on the variables.
15. Odd; Neither; Even
16. No 17. If f ( x) and g ( x) are both odd, then f ( x) f ( x)
and g ( x) g ( x) . Therefore, f ( x) g ( x) f ( x) g ( x) so the resulting function is even.
If f ( x) and g ( x) are both even, then f ( x) f ( x) and g ( x) g ( x) .
Therefore, f ( x) g ( x) f ( x) g ( x) so the resulting function is even.
18. If f ( x) is odd and g ( x) is even, then f ( x) f ( x) and g ( x) g ( x) .
Therefore, f ( x) g ( x) [ f ( x) g ( x)] so the resulting function is odd.
Transformations of Graphed Function
1.
2.
3.
4.
Answer Key
5.
6.
7.
9.
10.
11.
8.
12. right 2 13. up 2 14. vertical stretch 2, down 1 15. horizontal compression ½, down 3
12.
13.
14.
15.
16. vertical compression ½
16.
17. left 1, down 2 18. flip over the x-axis
17.
18.
19. verical stretch 2, flip over the x-axis, right 1, up 2
20. vertical compression ½, flip over the y-axis, down 3
Answer Key
19.
20.
21. flip negative part of function over the x-axis
22. flip negative part of function over the x-axis, down 2
23. right 1 and down 2, flip negative part of resulting function over the x-axis
21.
22.
23.
Graphical Transformations
1a. D: [0, 20], R: [0, 14] b. 14; 8 c. 20; 8 d. 3/4 e. 171 square units
2a. D: [0, 20], R:[2, 9] b. 9; 8 c. none; 6 d. 3/8 e. 125.5 square units
3a. D: [2, 22], R: [0, 7] b. 7; 10 c. 22; none d. -7/12 e. 85.5 square units
4a. D: [0, 20], R: [-7, 0] b. 0; 20 c. -7; 8 d. 85.5 square units e. [8, 20]
5a. D:[3, 23], R:[5, 26] b. 26; 11 c. 5; 23 d. none; none e. 356.5 square units f. [11, 23]
6. change: y-intercept, range, maximum value, area unchanged: domain, x-int, x of maximum
7. change: domain x-intercept, x of maximum unchanged: maximum value, area
need more info to determine: y-intercept, area
Transformation of Points without a Graph
1a.(2, 6) 1b. (2, 4) 1c. (2, -3) 1d. (2/3, 3) 1e. (-2, 3) 1f. (3, 3) 1g. (0, 3) 1h. (2, 0)
1i. (4, -1) 1j. (0, -4) 1k. (4, 14) 2a. f(x – 1) or f(2x) 2b. f(x – 5) + 1 2c. f(0.5x) or f(x + 2)
3. Amount of money Molly will earn from Thursday of one week to Friday of the next week
4.
5.
6.
7.
8.
9.
x
y
x
y
x
y
x
y
x
y
x
y
-7
-5
-6
-4
-3.5 2
-5 6
-7 6
-5 11
-4
-9
-3
-12
-2
-2
-2 8
-4 18
-2 27
-2
-18
-1
-30
-1
-11
0
12.5
-2 45
0
63
-1
-10
0
-14
-0.5 -3
1
8.5
-1 21
1
31
2
-4
3
-2
1
3
4
5.5
2
3
4
7
3
-1
4
4
1.5 6
5
4
3
6
5
-5
5
0
6
6
2.5 7
7
3.5
5
9
7
-9
6
5
7
16
3
12
8
1
6
24
8
-29
10.
x
y
11. f (0.5 x) 12. f ( x 1)
7
0
4
-4
13. f ( x) 14. 3 f ( x 4)
Answer Key
2
1
-2
-3
-5
-6
-13
-5
1
4
5
10
15. 0.5 f (4 x)
16. 2 f ( x 1) 3
Functions Transformations
1. O 2. D 3. K 4. B 5. G 6. M 7. F 8. J 9. A 10. H 11. C 12. I 13. N 14. L 15. E
Transformations of Parent Functions
1a. cubic b. left 1, down 2 c. y ( x 1)3 3 d. D: (, ) ; R: (, )
2a. quadratic b. right 1, up 2 c. y ( x 1) 2 2 d. D: (, ) ; R: [2, )
3a. square root b. flip over x-axis, right 2, up 1 c. y x 2 1 d. D:[2, ) ; R: (,1]
4a. absolute value b. vertical stretch 2, right 2 c. y 2 x 2 d. D: (, ) ; R: [0, )
5a. absolute value b.flip over x-axis, right 2, up 3 c. y x 2 3 d. D: (, ) ; R: (,3]
3
x 1 2 d. D:[1, ) ; R: [2, )
2
7a.quadratic b.flip over x-axis, vertical stretch 2, left 3, up 4 c. y 2( x 3) 2 4
d. D: (, ) ; R: (, 4]
1
4 d. D: (, 2) U (2, ) ; R: (, 4) U (4, )
8a. rational b. left 2, down 4 c. y
x2
9a. quadratic b. flip over x-axis, vertical stretch 3 c. y 3x 2 d. D: (, ) ; R: (, 0]
10a. greatest integer b. vertical stretch 2, down 1 c. y [ x] 1 d. D: (, ) ; R: {odd integers}
11. right 3, down 2 12. flip over x-axis, vertical stretch by 2 13. flip over x-axis, horizontal
compression by 0.5, up 1 14. vertical compression by 0.5, left 3 15. flip over x-axis, vertical
compression by 1/3, horizontal compression by 1/2, right 5/2, up 1 16. vertical stretch by 4,
horizontal compression by 1/2, left 3/2, down 2 17. vertical stretch by 2, left 4, up 1
18. right 1, down 2 19. left 2, down 1 20. vertical stretch by 2, horizontal stretch by 2, left 6,
up 1 21. vertical stretch by 1.5, left 1, up 4
6a. square root b.vertical stretch 3/2, left 1, down 2 c. y
Review
1. a) y x 2 b) flip over x-axis, vertical stretch by 2, up 4 c) y 2 x 2 4
d) D: (, ) ; R: (, 4]
2. a) y x 2 b) right 1, down 4 c) y ( x 1) 2 4 d) D: (, ) ; R: [4, )
1
3. a) y x 3 b) vertical compression ½, up 1 c) y x3 1 d) D: (, ) ; R: (, )
2
4. a) y x b) flip over x-axis, right 5, up 6 c) y x 5 6 d) D: (, ) ; R: (, 6]
5. a) y x b) vertical stretch by 2, right 5, up 2 c) y 2 x 5 2 d) D: (, ) ; R: [2, )
6. a) y x b) flip over x-axis, vertical stretch 4,up 2 c) y 4 x 2
Answer Key
d) D: [0, ) ; R: (, 2]
1
1
b) right 3, up 2 c) y
2 d) D: (,3) (3, ) ; R: (, 2) (2, )
7. a) y
x
x 3
1
3
b) vertical stretch 3, right 1, up 2 c) y
8. a) y
2
x
x 1
d) D: (,1) (1, ) ; R: (, 2) (2, )
9. quadratic parent; vertical stretch 2/3, right 5, down 1 10. square root parent; vertical stretch
by 3, right 1, down 4 11. cubic parent; flip over x-axis, vertical stretch of 2, right 1
12. absolute value parent; flip over the x-axis, right 2, up 4 13. quadratic parent; vertical stretch
by 1/3, left 6, up 2 14. absolute value parent; vertical stretch of 4/3, right 2, up 7
15. square root parent, right 4, down 2 16. rational parent; vertical stretch 3, left 2, up 4
17. exponential (power of 3) parent; flip over x-axis, right 1, down 4 18. y ( x 3) 2 5
1
x 2 3 21. y 2sin( 2.5) 1 (1st set of #22 – 27 are graphs)
19. y 3 x 5 20. y
2
22.
23.
26.
27.
24.
25.
22. -3 23. 1 – 2n 24. -32 25. 3x 2 24 x 44 26. 12 x 6 12 x3 1 27. 8 28. 8
29. x 2 x 6 30. 12 x 12 31. 6 x 6 32. -2 33. 6 x 3h 34. a 5.5 35. n 1
36. f ( x 1) 4 37. f ( x 1) 38. f (4 x) 5 39. function 40. not a function 41. function
42. not a function 43. (, 5) (5,5) (5, ) 44. (, 6) (6, 7) (7, ) 45. [4.5, )
46. (, ) 47. [4, 1) (5, ) 48. (, 3] [13, ) 49 and 50. answers may vary
51a. [4, 6) b. [3, 6] c. 2 and 4 d. 3 e. (2, 4) f. [4, 2); (4, 6) g. (4, 0), (1,3), (5, 6)
h. (0,1), (3,5) i. none j. 0 and 3 k. 3 and 6 l. 1 and 5 m. 2 and 2 n. 6 o. 3
52. even 53. odd 54. odd 55. neither
Answer Key
Section 2.2
1. they are equal 2. their product is -1 3. vertical lines 4. horizontal lines 5. -11/4, falls
6. 65/92, rises 7. 3/4, rises 8. 0, horizontal 9. -3/4, falls 10. undefined, vertical
11. line 1 is steeper 12. line 2 is steeper 13. k = 6 14. k = 4 15. k = 44 16. k = -1/9
17. yes, slopes are equal 18. no, slopes are not equal 19. pillar = 1000/21; pyramid = 55/17
20. 3 deg/hr; 73o 21. 225 mph 22. Since these lines are parallel, we know the slopes are equal.
If the slopes are equal, then the y-intercepts must be different for the lines to be distinct.
1 gal hr
1 gal hr
gal hr
ft
ft
ft
ft
23a.
24a. 52
b. 44
d. 52
c. 0
c. 84
b.
hr
sec
sec
sec
sec
9 hr
120 hr
25a. 8.4 b. 8.04 c. 8.004 26. 8
30. (0.5, 1.5) 31. 2 32.
m
27. 4 28.
sec
29a. -6 b. 0 c. 4 d. 2 e. 2
a.-3/2 b.-3/2 c.-3/2 d.-3/2 34.(1, 3.75) 35.-3/2
Section 2.3
1. m=-1/2, y-int=2, x-int=4 2. m=-2/3, y-int=2, x-int=-3 3. m=0, y-int=-3, x-int=none
4. m=2, y-int=-5, x-int=5/2 5. . m=-4, y-int=-3, x-int=-3/4 6. m=3/4, y-int=-3, x-int=4
7. m=undefined, y-int=none, x-int=6 8. . m=5/4, y-int=-2, x-int=8/5 9. k = -3 10. k = -5/3
11. k = 2 12. k=19/84 13a. V-int = 500,000, t-int = 6, t-int is the time when the value is $0
b. slope = -250,000/3 $/yr; slope represents the $ value the equipment drops each year
15a. 4.5x + 10y = 2500 b. (0, 250), (20, 241), (100, 105), (200, 160)
16a. b = 100 – 6.25w ; b = 175 – 12.50w b.
Your brother; his x-intercept
is smaller c. find the x-intercept of each by substituting 0 for y d. yes, now you pay off earlier
17. False, A = -7, B = 4, slope = 7/4 18. True, slope = 0 have one y-intercept, undefined slope
have one x-intercept and all other slopes have an x-intercept and a y-intercept. 19. False; x = 4
Section 2.4
1. y – 6 = 2(x – 7)
2. x = 3 3. y + 2 = 3/8(x – 7) 4. y = 9 5. y + 3 = -1/3(x – 2)
6. y + 3 = 1/5(x + 1) 7. y – 3 = -3(x – 5) 8. x = 5 9. y + 2 = -(x + 6) 10. y = 1
11. y – 10 = 2/3(x – 8) 12. 5x + y = 7 13. 2x + 3y = 20 14. 8x – y = 1 15. y = 2.5x + 12.5
16a. p = 2,839,974.1t + 152,271,417 b. yes within 5% of forecasted c. no, not within 5%
17a. about 4 b. h + 4; 5, 3, 4.1 c. 4 b/c the slope is approaching 4 as h approaches 0.
Answer Key
18. y = 2x + 0.5 19. 6x +4y = 21 20. y – 66 = 50(x – 4) 21. y – 66 = (-1/50)(x – 4)
22. y = -4x + 1.5 23. -2/27 24. S(0, 1/3) 25. R(9/2, 0) 26. 2 x 27 y 9 27. 0.75 units2
28. -1/4 29. x 4 y 3 30. 9/8 units2
Linear Models
1a. C 72q 21 b. $8.85 c. Pint matches, quart is higher d. 4.5 quarts e. cost of carton
f. cents/quart, the price of the milk itself 2a. C 4t 160 b. 200 chirps per minute
c. 70 F
d. 40 F ; temp when a cricket stops chirping e.
D: [40, 120]
44
inches
f. nothing, you cannot have a negative chirping rate 3a. s 3l 22 b. Size 14 c.
3
4a. there is constant rate of change; 30 cal
; C 30t 3630 b. 2130 calories/day
degree
c. 5130 calories/day d. 121C ; no b/c water boils at 100C 5a. constant rate of change
b. d 60t 4500 c. t 75 sec d. 4500 m e. you weren’t always at constant velocity
f. 216 km/hr 6a. y 7 x 90 b. 139 mph c. 1 uphill because it is positive
d. 90 mph max speed on 0 hill e. 12.857 ; steepness at which the car can no longer move
5
7a. V t 455 b. 555 cm3 c vary d. 955 cm3 e. 273C ; absolute zero
3
8a. the equation is in the form y mx b b. 615 ft c. 17 stories d. # of feet in each story
e. 35; it represents the extra cable needed f. {2, 3, …largest number of stories possible}
9a. t 0.33i 4478.50 b. $9761; $25,076.30 10. C = 1.25d + 2.50; $127.50
Circles and Tangents and Secants (Fix and error in #5 and 6, x + 1 should be x – 1)
3
3. -3/4 4. y 4 ( x 3) 5. no 6. 4 and -3
4
4
25
3
25
11. -3/4 12. y x
7. P(4, 3) and Q(-3, -4) 8. 3/4 9. -4/3 10. y x
3
3
4
4
13. no, because the line PQ does not pass through the center of the circle 14.(6, 14) 15. -5/12
1.
16. y 14
2. 4/3
5
( x 6)
12
17. (39.6, 0) 18.
19. (-4, -10)
Answer Key
20. (-11, 7) and (13, -3)
3
24. y 6 ( x 5)
2
21.(1, -2)
25. parallel
22. ( x 1) 2 ( y 2) 2 52
3
23. y 2 ( x 7)
2
26. -3/2
Section 2.5
1. yes; k 5
2. yes; k=8 3. yes; k=1.5 4. no 5. no 6. yes; k=4 7. no 8. yes, y = x/15
1
5
15
25
13. y x ; y
14. y 0.4 x ; y 2
9. yes 10. yes 11. yes 12. y x ; y
3
3
4
4
5
15. no, because there is a fixed fee 16. yes, the constant of variation is p 17a. m k
8
1
9
19. y 3.2 x 3 ; y 0.2048
b. yes you are speeding; 100 kph = 62.5 mph 18. y x 2 ; y
2
8
8 4
x ; y 31,104 21. it will be 4 times as large 22. (0, 0)
20. y
27
Direct Variation Word Problems
1a. c = 0.7p 1b. 35 cups 1c. 17 people 2a. p=2.2k 2b. 220 lb; 55 lb; 330 lb 2c. 75 kg
2d. # of kg in a lb 3a. p=0.43d 3b. 21.5 psi 3c. 151.163 ft 4a. r=(65/32)n 4b. $2.03
4c. 4923 cans 4d. cents per can refund 5a. t = 3d 5b. seconds per kilometer (it is the
reciprocal of the speed of sound) 5c. 3 sec; 7.5 sec; 30 sec 5d. 9.67 km 6a. V = 0.025T
6b. 10; 15; 22.5; 30 6c. at too high a temp balloon might melt; at too high a volume balloon
might burst; too low a temp balloon might become brittle and the air might liquefy
Section 2.6
1. Answers vary; about y = 4.05x – 3.3 2. Answers vary; about y = -1.125x + 2.3125
3a. negative b. y = -0.241x + 290.989 c. no; b/c the points lie in a curve not a line
4b. P(t) = 0.33t +15.63 c. P(18) = 21.57 million people d. 0.33 million people per year
e. P(-15) = 10.68 million people f. population is growing at a rate of 330,000 people per year
g. during the year 2062 5b. C(H) = 0.373H + 7.327 c. circumference inches/height inches
d. 17.035 inches e. 27.005 inches 8. 153.59 9. Given line not fit data well; linear reg line
from calculator is y = -1.098x + 39.161; square differences = 7.189; linear reg line is better fit
Piecewise Functions
1a. [-5, 4]
b. [-2.5, 2]
c. (-5, -2) d. (1, 4) e. (-2, 1)
1
2 x 5 x 2
f. y 2
2 x 1
4
10
x
1 x 4
3
3
Answer Key
( x 2) 2 3 5 x 0
1
2. y
x 1
0 x4
2
2 x 11 4 x 7
3
x 5 x 0
3a. y 2
2 x x 0
3
x x 0
b. y 2
or
2 x x 0
0 t 1
3t
3
2 7
2
t
1 t 4
3
3 3
3
x 5 x 0
y 2
4a. d (t )
c. s (t )
4t 7
0
2 x 5 x 0
5
5
5
20
7 t 10
x
3
3
3
x x 0
2 x 5 x 3
5. y
6. y
7a. 0.6 in/hr b. 0.64 in/hr
x x0
2 x 7 x 0
8.
9a. (, 1) ; [1, )
yes,(-2, 0), (0, 3), (2, -1)
b. D: (, ) ; R: (,1]
10a. k = 5 b. m = 7/6 11. a = 2
b.
12.a = -2
c. 115 miles
3
5 x 4
14. y x 2 1 4 x 2
3
2 x4
2( x 3)
c. -8
0 t 1
1 t 4
4t 7
7 t 10
c. 0 in/hr d. 1 in/hr
9.
d. -21
e. 2.5 and
0t 3
50t
3t 6
13a. d (t ) 150
60(t 6) 0 t 3
d. 8 hours 52 minutes
2( x 5)3 4 6 x 4
15. y 1
4 x 1
2
1 x 5
( x 2) 4
8
3
Answer Key
Section 2.7
1. True; because f ( x) x 2 is even so f ( x) f ( x)
2. False; order of process described is
wrong, need to flip first 3. True; moved f(x) left 2 and down 1 4. True; because 2 is positive
5a.D:[-4, 5]; R:[-3, 4] c.flip over x-axis, vertical stretch 3, left 3, down 3 d.D:[-7, 2]; R:[-15, 6]
6a. D: [-8, 8]; R: [-7, 5] c.flip over x-axis, vertical compression. by ½, right 6, up 5
d. D: [-2, 14]; R: [2.5, 8.5] 5b.
7.
6b.
8.
Section 2.8
1. dashed line through points (2, 4) and (-2, -3) shading below the line
2. dashed line through (-2, 1) and (2, 3) and shading below the line
3. solid absolute value graph with vertex at (2, 1) , opening up with a slope of 2 , shaded inside
4. dashed absolute value graph with vertex at (-2, 1.5) opening down with slope ½, shaded above
5. solid absolute value graph with vertex at (3, 3) opening down with slope 3, shaded below
2
3
3
1
1
10
8. y x 2 4 9. y x
10 y 4 x 12
6. y x 1 7. y x
3
4
2
2
3
3
2
37
11. y x
12. y 2 x 3 13a. 3m 2 f 88 13b. line with m-intercept of 88/3
5
5
and f-intercept of 44 13c. no, the score would have been 89 and the highest score was 88.
Answer Key
Sections 3.1 and 3.2
5
4 13
4. ,3 5. , 6. no solution
5
2
5
2 8
16 7
7. infinitely many solutions 8. , 9. , 10. two lines with the same slope
5 5
5 5
and different y-intercepts 11. the same line written in two different ways 12. two lines with
two different slopes 13. a = 2, b = 3 14a. p = 50, quantity = 2,100 b. 50 p 190
15. x 3, y 4 16. $5,000 at 6.5%, $15,000 at 8.5% 17a. demand increases and supply
decreases b. at $21.50, the supply and demand quantity will be 135,000
1. (3, 1)
2. (-2, -1)
Section 3.3
1.
3. (1, -4)
2.
3.
15 x 12 y 300
5a. 10 x 18 y 250
15 x 16 y 350
4.
b. x = 21, y = 3
Section 3.4
1. (2, 1, -3) 2. (3, -0.5, 2) 3. (1, -4, 2) 4. (1.5, -2, -1) 5. no solution 6. (1/3, -2/3, 1/3)
7. (4, -2, -5) 8. (7, -21, 18) 9. (0.5, 1, -1.5) 10. line x + y = 2 11. (8/15, -1/8, -4/11)
12. a = 1/2, b = -2, c = 0 so the quadratic equation is y 0.5 x 2 2 x
Systems of Equations
3b 10 s 48
1.
; b 6, s 3
7b 4s 54
4c 8t 52
2.
; c 5, t 4
3c 2t 23
f 100m 70
3.
; f 30, m 0.40 4.
f 4m 46
1.5( f c) 6
5.
; f 3.75, c 0.25
1.5( f c) 5.25
5( p w) 800
; p 130, w 30
8( p w) 800
t u 12
6.
; t 3, u 9 number = 39
10u t 15 2(10t u )
Answer Key
x y 20000
;
7. 1.5(0.08 x) 1.5(.06 y ) 2160
$12, 000 at 8%, $8, 000 at 6%
2c b 1.00
5c 5b 3.75 ;
8.
2b 2 p 2.50
b 0.50, c 0.25, p 0.75
a p s 12
h t u 18
9. 2a 3 p 16 ; a 2, p 4, s 6 10. 3t 5u 17 ; h 7, t 9, u 2 number = 792
3 p 2 s 24
2h 4u 22
$4, 000 at 5%
x y 10
x y z 10000
11. 5 x 6 y 7 z 61000 $1, 000 at 6% 12. .10 x .30 y (.15)(10)
$5, 000 at 7%
x y 3000
7.5 L of 10%, 2.5 L of 30%
Geometry Review
1
a) y x 1 b) y 3 x 19 c) (6, -1)
3
g) orthocenter is at (7, 2)
d)
40 2 10
e)
90 3 10
f) 30 units2
Section 3.5 and 3.6
42 9 20
54 22
1. 7 11
2 2. 16 102
42 30 16
1
16
5. x = -3, y = 6
6. x = -2, y = 0
7
4
1
4 8
11.
12. 1 0.5 13. 2
2
3
3 6.5
2
116
120
13 17 29
3.
4.
3
0
21 28
96
56
7. yes, 2x2 8. no 9. yes, 2x3 10. yes, 2x3
3
21
2
14 35
35 17
6 14. 35 14 15. 8 2 . 16. 30
22
13 18
8 30
5
1
22
17. x = 3, y = 4 18. x = -3, y = -1 19. yes 20. no 21. no 22. yes
23. A = 1,052 votes,B = 1,098 votes
Section 3.7
1. 22 2. -10 3. -14 4. 9 5. 9.375 6. x = 7, y = -8 7. x = -7, y = 3, z = 9
x1 y1 1
aw by az bz a b w x
8. cw dy cx dz c d y z 9. if collinear then x2 y2 1 0
x3 y3 1
adwz bcxy bcwz adxy
2 2 1
9a. 1
5
4 1 0 ; points collinear
12 1
3 8 1
9b. 4
2
1 1 7 ; points not collinear
4 1
10a. det A = 14, det B =-14 ; opposites 10b. det A = -43, det B = 43; opposites
11. both determinants are 0; If one column of a matrix is all zeros, its determinant is 0.
12. both determinants are 0; If two rows in a matrix are the same, the determinant is 0.
Answer Key
Section 3.8
5
1. 2
1
1
10
1
2.
20
3
5
3
2
1
2
1
5
1
10
1
1
5
1
2
1
4
1
16
3
3.
16
1
8
7
16
11
16
1
8
5
8
9
8
3
4
27 34 8
4.
9.5 12 3
1
1 7
10 10 9 1
2
1
2
2 1
5 5
6.
1
3 1 2 3 6
2 2 3 14 7
5 1 3 25 4
7.
x 6, y 7, z 4
1
x ; y 1
2
8. n = 9/4
6
1
11b.
1
0
2 x y 3 z 16
9. k = 4 10. 4 x 2 z 2
3 y 2 z 1
3 2
1 1
1 0
1 2
1 t 62
1 f 17
1 s 0
0 e 0
6t 3 f 2 s e 62
t f s e 17
11a.
e f t 0
f 2 s 0
11c. t = 8, f = 2, s = 1, e = 6
1
5
5. 7
13
10
Answer Key
Factoring
1. (c – 2d)(c+2d)
2. (4a – 3b)(4a +3b) 3. not factorable 4. 9(x – 2)(x + 2)
1
5. 2(3a-2b)(3a + 2b) 6. (3 x 1)(3 x 1) 7. (c 2)(c 2 2c 4)
9
2
8. (10 f 3g )(100 f 30 fg 9 g 2 ) 9. (c 2 2)(c 4 2c 2 4) 10. ( y 3 8 z 4 )( y 3 8 z 4 )
11. (10 9t 3 )(10 9t 3 ) 12. (11x 4 12 y 2 )(121x8 132 x 4 y 2 144 y 4 ) 13. (a 2b d )(a 2b d )
14. (u 9) 2 15. (3 x 8) 2 16. (5 x 4 y ) 2 17. 0.01(2c 3) 2 18. ( x 2)( x 3)
19. (t 9)(t 3) 20. ( x 8)( x 2) 21. ( f 11)( f 4) 22. ( x 1)( x 6)
23. not factorable 24. (3 y 5)( y 4) 25. (4a 3)(7 a 2) 26. (7 x 2 y )(2 x y )
27. (7 mn 8)(7 mn 9) 28. not factorable 29. (3ab 4c)(2ab c) 30. 3(2 w 3) 2
31. 3( x 9)( x 1) 32. (k 4)(k 1) 33. ( x 2)( x 2 2 x 4) 34. 2b(3m 5)(3m 1)
35. (v 4 4)(v 2 2)(v 2 2) 36. ( x 3) 2 ( x 3) 2 37. 2 p( p 10)( p 2 10 p 100)
1
38. x( x 2)( x 1) 39. 0.3(3 x 7)(2 x 5) 40. ( x a )( x b) 41. (2s 3t )(m 2n)
3
42. ( f g )( f g m) 43. (a b)(a 2 ab b 2 a b) 44. ( x 1)( x 2)( x 2)
45. ( x a y b )( x a y b ) 46. ( x n 7)( x n 4) 47. (2 x a y 4b 5 z c )(2 x a y 4b 5 z c )
48. (8 x n 2 5)(3x n 2 4) 49. (6 x 2 a 5 5 y a 3 )(7 x 2 a 5 3 y a 3 ) 50. 6( x 2 a 3 y 2b )( x 2 a 3 y 2b )
Section 4.1 and 4.2
4 6
, axis of symmetry: x = 2, vertex: (2, -3), Domain: (, ) ; Range:
2
2
8
2. y-int: 16, x-int: and , axis of symmetry: x = -1, vertex: (-1,
[3, ) , y 2( x 2) 2 3
3
3
2
25), Domain: (, ) , Range: (, 25] , y 9 x 18 x 16
1. y-int = 5, x-int =
Graph 2
2
3. f ( x) ( x 2) 2 1 , no x-intercepts 4. y 2( x 1)( x 4) 5. y 2( x 1) 2 8
3
6a. A x(400 2 x) 2 x 2 400 x b.(0, 200) c. 100 m d. 20,000 m2 7a. 400 ft b. 2.5 sec
c. 0 ft/sec d. at 7.5 sec e. 32 ft/sec 8a. R (0.45 0.05 x)(36 2 x) 8b. x = 4.5 is max
2
200 2x
100
200 2 x
b. A x
31.831
9a. d
x( x 100) c. x 50, d
Graph 1
Answer Key
10. x
8
16 3 96
3.749
33
3
3
2
Section 4.3, 4.4 and 4.5
1. x = 5 or -2 2. x = 1 or -1 3. x = -3/4 or 5/7 4. x = 7/5 or 2/3 5. x = 3, -3 or -2
6. x = 3/2, 6 or -6 7. x = -3 or 1 8. x = 10 or -1 9. x = 12 or 1 10. x = 3
11. x = 1
5
9
12. x =
2
2
9 2
13. 2
6
3
14. 3
42
6
15. 17
16. 0 or -15
17. 1 or 12 18. -1 or -7 19. 4/3 or -2 20. -3/2 or 3 21. 48 21 22. 12 42
23.
5 70
42
1
55
6 3
29 11 7
12 3 2 8 3 2 6
25.
26.
27.
28.
29. a > -5
3
3
4
3
3
30. a < 0 31. 3.092 ft 32. w = 100 yds; area = 10,000 yd2 33. x = 5 34. b 5.601; h 2.800
24.
Quadratic Models
a b c 6
1. 9a 3b c 26 y 3 x 2 2 x 5
4a 2b c 21
4a 2b c 41
2. 9a 3b c 72 y 4 x 2 11x 3
25a 5b c 48
16a 4b c 7.3
3. 36a 6b c 12.7 y 0.2 x 2 0.7 x 1.3
9a 3b c 1
16a 4b c 37
1
5. 4a 2b c 11 y x 2 7 x 1
2
0a 0b c 1
100a 10b c 40
4. 400a 20b c 160 y 0.4 x 2
25a 5b c 10
0a 0b c 0
6. a b c 7
y 2 x2 5x
36a 6b c 42
7. y 2( x 4) 2 3
8a. d 3t 2 78t 500 b. 500 km; she was 500 km from Mars’ atmosphere when she fired her
rockets c. 5 km; 20 km; pulling away d. she crashed because the distance is 0 at 11.5
minutes and 14.5 minutes
9a. y 0.4 x 2 36 x 1000
e.
b. 680 accidents
f. [0, 11.5]
c. 70 year old driver is safer d. 45 yr old
Answer Key
e. [16, 85] because f(85) = 830
10a. w 2.4t 2 24t 60 b. 60 liters c. 5 minutes
d. 0; yes
f. water does not drain at a constant rate
e.
2
11a. c 0.01r r 37 b.112 cents/km c. 40 r 60 d. no, vertex is at (50, 12) so 12 cents
is the minimum value of the function e. minimum = 50 kph and cost would be 12 cents/km
12a. y 80 x 2 120 x 610 12b. 530 m; 610 m 12c. at about -1.812 km (3.312 km would be
extraneous based on location of cannon and target) 12d. no, max height of projectile is 655 m
13a. 500 – 2x; 300 – 2x; they are linear
13b. A (500 2 x)(300 2 x) ; quadratic
2
2
13c. A(5)=142,100 yd ; A(10)=134,400 yd , A(15)=126,900 yd2 13d. about 34.169 yds
13f. (200 25 34) yds 54.226 yds
13e.
2
14a. p 0.01d 0.03d 0.45 14b. $3.85 14c. 25 inch 14d. $0.45; packaging?
14e. discriminant = -0.171; since it is negative there is no solution
14f. the slopes are not close to being equal, therefore a linear model is not a good choice
14g. vertex = (1.5, 0.4275)
Answer Key
15a.
15b. y 0.00025 x 2 0.6 x 365
15c. check the table in the calculator 15d. discriminant is negative, so there are no x-intercepts
15e. 9,006 meters 15f. 36.024 meters
Section 4.6
16 2i 16 2
i
5
5 5
1 1
10. i 11. 61 12.
3 3
1. 4 16i 2. 7 8i 3. 17 19i 4. 21 6i 5. 4 2i 6.
11 3i 11 3
47 21
54 80
39
i 8. i 9. i
17
17 17
50 50
17 17
3
83
14. 1 15. 1 16. 0 17. 1 i 18. x 2 25 0 19. x 2 6 x 17 0 20. real
13.
3
21. imaginary 22. real 23. neither 24. True; the imaginary part would have to be 0i.
25. True; ai bi (a b)i 26. True; (a bi)(a bi) a 2 b2i 2 a 2 b2
27. True; a bi a 2 b 2 ; a bi a 2 (b) 2 and a 2 b 2 a 2 (b) 2
28.
Sum is (a bi ) (c di ) (a c) (b d )i
Conjugate of sum is (a c) (b d )i
Sum of conjugates is (a bi ) (c di )
(a c) (b d )i
The conjugate sum and the sum of the conjugates are equal.
29.
Product is (a bi )(c di ) ac adi bci bdi 2
(ac bd ) (ad bc)i
The conjugate of the product is equal to the
Conjugate of product is (ac bd ) (ad bc)i
Product of conjugates is (a bi )(c di )
7.
ac adi bci bdi 2 ( ac bd ) (ad bc)i
product of the conjugates.
30. h0 0 ; which means it is not possible
Answer Key
Section 4.7
3 i 7
2 i 17
5 65
2. x 1 or 6 3. x
4. x
5. True
1. x
2
2
10
6. False; If d is a perfect square the solution will be rational. 7. False, x 2 is already a square
9
10. Mountain View to Capital City is about 382.5 miles; Capital City to
8. c 25 9. c
64
Rapid City is about 221.5 miles
Section 4.8
1. discriminant= 33; two real solutions
2. discriminant= -215;two complex solutions
1 i 19
3 3 33
2 i 17
1 4 22
2 34
3. 6 30 4.
5.
6.
7.
8.
2
8
2
9
2
2
2
9. 7 x 9 x 6 0 10. 8 x 3 x 11 0 11a. k < 0 or k > 1 11b. k = 0 or k = 1
1
1
1
1
1
1
12b. k or
12c. k or k
11c. 0 < k< 1 12a. k
3
3
3
3
3
3
2
2
2
13. discriminant = q 4 p ; two real solutions 14. discriminant = p 4q 2 ; two non-real
15b. max at 275000; p(275000)=$82.50
solutions 15a. P 0.0001x 2 55 x 150000
15c. not possible to have 10 million dollar profit because when you solve the equation
0.0001x 2 55 x 150000 10000000 you get two non-real solutions.
16a. t 15.22 sec 16b. t 2.62 sec
Section 4.9
1. x 3 or x 3
2. 6 x
6. no solution 7. 10 x
3
5
5
3
3. x 3 or x 4
8. x 2 or x 2
8
6
4. all real numbers 5. x
5
5
9. x 1
2
x 40 2
ft
5
2
2
3
2
5
100 25
0
1 x 12 x
8
2
2
3
(10 x)( x 2)
11. 2 < x < 8.0424 by solving the inequality
10. no solution
Answer Key
Section 9.1
1. (7, 0)
2. (9, 3)
3. (2 xm x1 , 2 ym y1 )
7 7 5 3 13 5
4. , ; , ; ,
4 4 2 2 4 4
3 1 3
3 9
5. , ; 1, ; ,
2 2 4
2 4
3x x 3 y y2 x1 x2 y1 y2 x1 3 x2 y1 3 y2
6. 1 2 , 1
,
,
;
;
4 2
2 4
4
4
x x y y
d 2 1 2 1
3 3
7a.. distance between endpoint and trisection point is
1
2
2
x2 x1 y2 y1
3
5
4
x 2 x2 y1 2 y2
4
2
9. , 2 ; , 1
7b. 2nd trisection point is 1
,
8. 2, ; 3,
3
3
3
3
3
3
3m 3
15
3 5 10b.
10a.
10d. The graph approaches y = 3. 11a. y x
5
m2 1
10
5
12. y 4 or 8 13. x 11 or 13 14. y 4 609
11b. y 1 11c. y ( x 4)
3
2
17a. 50t 17b. 250 miles
15. 17, 20 and 29 16. 2 triangles possible, vertices 2 3, 2
2
2
17c. after 2 hours 14d. 50 miles per hour
Parabolas - Sections 9.2 and 9.6
1. y 2 16 x 2. x 2 12 y 3. x 2 16 y 4. x 2 14( y 0.5) 5. ( x 2) 2 8( y 3)
6. ( y 2) 2 8( x 4) 7. ( x 3) 2 4( y 3) 8. ( y 2) 2 16( x 1) 9. y 2 4.5 x
10. V (0, 0); F (0, 1); d : y 1 11. V (0, 0); F (0.75, 0); d : x 0.75
12. V (2, 3); F (2, 4); d : y 2 13. V (5, 3); F (5.5, 3); d : x 4.5
14. V (0, 2); F (1, 2); d : x 1 15. V (3, 2); F (3, 1); d : y 3
49
1
193
23
17d.
16. ( y 1) 2 4( x 3) 17a. ( x 3) 2 ( y 4) 17b. 3,1 17c. y 6
6
24
24
24
193
17e.
17f. They are the same
18. 5 feet 19. 2304 7238.23 ft 2 20. y 4 x 2
24
21a. x 2 640 y 21b. 8 feet 22a. x t 2 22b.see table below 22c. yes
-3 -2 -1
0
1
2
3
t
9
4
1
0
1
4
9
x
-3 -2 -1
0
1
2
3
y
2
2
23a. y 0.5(t 1) 1 0.5t t 0.5 23b. see table below 23c. yes
-3 -2 -1
0
1
2
3
t
-3 -2 -1
0
1
2
3
x
7 3.5 1
-0.5 -1 -0.5 1
y
Answer Key
Circles - Section 9.3 and 9.6
1. ( x 2) 2 ( y 2) 2 5 2. ( x 3) 2 ( y 1) 2 9
3. ( x 3) 2 ( y 1) 2 49
2
3
( x 1)
5b. y 3 ( x 5)
4
4
6a. C (2, 1); r 5
6b. x int . 2 2 6; y int . 1 21 6c. D :[7,3]; R :[4, 6]
16 7
6d. not a function 7. (0, 3) and , 8a. A( x) (4 x) 16 x 2 8b. x 2
5
5
9.
10.
11.
12.
5a. y 2 2
4. ( x 4) 2 ( y 5) 2 25
13a. length = 2x; width = 2 y 2 36 x 2 13b. A 4 x 36 x 2 13c. 72 units2
14a. slope = -2/3; equation of line: y = (3/2)x 14b. (6, 9) 14c. x 2 y 2 117
15a. y
5
169
12 169 x 2
x
15b. mx
x5
12
12
2
15d. mx approaches a value of about 0.4; this is
2
2
41 1300
almost equal to the slope 16. x y
7
7
49
Ellipses - Section 9.4 and 9.6
vertices: (0, 5/2);
co-vertices: ( 4/3, 0);
foci: (0, 161 )
6
vertices: ( 3, 0);
co-vertices: (0, 1);
foci: (2 2, 0)
1.
2.
vertices: ( 10, 0)
co-vertices: (0, 2 2) ;
foci: ( 2, 0)
3.
2
2
x
y
1
9 49
x2 y2
1
9.
16 36
5.
4.
x
y
x2 y2
6.
1
7.
1
169 31
36 24
x2
4x2 4 y2
4 y2
10.
1
or
1
9 2 1 2
9h 2 h 2
h
h
4
4
2
vertices: ( 9/2, 0);
co-vertices: (0, 5/3);
foci: ( 629 , 0)
6
2
8.
x2 y 2
1
169 81
Answer Key
standard
equation
x–radius
y–radius
focal
radius
center
vertices
foci
#11
x2 y2
1
25 16
5
4
3
#12
( x 3) 2 ( y 4) 2
1
4
9
2
3
5
(0, 0)
(5, 0), (–5, 0),
(0, 4), (0, –4)
(3, 0), (–3, 0)
(3, 4)
(5, 4), (1, 4),
(3, 1), (3, 7)
(3, 4 5 )
none
none
#17
( x 1)
( y 2)
1
16
9
( x 5) 2 ( y 2) 2
1
4
9
( x 6) 2 ( y 3) 2
1
36
4
4
2
2 3
4
3
2
3
6
2
4 2
(–3, 0)
(1, 0), (–7, 0),
(–3, 2), (–3, –2)
(–3 2 3 , 0)
(1, 0), (–7, 0)
(–1, 2)
(–5, 2), (3, 2),
(–1, 5), (–1, –1)
( 1 7 , 2)
(–5, 2)
(–3, 2), (–7, 2),
(–5, 5), (–5, –1)
5,2 5
34 5
,0
3
8 3 15
0,
4
15 2 5
,0
3
none
(5, 0), (–5, 0)
(0, 4), (0, –4)
#15
#16
standard
equation
( x 3)
y
1
16
4
x–radius
y–radius
focal
radius
center
vertices
x–int.
y–int
11.
2
(0,
2
2
2
(0, 0)
(10, 0), (–10, 0),
(0, 5), (0, –5)
( 5 3 , 0)
(10, 0) (–10, 0)
(0, 5) (0, –5)
#14
x2 y2
1
25 4
5
2
21
(0, 0)
(5, 0), (–5, 0),
(0, 2), (0, –2)
( 21 , 0)
(5, 0) (–5, 0)
(0, 2) (0, –2)
#18
x–int.
y–int.
foci
#13
x2
y2
1
100 25
10
5
5 3
7
7
4
)
12.
5
13.
(–6, –3)
(0, –3), (–12, –3),
(–6, –5), (–6, –1)
( 6 4 2 , –3)
none
(0, –3)
Answer Key
14.
15.
17.
18.
16.
.
19.
( x 4) 2 ( y 12) 2
1
5
9
20.
x2 y2
1
16 12
21.
( x 5) 2 ( y 7) 2
1
4
25
r2 94
22.
x2 y2
1
8
9
23.
25a. A a(20 a)
a
7
8
( x 1) 2 ( y 2) 2
1
25
9
25b.
x2 y 2
1
196 36
9
10
24.
r 2 36
r 2 36
r6
25c. see table below 26. 32
11
12
13
A 285.88 301.59 311.02 314.16 311.02 301.59 285.88
25c. Because the maximum area occurs when a 10 and a b 20, b is also 10. So, the
x2
y2
1 x 2 y 2 100 which is the equation of a circle.
equation becomes
100 100
Hyperbolas - Lesson 9.5 and 9.6
3
1. C (0, 0); Tran. axis : x axis;V (5, 0); F ( 34, 0); Asy.: y x
5
2. C (0, 0); Tran. axis : y axis;V (0, 4); F (0, 2 5); Asy.: y 2 x
3. C (0, 0); Tran. axis : y axis;V (0, 3); F (0, 10); Asy.: y 3 x
4. C (0, 0); Tran. axis : x axis;V ( 2, 0); F ( 6, 0); Asy.: y 2 x
Answer Key
O
O
1.
2.
O
O
3.
4.
y
x
x
y
y
x
x2 y 2
( y 4) 2 ( x 3) 2
1 6.
1 7.
1 8.
1 9.
1
16 20
4 5
36 9
9 16
4
12
( y 4) 2
( y 7) 2
( x 4) 2
10. ( x 1) 2
1 11. ( x 5) 2
1 12. ( y 3) 2
1
8
3
8
( x 1) 2 ( y 1) 2
( y 1) 2 9( x 1) 2
13.
1 14.
1
4
9
4
16
5
15. C (3,1); Tran. axis : y 1; V (7,1), (1,1); F (3 41,1); Asy.: y 1 ( x 3)
4
2
2
2
2
2
2
5.
9
16. C (1, 2); Tran. axis : x 1; V (1, 11), (1, 7); F (1, 2 106); Asy.: y 2 ( x 1)
5
2
( y 2)
17. ( x 1) 2
1; C (1, 2); Tran. axis : y 2; V (2, 2), (0, 2); F (1 3, 2);
2
( y 1) 2
( x 2) 2 1; C (2,1); Tran. axis : x 2;
Asy.: y 2 2( x 1) 18.
4
V (2,3), (2, 1); F (2,1 5); Asy.: y 1 2( x 2)
19. a) Q(5, 0), R(5, 0) b) PR PQ 2a 2 4 8, then PR PQ 8 5 8 13
Answer Key
20
21.
22.
.
24.
25.
26.
#20
standard
equation
x
2
25
#21
y
2
1
16
x
23.
#22
2
9
y
2
1
16
( y 3)
1
25
2
x–radius
y–radius
focal radius
center
vertices
5
4
foci
( 41 , 0)
horiz.,
length = 10
vert., length
=8
vert., length =
8
horiz., length
=6
4
y x
5
4
y x
3
x–
intercepts
(5, 0), (–5,
0)
none
5
x 2
6
6 34
, 0
2
5
y–
intercepts
none
(0, 4), (0, –4)
none
transverse
axis
conjugate
axis
asymptotes
41
(0, 0)
( 5 , 0)
3
4
5
(0, 0)
( x 2)
36
#23
2
0, 4
0, 5
6
5
61
(2, 3)
(8, 3) and (–4, 3)
(2 61 , 0)
horiz., length =
12
vert. , length = 10
y 3
( x 3)2
36
( y 6) 2
1
64
6
8
10
(3, –6)
(3, 2) and (3, –14)
(3, 4) and (3, –16)
vert., length = 16
horiz. , length = 12
y6
4
x 3
3
none
0, 6 3 10
Answer Key
#24
standard
equation
( x 1)
16
x–radius
y–radius
focal radius
center
vertices
foci
transverse
axis
conjugate
axis
asymptotes
4
1
x–
intercepts
y–
intercepts
27.
x2
9
( y 4)
1
1
2
( y 2)
1
36
2
x2 y 2
1
144 25
(1, 4)
(5, 4) and (–3, 4)
( 5 17 , 4)
horiz., length = 8
(3, –2)
(7, –2) and (–1, –2)
( 3 2 13 , 4)
horiz., length = 8
12
5
13
(0, 0)
(12, 0) and (–12, 0)
(13, 0) and (–13, 0)
horiz., length = 24
vert. , length = 2
vert. , length = 122
vert., length = 10
y
(2, 0) and (0, 0)
3
x 3
2
4 10
, 0
3
3
(12, 0) and (–12, 0)
none
none
none
2 13
y6
( x 3)
16
#26
2
4
6
17
y2
1
7
( x 2)2
31.
36
#25
2
28.
1
x 1
4
x2
16
y2
1
4
y2
29.
( x 2) 2
39
( y 1)2
1
25
5
x
12
30.
( x 3)2
9
( y 3)2
1
16
( y 5)2
1
64
Conics Challenges and Extras
1. y 2 4h( x h) , where h is the x-coordinate of the vertex and the focal radius
( x 6) 2 ( y 2) 2
2.
1 3. if 8, then the graph would be the point (1, – 2); if greater than 8,
9
7
there would be no graph since the equation would have no solution
25
5. 0, 6. length = 4p
4. 2 x (3 3) y 12
3
Answer Key
7a. Endpoints of latera recta:
9
9
9
9
, 7 , , 7 ; , 7 ; , 7
4
4
4
4
7b. Endpoints of latera recta:
4
4
4
4
, 5 , , 5 ; , 5 ; , 5
3
3
3
3
7c. Endpoints of latera recta:
3 5
3 5
3 5
3 5
2,
, 2,
; 2,
; 2,
5
5
5
5
x2 y 2
x2
1
to
obtain
the
positive
solution
y
b
1
.
a 2 b2
a2
This represents half the length of the latus rectum. Because the focus (c, 0) and the latus rectum
directly above the focus have the same x–coordinate, substitute c for x in the equation for y to get
7d. First solve for y in the equation
y b 1
c2
a2 c2
.
Add
the
expressions
underneath
the
radical
to
get
. Then notice
y
b
a2
a2
that for an ellipse, a2 c2 b2. So, the equation for y simplifies to y b
b2
b2
. Therefore,
a2
a
2b 2
c
8. The eccentricity is e . Solve this equation for c
a
a
2 2
2
and then square each side to obtain e a c . You know that for an ellipse, c 2 a 2 b 2 .
Substitute e2a2 for c2 and solve for b2 to obtain b 2 a 2 (1 e 2 ) . Now substitute this expression for
the length of the latus rectum is
x2
y2
b in the equation for an ellipse to obtain 2 2
1 . As e approaches 0 and a remains
a
a (1 e 2 )
fixed, the ellipse approaches the shape of a circle. 9a. apogee: about 405,508 km
9b. perigee: about 363,292 km 9c. The apogee A is A a c. The perigee P is P a c.
A P a c ( a c ) 2c c
Substitute these values for A and P into e
A P
acac
2a a
Lesson 9.7
1. (5, 9), (3, 7) 2. (3, 1), (1, 3) 3. no points of intersection 4. ( 10, 2), ( 10, 2)
2
9 4 6 9 4 6
5. ,
, ,
6. (1.5, 1.5), (6, 69) 7. (), (), (), () 8. about 53.2 mi
5 5 5
5
9. yes; about 2.39 ft by 2.39 ft by 6.13 ft; or 4.35 ft by 4.35 ft by 1.85 ft
10. b 4 5
0
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