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Answer Key Algebra I Regents Course Workbook
2020-21 Edition Donny Brusca
ISBN 978-1-67816-250-4 © 2020 Course Workbooks. All rights reserved. www.CourseWorkbooks.com
3
Unit Page Chapter 1. Equations and Inequalities ............................................................. 4 Chapter 2. Verbal Problems ........................................................................... 12 Chapter 3. Linear Graphs................................................................................ 17 Chapter 4. Linear Systems .............................................................................. 24 Chapter 5. Polynomials .................................................................................. 32 Chapter 6. Irrational Numbers ....................................................................... 35 Chapter 7. Univariate Data ............................................................................. 39 Chapter 8. Bivariate Data ............................................................................... 44 Chapter 9. Introduction to Functions ............................................................. 49 Chapter 10. Functions as Models ..................................................................... 52 Chapter 11. Exponential Functions .................................................................. 56 Chapter 12. Sequences ..................................................................................... 60 Chapter 13. Factoring ....................................................................................... 62 Chapter 14. Quadratic Functions ..................................................................... 65 Chapter 15. Parabolas ...................................................................................... 74 Chapter 16. Quadratic-Linear Systems ............................................................. 81 Chapter 17. Cubic and Radical Functions ......................................................... 85 Chapter 18. Transformations of Functions ....................................................... 87 Chapter 19. Discontinuous Functions .............................................................. 89
4
Chapter 1. Equations and Inequalities
1.1 Properties of Real Numbers 1. (3) 2. (1) 3. associative property of multiplication 4. distributive property 5. (1) 6. (3) 7. 23− 8. 32 9. a b− + 10. –ab 11. (2 1) (1 2)÷ ≠ ÷ or any similar counterexample. 12. No. For example, when we subtract the whole number 5 from the whole number 2, the result is –3, which is not a whole number. 13. commutative property of multiplication 14. associative property of addition 15. No. For example, when we divide the integer 1 by the integer 2, the result is 12 , which is not an integer. 16. If a
b and c
d are rational numbers and
a, b, c, and d are non-zero integers, then a c a d adb d b c bc
÷ = × = . Since the set of integers is closed under multiplication, ad and bc are integers, so adbc
is rational. 17. 5 25x + 18. 4 16b − 19. 2 2x− + 20. 3 3a b− + 21. 1 y− − 22. 1a + 23. ( )rs rt r s t+ = + 24. 2 10 2( 5)x x+ = + 1.2 Solve Linear Equations in One Variable 1. 3( 2) 183 6 183 248
mm
mm
− =− =
==
2. 4 123 124n nnn
− = −= −= −
5
3.
( )2 4 7 32 8 7 32 1 32 42xx
xxx
− + =
− + =− =
==
4. ( )0.2 6 2.80.2 1.2 2.80.2 420
nn
nn
− =
− ===
5. 5 ( 1)5 15 2 14 22
y yy y
yy
y
− = − + −− = − − −− = − −− = −
=
6. 15 3(3 4) 615 9 12 66 12 66 183
x xx x
xxx
− + =− − =
− ===
7. 3 8 58 24x x
xx
+ ===
8. 3 2 5 93 3 912 34
g ggg
g
+ = −= −==
9. 8 2 4 104 2 104 123
p pp
pp
+ = −+ = −
= −= −
10. 5 1 2 203 1 203 217
p pp
pp
− = +− =
==
11. 0.06 200 0.03 3500.03 200 3500.03 1505000
y yy
yy
+ = ++ =
==
12. 5 2 4 75 2 72 126
x xxxx
− = − −+ = −
= −= −
13. 5(2 7) 15 1010 35 15 1035 5 1025 55
x xx x
xx
x
− = −− = −− = −− =
− =
14. 5( 2) 2(10 )5 10 20 23 10 203 3010
x xx xx
xx
− = +− = +− =
==
6
15.
( ) ( )2 4 4 2 12 8 8 48 6 412 62x xx x
xx
x
− = +
− = +− = +
− =− =
16.
( ) ( )3 1 5 12 6 73 3 5 12 6 73 2 19 63 4 194 164
x x xx x x
x xxxx
+ − = − −
+ − = − +− = −+ =
==
17. 4( 3) 5(2 6)4 12 10 3012 14 3042 143
y yy y
yy
y
− − = −− + = −
= −==
18.
3( 2) 2( 1) 5( 4)3 6 2 2 5 208 5 208 4 2012 43x x x
x x xx x
xx
x
− − + = −− − − = −
− = −− = −
==
1.3 Solve Linear Inequalities in One Variable 1. 4x ≤ 2. 1x > − 3. 2 5 112 168x
xx
− ≤≤≤
4. 6 1 256 244yyy
− + >− >
< −
5. 4 2( 3)4 2 62 21 1
rrr
rr
− > −− > −
>><
6.
43 43 43( 3) 124 1284 246
x
x
xxx
− − ≥
− + ≥
− ≥− ≥
≤ −
7.
6 17 8 2517 14 2542 143 3
x xxx
xx
− − ≥ +− ≥ +− ≥
− ≥≤ −
8. 5 35 155 204x
xx
− + <− < −
>
7
9. 2 5 32 84
xxx
− <<<
Graph (1) 10.
( )3 2 1 4 76 3 4 72 3 72 105m mm mm
mm
− ≤ +
− ≤ +− ≤
≤≤
11.
4(2 6) 3 48 24 3 47 24 3 424 10 420 102 2
m m mm m mm m
mm
mm
− − + > +− + + > +− + > +
> +>
><
12.
5( 1) 115 5 115 4 1116 44 4p p
p pp
pp
p
− + ≥ − +− − ≥ − +− ≥ +− ≥− ≥
≤ −
1.4 Compound Inequalities 1. 4 2x− ≤ < 2. 2 3x− < ≤ 3. −3 < 𝑥 ≤ 4 4. 1x < − or 4x > 5.
6. 3 2 1 92 2 81 4xx
x
≤ + <≤ <≤ <
7. 2 3 4 106 3 62 2x
xx
− < + ≤− < ≤− < ≤
There are 2 positive integer solutions, {1, 2}
8. (3)
8
1.5 Solve Equations with Fractions 1.
1 116 4 21 116 16 1616 4 24 84x
x
xx
+ =
+ =
+ ==
2.
22 66 6 6(2)2 63 124 123
x x
x x
x xxx
+ =
+ =
+ ===
3. ( )
2 53 626 6 6 53 64 305 306
x x
x x
x xxx
+ =
+ =
+ ===
4.
3 2 45 53 25 5 5(4)5 53 2 203 186
x
x
xxx
+ =
+ =
+ ===
5.
3 52 64 43 54 4(2) 4 4( 6)4 43 8 5 248 2 2432 216
x x
x x
x xxx
x
+ = −
+ = + −
+ = −= −==
6.
2 14 2
2 1 53 2 62 1 56 6 63 2 64 3 54 2x
x
xxx
+ =
+ =
+ ==
= =
7.
3 1 54 33 112 12 12(5)4 39 4 605 6012
x x
x x
x xxx
= +
= +
= +==
8. ( )
( )
13 216 6 63 22 3 1 62 3 3 65 3 63
x x x
x x x
x x xx x x
x xx
++ =
+ + =
+ + =
+ + =+ =
=
9
9.
2 1 7 25 3 152 1 7 215 15 155 3 156 5 7 25 27
x x
x x
x xxx
−+ =
− + =
+ = −= −=
10.
1 2 15 37 3 211 2 15 321 21 217 3 213 14 15 33 36
x x
x x
x xxx
−+ =
− + =
+ = −= −=
11. [ ]
3( 3) 9434 ( 3) 4 943( 3) 363 9 363 279
x
x
xx
xx
+ =
+ = + =+ =
==
12.
( )
( ) [ ]
( )
3 2 4535 2 5 453 2 5 203 6 5 206 2 2026 213
x x
x x
x xx x
xx
x
+ = −
+ = − + = −
+ = −= −==
13.
1 1(18 5 ) (6 4 )2 31 16 (18 5 ) 6 (6 4 )2 33(18 5 ) 2(6 4 )54 15 12 854 12 742 76
x x
x x
x xx x
xx
x
− = −
− = − − = −− = −
= +==
14. [ ]
2 12 133 22 13 2 3 133 212 2 3924 1 394 4010
x
x
x
xxx
− =
− = − =
− ===
15.
( ) ( )
( )
3 1 2 35 23 3 2 65 23 310 10 10 2 65 22 15 15 20 6017 15 20 6045 315
mm m
m m m
m m m
m m mm m
mm
−+ = −
−+ = −
− + = −
+ − = −− = −
==
10
1.6 Solve Literal Equations and Inequalities 1. 2 2 162 2 168m pp mp m
+ == − += − +
2. 2 22bx Kbx K
Kxb
− == +
+=
3. 222
c m dc d mc d m
= +− =− =
4. 3 33bx a cbx a c
a cxb
− == +
+=
5. V lwh
V wlh
=
= 6.
222bhA
A bhA h
b
=
=
=
7. 5 055
abxabx
xab
− ==
=
8. 2 22 222y w x
w x yx yw
+ == −
−=
9.
2222
x tsr
sr x tsr t xsr t x
+=
= +− =− =
10. 1333
V Bh
V BhV hB
=
=
=
11. 2
22
1222v at
v atv a
t
=
=
=
12. ( )
ey k tn
ey t kney n t key nt nk
nt nkye
+ =
= −
= −
= −−=
11
13. 3(3 ) 3x ax b
x a bbx
a
− =− =
=−
14. ( )bc ac abc b a ab
abcb a
+ =+ =
=+
15. ( )
333
k am mxk m a x
k ma x
= += +
=+
16. 2 12 1(2 ) 1 12
ax bxax bx
x a b
xa b
= − ++ =+ =
=+
17.
( )
3 73 744 4ax bx
ax bxax bx
x a b
xa b
+ = −+ + =
+ =+ =
=+
18. ( )
2 22
11
z y x xy
z y x y
z y xy
+ = +
+ = +
+ =+
12
Chapter 2. Verbal Problems
2.1 Translate Expressions 1. (3) 2. (4) 3. 7 5x − 4. 2( 8)x − 5. 33 g− 6. 20 2d− 7. 4 10x + 8. 12n 9. 𝑥𝑑 10. 𝑑 − 2ℎ 11. 5 𝑥 + 4 , or 5𝑥 + 20 12. 3𝑥 − 4 13. 3𝑑 − 1200 14. 280 + 0.05𝑛 15. 1 2 34 6y y y yy+ + + + + +
+ 16. 3 5 73 15x x xx+ + + + ++
17. 2 43 6x x xx+ + + ++
18. 𝑥(𝑥 + 1) 𝑥 + 𝑥 19. t = Tommy's age 4t − = Donny's age 4 7 3t t− + = + = Camille's age Sum is 3 1t − 20. h = horse's lifespan 70h + = stork's lifespan 4( 70) 4 280h h+ = + = whale's lifespan Sum is 6 350h+ 21. a = number of cookies eaten by Alice 𝑎 + 4 = number of cookies eaten by Carl 2(𝑎 + 4) cookies were eaten by Bob 22. x bags of chips 3𝑥 bags of pretzels 3𝑥 − 2 bags of nachos 𝑥 + 3𝑥 + 3𝑥 − 2 7𝑥 − 2
2.2 Translate Equations 1. 9ℎ + 60 = 375 2. 3(𝑥 + 4) = 5𝑥 − 2 3. 𝑙 − 4 4. 𝑥(2𝑥 − 3) = 43 5. 2(3 2) 22x + = 6. 0.30( 4) 0.50 3.60n n+ + = 7. 0.05𝑛 + 0.10(𝑛 + 6) = 1.35 8. 0.10(72 − 𝑞)+ 0.25𝑞 = 14.70 9. 𝑥(𝑥 + 1) = 20 10. 𝑥 + 𝑥 + 2 + 𝑥 + 4 = −3
13
2.3 Translate Inequalities 1. 3𝑥 − 8 > 15 2. ℎ ≥ 48 3. ( 9) 144b b+ + < 4. 3 120h h+ ≤ 5. 𝑥 + 2𝑥 ≥ 90 6. 0.75 1.25 25a b+ ≤ 7. 30 + 2𝑤 ≤ 50 8. 𝑤(2𝑤 − 3) ≤ 30 9. 155 ≤ ℎ ≤ 190 10. 1500 1800c≤ < 11. 0.75(200) + 1.25𝑥 ≥ 250 2.4 Linear Model in Two Variables 1. the number of hours of tutoring 2. the number of miles driven 3. 80 75c x= + 4. ( ) 5 100P y y= + 5. 20 0.50m g= − 6. ( ) 30,000 2000h m m= − 7. 2( 1) 5c n= − + 8. ( ) 30( 40) 800w h h= − + 2.5 Word Problems – Linear Equations 1. 5 192 5 192 147
a aa
aa
+ + =+ =
==
5 12a + =
Jamie is 12 years old.
2. 2 5613 561187c ccc
+ ===
2 374c = There are 187 crickets and 374 grasshoppers. 3. 3 204 205c c
cc
+ ===
3 15c = There were 15 robins.
4. 2 4 163 4 163 124
f ff
ff
+ + =+ =
==
2 4 12f + =
There are 4 freshmen and 12 sophomores. 5.
2 3 152 126x
xx
+ ===
He bought 6 pizzas last year.
6. 2(2 ) 2 454 2 456 457.50
x xx x
xx
+ =+ =
==
Each CD costs $7.50.
14
7. 4 8 284 369mmm
− ===
Minnie owns 9 video discs. 8. 7x deer, 3x foxes 3 21070x
x==
So, 7 490x = deer 9. ( )2 3 423 3 423 3913
b bb
bb
+ + =
+ ===
There are 13 black marbles, so there are 42 − 13 = 29 red marbles.
10. 3 5 173 124x
xx
+ ===
4 rides 11.
2.25 3.50( 1) 44.252.25 3.50 3.50 44.253.50 1.25 44.253.50 45.5013x
xx
xx
+ − =+ − =
− ===
13 miles
12. ( )3 4 483 522 5226
x xx xxx
= + +
= +==
26 years old
13. 7x boys, 10x girls 7 10 35717 35721x xxx
+ ===
So, 7 147x = boys 14.
4( 100) 12 30564 400 12 305616 400 305616 2656166100 266m mm m
mmm
m
+ + =+ + =
+ ===
+ =
There were 266 balcony tickets sold. 15.
0.10(3 ) 0.25( 4) 0.05 4.600.3 0.25 1 0.05 4.600.6 1 4.600.6 3.606n n n
n n nn
nn
+ + + =+ + + =
+ ===
6 nickels, 18 dimes, 10 quarters
16. 6.50 9.00(150 ) 1180.006.5 1350 9 11802.5 1350 11802.5 17068
s ss s
sss
+ − =+ − =
− + =− = −
=
68 small and 82 large
2.6 Word Problems – Inequalities 1. 2 5 232 2814nnn
− >>>
Smallest integer is 15. 2. 5 5511x
x<<
Largest integer is 10
15
3. 7 608 607.5n nnn
+ ≤≤≤
Largest two integers are 7 and 49. 4. 375 155 900155 5253.387...
www
+ ≥≥≥
He needs to work 4 weeks. 5. 5.95 21536.1344...hh
≥≥
He needs to work 37 hours. 6. 6 3 303 3010n n
nn
> +>>
They need to make 11 toys. 7. 13.95 0.49 50.000.49 36.0573.5714...xxx
+ ≤≤≤
She can buy 73 songs. 8. 19.00 0.07 29.500.07 10.50150
xxx
+ ≤≤≤
She can use 150 minutes. 9. Convert $1.50 per 30 mins. to $3/hr. 3( 1) 5 12.503 3 5 12.503 2 12.503 10.503.5
hh
hhh
− + ≤− + ≤
+ ≤≤≤
She can park 3.5 hours.
10.
132 (150 0.50 ) 5002 150 0.50 5001.5 150 5001.5 650433
n nn n
nnn
− + ≥− − ≥
− ≥≥≥
They must sell 434 programs.
2.7 Conversions 1. 20 in 2.541 cmin
× 50.8 51cm= ≈ 2. 8900 ft 15280mift
× 1.7mi≈ 3. 1680 oz 1 lb× 16 oz
15baglbs
× 21bags= 4. 0.75 tsp 13Tbsntsp
× 5 1.25 Tbsn× = 5. 2.625 in 2.54 cm× 1 in
101 mmcm
×2.625 2.54 10 66.675 67mm mm mm
=
× × = ≈ 6.
6 furlongs 1 mile× 8 furlongs1.611 km
mile×6 1.61 1.2075 1.218 km km km
=
× = ≈
16
7. 48 .in $3.751 .yd×
1 .yd× 3 .ft
1 .ft× 12 .in
$5.00= 8. 60 ft 12 in× 1 ft
2.54 cm× 1 in1100m
cm×60 12 2.54 18.288 18.3100 m m m
=
× × = ≈
9. 1501.5 mmin
60 min× 90001 1.56000m
hr hrm per hr
=
= 10. 3441 m
s60 s× 1 min
60 min× 11,238,400 hrm per hr=
11. 43 mi1 g
1.611 kmmi
×1 g
×3.7943 1.61 18.33.79l
km kmper literl
=
× ≈
12. $1.502 l3.79 l× $1.50 3.791 2$2.84 g g
per g
×= ≈ The 1-gallon bottle is the better buy. 13.
8000 mi1 yr17601 yds
mi×
1 yr× 36514,080,000 38,575365
days
yds yds per daydays
=
≈
14. 30 mi1 hr
52801 ftmi
× 1 hr×60 mins1 min× 6030 5280 44 /60 60 s
ft ft ss
=
× =×
15. 100 yd11 s
3 ft×1 yd
15280mift
× 60 s×1 min60 mins× 100 3 60 60 18.61 11 5280 mi mph
hr hr× × ×= ≈
×
17
Chapter 3. Linear Graphs
3.1 Determine Whether a Point is on a Line 1. Yes. 7 3(3) 2 ?7 9 2 ?7 7= −= −=
2. No. 129 (4) 5 ?9 2 5 ?9 7
= += +≠
3. Yes. 0 4(0) ?0 0=
= 4. Yes. 2( 4) 3( 2) 2 ?8 6 2 ?2 2− − − = −
− + = −− = −
5. No. 4( 4) (3) 13 ?16 3 13 ?19 13− − = −
− − = −− ≠ −
6. Yes. 5( 2) 2( 4) 2 ?10 8 2 ?2 2− − − = −− + = −
− = −
7. No. 2( 5) ( 1) 11 ?10 1 11 ?9 11− − − = −− + = −
− ≠ − 8. Yes. 4(3) 3( 2) 18 ?12 6 18 ?12 12= − +
= − +=
9. 2 6( 2) 42 12 42 168
xx
xx
+ − =− =
==
10. 4 (3) 94 123k
kk
+ = −= −= −
11. ( )2 3 26 28
kk
k
− − = −
+ = −= −
12. ( )2 5 910 91
kkk
+ =
+ == −
3.2 Lines Parallel to Axes 1. (1) 2. (2) 3. 9x = 4. 1y = 5. 0x = 6. 0y =
18
7. (5,0) 8.
9.
10. 4 13yy
− = −=
3.3 Find Intercepts 1. x-intercept: 2 y-intercept: –4 2. x-intercept: –2 y-intercept: –3 3. x-intercept: y-intercept: 3(0) 2 62 63x
xx
+ ===
3 2(0) 63 62yyy
+ ===
4. x-intercept: y-intercept: 3 4(0) 123 124x
xx
− ===
3(0) 4 124 123yyy
− =− =
= −
5. x-intercept: y-intercept: 52
(0) 2 55 2xx
x
= − +− = −
= 2(0) 55y
y= − +=
6. x-intercept: y-intercept:
599 6(0) 5 09 5 09 5x
xxx
− + =+ =
= −= −
569(0) 6 5 06 5 06 5y
yyy
− + =− + =
− = −=
19
3.4 Find Slope Given Two Points 1. 4 18 2m = = 2. 2 16 3m = − = − 3. 43m = 4. 23m = − 5. 3 13m = − = − 6. 5 15m = = 7. 13 3 10 55 1 4 2m −= = =
− 8. 8 ( 6) 14 71 3 2m − −= = = −
− −
9. 3 5 8 20 4 4m − − −= = =− −
10. 2 ( 2) 0 02 ( 4) 6m − − −= = =− −
11. 3 5 27 2 5m −= = −
− 12. 2 5 3 32 3 5 5m − −= = =
− − −
3.5 Find Slope Given an Equation 1. Slope is 25 . 2. 3 13 1y x
y x− =
= + Slope is 3. 3. 522 5 42y xy x
= += +
Slope is 52 . 4. 5 10 155 10 152 3y x
y xy x
− = −= −= −
Slope is 2. 5. 434 3 123 4 124
x yy xy x
+ == − +
= − +
Slope is 43−
6. 122 42y xy x
= −
= −
Slope is 12
20
7. 323 2 122 3 126x y
y xy x
− =− = − +
= −
Slope is 32 . 8. 34
3 4 16 03 4 164 3 164x y
x yy xy x
− − =− =− = − +
= −
Slope is 34 . 9. 2 2y x= − + 10. 12y x= 11. (1) Same slope of –3 12. 23
2 3 93 2 93x y
y xy x
− =− = − +
= −
Choice (1) 13. The first equation: 2 2 62 2 63y x
y xy x
+ == − += − +
14. The first equation: 523 6
4 6 56 4 5x yy xy x
+ == − += − +
523 33 2 5y xy x
− = += − −
3.6 Graph Linear Equations 1.
2.
21
3. 3 43 4y xy x
− == +
4. 122 2 22 2 1y x x
y xy x
+ = −= − −= − −
5. (4) 2 4 102 5y xy x
= − −= − −
3.7 Write an Equation Given a Point and Slope 1. 4 2(1)4 22
y mx bb
bb
= += += +=
2 2y x= + 2. 5 5( 6)5 3035
y mx bb
bb
= += − += − +=
5 35y x= + 3. 132 ( 3)2 13
y mx bb
bb
= += − += − +=
13 3y x= + 4. 343 (8)3 69
y mx bb
bb
= +− = +− = +− =
34 9y x= − 5. 344 ( 8)4 610
y mx bb
bb
= +
= − +
= − +=
34 10y x= + 6. ( )437 37 43
y mx bb
bb
= +
− = − +
− = − +− =
𝑦 = − 𝑥 − 3
22
3.8 Write an Equation Given Two Points 1. 6 2 4 15 1 4m −= = =−
2 1(1)2 11
y mx bb
bb
= += += +=
1y x= + 2. 4 ( 1) 5 53 2 1m − −= = =
−
1 5(2)1 1011y mx b
bb
b
= +− = +− = +
− =
5 11y x= − 3. 2 0 2 13 ( 3) 6 3m − − −= = = −
− −
130 ( 3)0 11y mx b
bb
b
= += − − += +
− =
13 1y x= − − 4. 4 4 0 02 ( 2) 4m −= = =
− −
4 0( 2)4y mx bb
b
= += − +=
4y = 5. 5 3 28 1 7m −= =
− 𝑦 − 3 = (𝑥 − 1)
6. 250 45 5m −= =
− − (a) 254 (5)4 22
bb
b
= += +=
25 2y x= + (b) 𝑦 − 4 = (𝑥 − 5)
3.9 Graph Inequalities 1. (1) 2. (2) 3. (3) 2 6 42 4 62 3
y xy xy x
+ >> −> −
4. 𝑦 ≤ 𝑥 − 1 5. 𝑦 ≤ 𝑥 − 4 6. 3y < 7. 32 2y x> +
23
8.
9.
10.
11.
12. 3 3x yy x
+ ≤ −≤ − −
13. 1 11x yy xy x
− ≤ −− ≤ − −
≥ +
14. 2 6 102 6 103 5y xy xy x
− >> +> +
15. 13 139 33 3
x yx y
y x
− ≥− ≥
≤ − +
24
Chapter 4. Linear Systems
4.1 Solve Linear Systems Algebraically 1. 3 844 123
x yx y
xx
− =+ =
==
43 41x yyy
+ =+ =
=
2. 2 3 193 3 215 408
x yx y
xx
− =+ =
==
2(8) 3 1916 3 193 31yyyy
− =− =− =
= −
3. 3 2 125 2 48 162
x yx y
xx
+ =− =
==
3(2) 2 126 2 122 63yyyy
+ =+ =
==
4. 2 5 112 3 92 21
x yx y
yy
− =− + = −
− == −
2 5( 1) 112 5 112 63x
xxx
− − =+ =
==
5. 2 4 122 93 31
x yx y
yy
− =− + = −
− == −
2 1 92 84xxx
− − = −− = −
=
6. 3 042 42
x yx y
xx
+ =− − = −
= −= −
3( 2) 06 06yyy
− + =− + =
=
7. 3 2 42 2 24 ( 1)x y
x y+ = →
− + = × − 3 2 42 2 245 204x yx y
xx
+ =− = −
= −= −
3( 4) 2 412 2 42 168
yyyy
− + =− + =
==
8. 2 3 62 2 ( 1)x y
x y+ = →+ = − × − 2 3 62 22 84
x yx y
yy
+ =− − =
==
9. 3 4 11 26 5 16x y
x y− + = ×
− = − → 6 8 226 5 163 62x yx y
yy
− + =− = −
==
3 4(2) 113 8 113 31
xx
xx
− + =− + =
− == −
10. 2 3 7 2 3 73 ( 3) 3 3 922
x y x yx y x y
xx
+ = → + =+ = × − − − = −
− = −=
2(2) 3 74 3 73 31
yyyy
+ =+ =
==
25
11. 2 8 33 3x y
x y+ = ×
− = − → 6 3 243 37 213x yx y
xx
+ =− = −
==
2(3) 86 82y
yy
+ =+ =
=
12. 2 93 2x y
x y+ = →
− = × 2 92 2 63 155x yx y
xx
+ =− =
==
(5) 322y
yy
− =− = −
=
13. 3 2 4 3 9 6 124 3 7 ( 2) 8 6 142x y x yx y x y
x
+ = × + =+ = × − − − = −
= − 3( 2) 2 46 2 42 105
yyyy
− + =− + =
==
14. 3 4 9 35 6 21 ( 2)x y
x y+ = ×+ = × − 9 12 2710 12 421515
x yx y
xx
+ =− − = −
− = −=
3(15) 4 945 4 94 369
yyyy
+ =+ =
= −= −
15.
4 10 55 10 55 153x xx
xx
− = −− =
==
5 32yy
= −=
16. (10 3 ) 24 82x x
xx
= − −==
10 3(2)4yy
= −=
17. 3(9 2 ) 2 1127 6 2 1127 8 118 162
x xx x
xxx
− − =− − =
− =− = −
=
9 2(2)5yy
= −=
18. 4 84 8x yx y
− = −= − 7(4 8) 3 6828 56 3 6831 56 6831 1244
y yy y
yyy
− + =− + =
− ===
4(4) 816 88xx
x
− = −− = −
=
19. ( )122 6 3 1212 3 124 246
b b
b bbb
− + =
− + ===
( )12 6 63a
a
= −
= −
20. ( )4 6 3 87 6 87 142
d dd
dd
− + =
− ===
( )4 2 62cc
= −
=
21. 2 52 52 5
x yy xy x
− =− = − +
= − Choice (1)
26
4.2 Solve Linear Systems Graphically 1. (3) 2. (3) 3. (–2,3) 4.
Solution: (–1,–5)
5. 2 2x yy x
+ == − + 4 44x y
y xy x
− =− = − +
= −
Solution: (3,–1) 6. 353 5 155 3 153x y
y xy x
− =− = − +
= −
Solution: (–5,–6)
7. 133 33 3 1x y
y xy x
+ = −= − −= − −
Solution: (–6,1)
27
8. 4 2 102 4 102 5x y
y xy x
− =− = − +
= −
Solution: (1,–3)
9. 2 52 5x yy x
+ == − +
Solution: (1,3) 4.3 Solutions to Systems of Inequalities 1. (1) (1,1) 2. (4) (–9,0) 3. (4) (4,0) 4. (2) (2,0) 5. (4) (2,–2) 6. (2) (2,–1) 4.4 Solve Systems of Inequalities Graphically 1.
2.
28
3.
4.
5.
6.
7.
infinitely many solutions such as (3,–3)
8.
infinitely many solutions such as (6, 6) 9.
infinitely many solutions, such as (7,1)
10.
infinitely many solutions, such as (4,0)
29
11.
infinitely many solutions such as (–1,–1)
12.
infinitely many solutions such as (2,4) 13. Yes 14. No 15. (1) (1,–4) 16. (2) (–2,2) 4.5 Word Problems – Linear Systems 1. x and y are the two numbers. 5592 6432
x yx y
xx
− =+ =
==
(32) 5927yy
+ ==
The numbers are 32 and 27.
2. a = smaller number b = larger number 47152 6231
b ab a
bb
+ =− =
==
The larger number is 31. 3. p = cost of one bag of popcorn c = cost of one cookie
2 5 2 54 6 ( 1) 4 62 10.50p c p cp c p c
cc
+ = → + =+ = × − − − = −
− = −=
One cookie costs $0.50.
4. d = cost of a doughnut c = cost of a cookie 2 3 3.30 25 2 4.95 ( 3)d cd c
+ = ×+ = × −
4 6 6.6015 6 14.8511 8.250.75d cd c
dd
+ =− − = −
− = −=
2(0.75) 3 3.301.50 3 3.303 1.800.60cccc
+ =+ =
==
Doughnuts cost 75¢ and cookies cost 60¢.
30
5. p = cost of a pizza slice c = cost of a cola 3 2 6.00 32 3 5.25 ( 2)p cp c
+ = ×+ = × −
9 6 18.004 6 10.505 7.501.50p cp c
pp
+ =− − = −
==
3(1.50) 2 6.004.50 2 6.002 1.500.75cccc
+ =+ =
==
Pizzas cost $1.50 and colas cost $0.75.
6. s = hourly rate for the sprayer g = hourly rate for the generator 6 6 90 24 8 100 ( 3)s gs g
+ = ×+ = × −
12 12 18012 24 30012 12010s gs g
gg
+ =− − = −
− = −=
6 6(10) 906 60 906 305s
sss
+ =+ =
==
Sprayer costs $5/hr; generator costs $10/hr. 7. f = number of fancy shirts bought p = number of plain shirts bought 28 15 1317 ( 15)f p
f p+ = →
+ = × −
28 15 13115 15 10513 262f pf p
ff
+ =− − = −
==
(2) 75pp
+ ==
She bought 2 fancy and 5 plain shirts.
8. n = cost of one notebook p = cost of one pencil
2(3 4 8.50) 6 8 17.005 8 14.502.502.50n p n p
n pn
n
− + = → − − = −+ =
− = −= 3(2.50) 4 8.507.50 4 8.504 1.000.25
pppp
+ =+ =
==
$2.50 per notebook, $0.25 per pencil 9. a = number of apples sold last week o = number of oranges sold last week 3( 108) 3 3 3245 3 4522 12864
a o a oa o
aa
− + = → − − = −+ =
==
64 10844o
o+ =
= 64 apples and 44 oranges
10. t = tens digit; u = units digit 10 10 99 10 99 9 9u t t uu t t
u t
+ = + ++ = +
− = 7 99 9 9u t
u t+ = ×
− = →
9 9 639 9 918 724u tu t
uu
+ =− =
==
(4) 73tt
+ ==
The number is 34.
31
4.6 Word Problems – Systems of Inequalities 1. d = number of dog-walking hours c = number of car wash hours 207.50 6.00 92.00d cd c
+ ≤+ ≥
2. s = number of bags of soil p = number of plants 4 10 1005s p
p+ ≤
≥ 3. s = number of boxes of small books l = number of boxes of large books 15 8 35035s l
s l+ ≥
+ ≥
4. Therefore, (a) 355td t
≤≤
55(3)165dd
≤≤
(b) Yes 5. Ans: 10x ≤ , 12y ≤ and 16x y+ ≤
32
Chapter 5. Polynomials
5.1 Polynomial Expressions 1. a) 3 b) 4 c) 3 d) –1 2. a) 2 2 3x x− + b) 2 c) 1 d) 3 3. ( )3 23 2 23 2
2 5 2 5 152 5 2 10 155 3 8 15x x x x x
x x x x x
x x x
− − + + −
= − − − + −
= − − − +
5.2 Add and Subtract Polynomials 1. 28 1x − 2. 24 1x x+ − 3. 7𝑥 + 9𝑥 − 3𝑥 − 8 4. −8𝑥 − 𝑥 + 5 5. 2𝑥 + 8𝑥 + 3 6. 5𝑛 − 9𝑛 + 3 7. 2 22 2 2
(3 2 7) (6 4 3)3 2 7 6 4 33 6 4x xy x xy
x xy x xyx xy
+ + − − + =+ + − + − =
− + +
8. 2 22 2 2
( 1) (3 2 5)1 3 2 52 3 6a a a a
a a a aa a
+ − − − + =+ − − + − =
− + −
9. 2 22 22
( 3 2) (2 6)3 2 2 62 8x x x x
x x x xx x
− − − − + =− − − + − =
− − −
10. 2 22 2 2
( 1) (3 4 1)1 3 4 12 4 2x x x
x x xx x
+ − + − =+ − − + =
− − +
11. ( ) ( )2 2
2 2 29 2 3 4 7 59 2 3 4 7 55 9 8
x x x x
x x x x
x x
− + − + − =
− + − − + =
− +
12. ( ) ( )2 2
2 229 3 4 5 7 69 3 4 5 7 64 10 2
x x x x
x x x x
x x
+ − − − − =
+ − − + + =
+ +
13. ( ) ( )2 2
2 226 3 2 2 5 86 3 2 2 5 84 8 10
x x x x
x x x x
x x
+ − − − + =
+ − − + − =
+ −
14. ( ) ( )2 2
2 2 23 6 7 6 13 123 6 7 6 13 129 19 5
x x x x
x x x x
x x
− + + − − + =
− + + − + − =
− + −
33
15. ( ) ( )3 2 2
3 2 23 23 2 3 43 2 3 42 5 4
x x x x x
x x x x x
x x x
+ − − + − =
+ − − − + =
+ − +
16. ( ) ( )5 4 5 45 4 5 4 8
x xx x
− − + =
− − − =−
5.3 Multiply Polynomials 1. 47 7x x− 2. 36 15r r− 3. −15𝑥 𝑦 − 3𝑥 𝑦 4. 4𝑥 + 12𝑥 + 8𝑥 5. ( )( )23 73 7w w
w w
− =
− 6. 2 2
( 8)( 5)5 8 403 40c c
c c cc c
+ − =− + − =
+ −
7. ( )( )2 2
3 2 73 21 2 143 19 14x x
x x x
x x
+ − =
− + − =
− −
8. 2 2
( 7)(2 3)2 3 14 212 11 21x x
x x xx x
− + =+ − − =
− −
9. 2 22a ab b+ + 10. 2 2( 6)( 6)6 6 3612 36
x xx x x
x x
− − =− − + =
− +
11. x 3 x 2x 3x –y xy− 3y− –1 x− –3
12. (3) ax by+ 13. 23 2 23 2
( 1)(2 2)2 2 2 22 3 2x x x
x x x x xx x x
− + − =+ − − − + =
− − +
14. 2 24 3 2 24 3 2
( 2)( 2 1)2 2 4 22 3 4 2x x x
x x x x xx x x x
+ − + =− + + − + =
− + − +
34
5.4 Divide a Polynomial by a Monomial 1. 2 4 2 4 22 2 2x x x+ = + = + 2. 2 22 2 2x x x x xx x x+ = + = + 3. 14 28 14 28 214 14 14ab b ab b a
b b b+ = + = + 4. 3 2 3 2
26 9 3 6 9 33 3 3 32 3 1x x x x x x
x x x xx x
+ + = + + =
+ +
5. 3 2 3 2
212 6 2 12 6 22 2 2 26 3 1
x x x x x xx x x x
x x
− + = − + =
− +
6. 3 2 3 22
16 12 4 16 12 44 4 4 44 3 1x x x x x x
x x x xx x
− + = − + =
− +
7. 6 4 2 6 4 22 2 2 24 22 18 2 2 18 22 2 2 29 1
x x x x x xx x x x
x x
− + = − + =
− +
8. 5 4 3 223 2
8 2 4 624 2 3x x x x
xx x x
− + − =
− + −
9. 4 3 3 2 2245 90 3 615a b a b a b a
a b− = − 10. 2 6 6 2 2 4 5224 16 4 6 4 14x y x y xy xy x
xy− + = − +
35
Chapter 6. Irrational Numbers
6.1 Simplifying Radicals 1. 12 2 2 3 2 3= ⋅ ⋅ = 2. 50 2 5 5 5 2= ⋅ ⋅ = 3. 32 2 2 2 2 2 2 2 2 4 2= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ = 4. 4 75 4 3 5 5 4 5 3 20 3= ⋅ ⋅ = ⋅ = 5. 5 20 5 2 2 5 5 2 5 10 5= ⋅ ⋅ = ⋅ = 6. 3 45 3 3 3 5 3 3 5 9 5= ⋅ ⋅ = ⋅ = 7. 5 72 5 2 2 2 3 35 2 3 2 30 2= ⋅ ⋅ ⋅ ⋅ =⋅ ⋅ =
8. 2 128 2 2 2 2 2 2 2 22 2 2 2 2 16 2= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =⋅ ⋅ ⋅ =
9. 3 48 3 16 3 3 4 3 12 3− = − ⋅ = − ⋅ = − 10. 98 49 2 7 2− = − ⋅ = − 11. 2 108 2 36 3 2 6 3 12 3= ⋅ = ⋅ = 12. 3 250 3 25 10 3 5 10 15 10= ⋅ = ⋅ = 13. 32 4 2 24 4= = 14. 7 18 7 3 2 7 23 3⋅= = 6.2 Operations with Radicals 1. 75 3 5 3 3 6 3+ = + = 2. 27 12 3 3 2 3 5 3+ = + = 3. 50 32 5 2 4 2 9 2+ = + = 4. 27 108 3 3 6 3 9 3+ = + = 5. 28 63 2 7 3 7 5 7+ = + = 6. √150 + √24 = 5√6 + 2√6 = 7√6 7. 3√2 + √8 = 3√2 + 2√2 = 5√2 8. √72 − 3√2 = 6√2 − 3√2 = 3√2 9. 5 7 3 28 5 7 6 7 11 7+ = + = 10. 2 50 2 10 2 2 9 2− = − = 11. 6√50 + 6√2 = 30√2 + 6√2 = 36√2 12. √25 − 2√3 + √27 + 2√9 = 5 − 2√3 + 3√3 + 6 = 11 + √3 13. 6 15 90 3 10⋅ = = 14. 4√2 ∙ 2√6 = 8√12 = 16√3 15. 90 40 8 183600 14460 12 48⋅ − ⋅ =
− =− =
16. 3 20(2 5 7)6 100 21 2060 42 5
− =− =
−
36
17. ( )
( )3 7 14 4 563 7 14 8 143 98 24 98 27 9827 36 2 189 2
+ =
+ =
+ = =⋅ =
18. ( )( )3 5 3 59 3 5 3 5 59 5 4
+ − =
− + − =− =
19. ( )3 4 2 3 33 4 2 3 32 3 4 2
y y
y yy
− + =
− − =− −
20. 3 8 6 2= ( ) ( )2 6 2 2 2 2 2 112 2 4 4 2 2 16 2 6P = + + +
= + + + = +
( )( )6 2 2 2 2 112(2) 6 2 4 2 2 10 2 26A = + +
= + + + = +
21. 65 135 = 22. 20 100 5 50 25 24 2 = = 23. 1 12 284 28 2 7 72 3 = = ⋅ = 24. 6 20 2 4 2 2 43 5 = = ⋅ = 25. 3 75 27 15 3 3 33 318 3 6 33
+ += =
=
26. 16 21 5 122 78 3 10 3 2 3− =
− = −
27.
48 5 27 2 75316 5 9 2 254 15 10 1− + =
− + =− + = −
28. 27 75 3 3 5 3 8 3 412 2 3 2 3+ += = =
6.3 Rationalizing Denominators 1. 1 7 777 7 ⋅ =
2. 6 2 6 2 3 222 2 ⋅ = =
37
3. 5 10 5 10 1010 210 10 ⋅ = =
4. 6 21 6 21 2 2121 721 21 ⋅ = =
5. 8 6 8 6 8 6 4 63(6) 18 93 6 6
⋅ = = =
6. 10 2 5 10 10 2 1055 5 ⋅ = =
7. 2 2 2 253 5 32 2 3 2 2 3 2 65(3) 155 3 3
× =
⋅ = =
8. 16 16 43 3 34 3 4 333 3
= =
⋅ =
9. Rationalize denominators of both fractions. 1 3 333 3
⋅ =
1 2 222 2 ⋅ =
LCD is 6. 3 2 2 3 2 3 3 23 2 2 3 6 62 3 3 26 ⋅ + ⋅ = +
+=
10. Rationalize denominators of both fractions. 1 2 222 2 ⋅ =
3 5 3 555 5 ⋅ =
LCD is 10. 2 5 3 5 2 5 2 6 52 5 5 2 10 10 ⋅ + ⋅ = +
5 2 6 510+= 11. Rationalize denominators of both fractions. 3 5 3 555 5
⋅ =
4 6 4 6 2 66 36 6 ⋅ = =
LCD is 15. 3 5 3 2 6 5 9 5 10 65 3 3 5 15 15 ⋅ + ⋅ = +
9 5 10 615+=
12. ( )
233 3 83 8 3 33 33 3 24 3 3 2 6 3 63 3
− − ⋅ = =
− −= = −
38
13.
4 3 4 3 2 33 4 23 4 32 3 3 2 3 32 3 23 32 3 2 3 3 4 3 3 3 33 2 2 3 6 6 6
− = − = − =
⋅ − = − =
⋅ − ⋅ = − =
6.4 Closure 1. (2) 8 16 4− = − , 64 8= , 1 164 8= 2. Irrational. 3 is not a perfect square.
3. Irrational. π is irrational, and the quotient of an irrational number and a non-zero rational number is irrational. 4. Irrational. 29 is irrational since 29 is not a perfect square. The numerator is the difference of a non-zero rational and irrational, so the numerator is irrational. The fraction is the quotient of an irrational and a non-zero rational, so it is irrational. 5. x could be 0, 3 , 12 , 13 , etc. 6. 3.141593π ≈ and 227 3.142857≈ 3.14 0.001593π − ≈ 227 0.001264π− ≈ So, 227 is a closer approximation.
39
Chapter 7. Univariate Data
7.1 Types of Data 1. (4) 2. (3) 7.2 Frequency Tables and Histograms 1. Add the frequencies: 2 + 4 + 5 + 4 + 1 = 16 2. Add the frequencies: 7 + 10 + 3 + 5 = 25 3. 25 – 18 = 7 4. 20 5. 3; 0; 20 6.
7.
40
8.
9.
10.
7.3 Central Tendency 1. mode 2. median 3. (a) 4. They are all divided by two as well.
41
5. The mean increased by five and the range remained the same. 6. mean ≈ 11.4, median = 12, mode = 7, they all increase by 5 7. City A (22) 8. mean = 79, median = 79, mode = 78 9. (2) mode = median = 6 10. (1) mean = 17, median = 18, mode = 22 11. (3) mean ≈ 85.6, median = 88, mode = 92 12. an outlier such as a very low score could greatly affect the range without affecting the median 13. mean = 22, median = 20, mode = 20 14. 131 – 150 There are 44 total scores, so the median would be the average of the 22nd and 23rd highest scores. 15. mean = 225000, median = 175000, the median because the mean is higher than all but one of the values (an outlier) 16. 71-80; out of 31 students, the 16th lowest value is the median, which is within the 41-80 interval, or 71-80 interval on the related frequency table 7.4 Distribution 1. Skewed to the right. 2.
3. left-skewed; no outliers 4. symmetrical, but with outliers at 9.45 7.5 Standard Deviation 1. The population is all the bolts in the shipment. The sample is the 100 selected bolts. 2. The population is all the mall shoppers. The sample is every sixth person within the 3 hour period. 3. (2) 4. (1) 5. The first set, as shown by the smaller SD. 6. McCrane; a larger SD means more variability 5. mean = 66, SD ≈ 30.4 6. mean ≈ 60.7, SD ≈ 15.1 7. SD ≈ 16.8 8. SD ≈ 0.88
42
7. mean = 9.46; standard deviation = 3.85 8. mean = = 44 (51 − 44) = 49, (48 − 44) = 16, etc. = = 16. 2 SD = 16. 2 ≈ 4.0 9. mean = $610, SD ≈ 14.7 10. SD ≈ 8.1 7.6 Percentiles and Quartiles 1. 25% of 40 = 10 students 2. 95,000 76%125,000 = , so the 76th percentile. 3. 22 130 373 %= , so the 73rd percentile. 4. 5 0.510bp
n= = = , so 70 is the 50th percentile. 5. second quartile = median = 35+452 = 40 6. 5, 6, 7, 8, 12, 14, 17, 17, 18, 19, 19 Q1 = 7, Q2 = 14, Q3 = 18
7. 3, 6, 7, 7, 8, 9, 9, 9, 10, 12, 13, 15 Q1 = 7, Q2 = 9, Q3 = 11 8. 21, 28, 28, 32, 33, 41, 45, 50, 53 Q1 = 28, Q2 = 33, Q3 = 47.5, IQR = 19.5 9. 71, 71, 72, 74, 74, 75, 78, 79, 79, 83 Q3 = 79 and Q1 = 72, so IQR = 7 10. Q1 = 70, Q2 = 80, Q3 = 90 11. The corresponding frequency table would show:
Minutes Used Frequency 31–40 2 41–50 3 51–60 5 61–70 9 71–80 11 25% of 30 is 7.5, so the first quartile would be between the 7th and 8th smallest values out of 30. This falls within the 51–60 interval.
43
7.7 Box Plots 1. 81 2. 75 3. 10 4. 84 5. 30 6. 4 7. 75 – 15 = 60 8. 25% 9. (4) 75–88 10. Q1 = 77, Q2 = 87, Q3 = 91
11. Q1 = 70, Q2 = 82, Q3 = 90 12.
13.
44
Chapter 8. Bivariate Data
8.1 Two-Way Frequency Tables 1. 15113 13.3%≈ of the students are undecided. 3160 51.7%≈ of the 9th graders are watching. 2. Fiction Nonfiction Total Hardcover 28 52 80 Paperback 94 36 130 Total 122 88 210 Fiction Nonfiction Total Hardcover 13.3% 24.8% 38.1% Paperback 44.8% 17.1% 61.9% Total 58.1% 41.9% 100%
3. Given data in bold below. Coca-Cola Sprite Total Table 16 14 30 Garbage 34 8 42 Total 50 22 72
8.2 Scatter Plots 1. (2) 2.
3.
8.3 Correlation and Causality 1. (3) 2. (2)
45
3. A. positive: children usually gain weight as they age and grow B. negative: as the volume of water increases, the remaining space decreases C. none: shoe size and hair length are unrelated D. positive: more people go to the beach when the temperature is higher
4. A. positive, causal B. negative, causal C. positive, not causal, the size and severity of the fire, which results in more firefighters being called D. negative, not causal, the degree of civilization and industrialization over time 8.4 Identify Correlation in Scatter Plots 1. (1) 2. positive correlation 3. negative correlation 4. negative correlation 5. positive correlation 6. no correlation 7. positive correlation 8.5 Lines of Fit 1. (a) 80 wpm (b) 9 wpm 2. Line A. Most of the points are closer to Line A than to Line B. 3.
4. 𝑦 = 5𝑥 + 25 5. 0.2 7.5y x= + 6. 2 5.14y x= + 7. (3) 1,000 15,000y x= + 8. The prediction for the 35 year old is more likely to be accurate, since it is an interpolation rather than an extrapolation. 9. (4) 𝑦 = 𝑥 + 1 10. (2) 72 11. (3) 480 12. (3) $42,500
46
13.
No, the line crosses near (18,13)
14.
0.56 162.79y x= + 15. 0.112 23.448y x= − + ; –5°C 16. 35.5 457.5y x= − + ; 103 8.6 Correlation Coefficients 1. (1) 0.89 2. (4) 0.90 3. (4) 4. (2) –0.24 It is a weak correlation. 5. a. 0.90 b. –0.40 c. 0.99 d. –0.85 e. 0.50 f. 0 6. 𝑟 = 1 7. 𝑟 ≈ 0.986 8. 𝑟 ≈ −0.999 8.7 Residuals 1. Actual value – Predicted value = 12,550 – 14,050 = –1,500 2. Study Time in Hours (x) Test Score (y) Predicted Test Score Residual 0.5 63 62.8 0.2 1 67 67.2 -0.2 1.5 72 71.6 0.4 2 76 76.0 0 2.5 80 80.4 -0.4 3 85 84.8 0.2 3.5 89 89.2 -0.2
47
3. a) ( )0.75 22 0.25 16.25y = − = . (Since scores cannot be fractional, 16 is a valid answer.) b) ( )0.75 34 0.25 25.25y = − = Residual = 32 – 25.25 = 6.75 c) ( )0.75 28 0.25 20.75y = − = 𝑝 − 20.75 = −0.75 𝑝 = 20 They scored 20 points. 4. x y Predicted Value Residual 5 3 2.5 0.5 10 4 5.0 -1 15 9 7.5 1.5 20 7 10.0 -3 25 13 12.5 0.5 30 15 15.0 0 Yes. There is no clear pattern in the residual plot. 5. x y Predicted Value Residual 2 5 15.5 -10.5 4 15 14.7 0.3 6 26 13.9 12.1 8 23 13.1 9.9 10 11 12.3 -1.3 12 3 11.5 -8.5 No. There appears to be a parabola-like pattern in the residual plot.
48
6. (a) 0.117 83.267y x= + (b) and (c) (d) Distance (miles)
Airfare ($)
Predicted Price ($)
Residual 576 178 150.7 27.3 370 138 126.6 11.4 612 94 154.9 -60.9 1,216 278 225.5 52.5 409 158 131.1 26.9 1,502 258 259.0 -1.0 946 198 193.9 4.1 998 188 200.0 -12.0 189 98 105.4 -7.4 787 179 175.3 3.7 210 138 107.8 30.2 737 98 169.5 -71.5
49
Chapter 9. Introduction to Functions
9.1 Recognize Functions 1. (2) 2. (2) 3. (1) 4. (2) 5. Yes, each x entry is mapped to a unique y 6. (1) 7. (2) 8. (1) 9. (1) 10. (3) 9.2 Function Graphs 1.
x ( )f x 0 0 1 3 2 4 3 7 4 2 5 0
2. (a) (9) 1f = (b) { 12 , 3}
9.3 Evaluate Functions 1. 2(3) 2(3) 3(3) 618 9 6 33f = − − − =− − − = −
2. 2( 3) ( 3) 2( 3) 19 6 1 16f − = − − − + =+ + =
3. 2(0) (0 3) 9f = − = 4. 2(2) 0.5 0.25f = = 5. [ ] 2(3) (2) 3(3) 4 (2) 1f g − = − − = 6. (0) 2(0) 1 1( 2) 2( 2) 1 5(0) ( 2) ( 1)( 5) 5h
hh h
= − = −− = − − = −
⋅ − = − − =
7. 10 4 212 43 xx
x
− = − +− = −
= 8. 12 = 𝑘(2) 12 = 4𝑘 𝑘 = 3
50
9. 2 2(4 ) 2(4 ) 6(4 ) 332 24 3g a a aa a
= + − =+ −
10. 22 2
( 2) ( 2) 2( 2) 14 4 2 4 16 7f a a a
a a aa a
+ = + + + − =+ + + + − =
+ +
11. 𝑃(125) = 0.0089(125) + 1.1149(125) + 78.4491 ≈ 356.9 9.4 Features of Function Graphs 1. (a) 0 2x< < and 4 8x< < (b) 2 4x< < (c) 8 10x< < 2. (a) (–3,5) (b) (1,–3) 3. (a) positive at 2x < − and 0 2.5x< < ; negative at 2 0x− < < and 2.5x > (b) increasing at 1 1x− < < ; decreasing at 1x < − and 1x > (c) relative maximum at (1,2); relative minimum at (–1,–2) 9.5 Domain and Range 1. {2, 3, 22, 51} 2. { }| 0x x ≠ 3. domain: 1 < 𝑥 ≤ 4 range: 1 < 𝑓(𝑥) ≤ 7 4. domain: 5 8x− ≤ ≤ Range: 3 2y− ≤ ≤ 5. 4 ≤ 𝑥 ≤ 13 6. 0 ≤ 𝑦 ≤ 100 7. 0 ≤ 𝑥 ≤ 12 8. 30 ≤ 𝑦 ≤ 80 9`. the set of counting (natural) numbers 10. (5) 25f = and (10) 40f = , so the range is 25 ( ) 40f x≤ < 11. (a) ( ) 0f x ≥ (b) 0 ( ) 9f x≤ ≤ 12. (a) ( ) 5f n n= (b) whole numbers 20n ≤ (c) {0, 5, 10, 15, … 100} 9.6 Absolute Value Functions 1. (4) 2. (1)
51
3.
4.
5.
6.
7.
52
Chapter 10. Functions as Models
10.1 Write a Function from a Table 1. ( ) 4 9f x x= + 2. 53( ) 10f x x= + 3. ( ) 3( 1) 7f x x= − + ( ) 3 4f x x= + 4. 12( ) ( 1) 5f x x= − − 1 12 2( ) 5f x x= − 5. ( ) 2( 2) 9f x x= − − + ( ) 2 13f x x= − + 6. ( ) 5( 11)f x x= − ( ) 5 55f x x= − 10.2 Graph Linear Functions 1. 𝑐(𝑔) = 5𝑔 + 25
2. 𝑐(𝑥) = 𝑥 + 25
53
3. 𝑦 = 2𝑥 − 40
4. Line through points (59,76) and (65,100) has a slope of 246 4= and y-intercept 𝑏 = 𝑦 −𝑚𝑥 = 100 − 4(65) = −160. ( ) 4 160c t t= −
10.3 Rate of Change for Linear Functions 1. negative 2. positive 3. negative 4. positive 5. positive; 348 232 116 586 4 2m −= = =
− mph 6. negative; as the distance travelled increases, the gas in the tank decreases.
10.4 Average Rate of Change 1. (a) 9 1 8 43 1 2m −= = =−
(b) 4 1 3 12 ( 1) 3m −= = =− −
2. (a) 5.06 3.91 1.15 0.0961999 1987 12− = ≈
− (b) 7.50 5.06 2.44 0.2442009 1999 10− = ≈
− (b) has the higher average rate of change.
54
3. 2 2(5) 5 2 27(15) 15 2 227227 27 200 2015 5 10
ff
m
= + == + =
−= = =−
4. 22( 3) ( 3) 10( 3) 16 5(3) 3 10(3) 16 5555 ( 5) 60 103 ( 3) 6
ff
m
− = − + − + = −= + + =
− −= = =− −
10.5 Functions of Time 1. (3) 2. (2) 3. 30 secs. 4. 7 minutes From 7:04 to 7:07 and 7:20 to 7:24 5. a) point B because it is the only point after which her distance from home decreases; b) 5 mins, from point D to point E
6. (a) Spencer starts at (0,20) and McKenna starts at (0,0). (b) McKenna speeds up, as the graph curves upward. The average rate of change increases. (c) At about 3.2 hours. They traveled about 41 miles. 7.
Runner A is decreasing the distance to Slopes the finish line at 8 mph and Runner B is running at 3.5 mph. The two runners meet after about 2/3 hour Point of intersection at about 234 miles from the finish line. Runner A finishes at 12:15 and Runner B x-intercepts finishes at 1:00.
55
10.6 Systems of Functions 1. (a) $50 (b) 5 months; $125 (c) slope = 125 75 505 0 5−− = = $10 2. Tasha: 𝐴(𝑥) = 60 + 5𝑥 Tyson: 𝐵(𝑥) = 135 − 10𝑥 60 5 135 1015 755x x
xx
+ = −==
5 weeks 3. (a) ( ) 25R x x= (b) ( ) 20 50000C x x= + (c) 25 20 500005 5000010000x xxx
= +==
10,000 widgets
4. (a) ( ) 36 50f h h= + (b) ( ) 39 35g h h= + (c) 36 50 39 3550 3 3515 35
h hhh
h
+ = += +==
5 hours
10.7 Combine Functions 1. 2 2( ) ( 1) ( 5)2 4h x x x xx x
= + + + − =+ −
2. 2( ) (2 1)( 2)2 3 2h x x xx x
= + − =− −
3. (a) 𝑅(𝑐) = 20𝑐 + 500 (b) 𝐸(𝑐) = 6𝑐 (c) 𝑃(𝑐) = 𝑅(𝑐) − 𝐸(𝑐) = (20𝑐 + 500)− (6𝑐) = 14𝑐 + 500
56
Chapter 11. Exponential Functions
11.1 Exponential Growth and Decay 1. 20(1.1)x 2. (0.98)nx 3. 2500(1.03) 4. 10,000(1.2) 5. 6$1500(1.05) $2010.14≈ 6. 51000(1.03) 1159.27≈ 7. 42000(1.035) $2295≈ 8. 430,000(0.95) 24,435.19≈ 9. 325,000(0.8) 12,800= 10. 53,810(1.035) 4,525≈ 11. 20,000(0.88) = 13,629.44 12. 3$11,900(0.87) $7,800≈ 13. 12$1.39(1.005) $1.48≈ 14. 3256(0.25) 4= 11.2 Graphs of Exponential Functions 1. (4) 2.
3.
57
4.
5.
6.
7.
8.
9.
It is shifted (translated) down by 5 units. 10. 0.1(4)xy = 11. ( )14 xy = 12. 0.488(1.116)xy = 13. 733.646(0.786)xy = ; for 12, 41x y= ≈
58
14. (a) ( ) 0.1(2)nt n = (b)
15. (a) ( )12( ) 30 nh n = (b)
59
16. (a)
(b) 276.67(1.21)xy = (c) 21276.67(1.21) 15,151y = ≈ , or $15,151,000 11.3 Rewrite Exponential Expressions 1. ( )2 25 5 25xx x= = 2. ( )5 510(1.1) 10 1.1 10(1.61051)xx x= = 3. ( ) ( )3 2 3 22 2 2 4(8)xx x+ = ⋅ = 4. ( )1 14(3) 4(3) 3 12(3)x x x+ = = 11.4 Compare Linear and Exponential Functions 1. (a) linear (c) exponential (b) exponential (d) neither 2. ( ) 3f n n= 3. ( ) 3nf n =
60
Chapter 12. Sequences
12.1 Arithmetic Sequences 1. 3 2. –4 3. 1 ( 1)15 ( 1) 515 5 55 10
nnnn
a a n da na na n
= + −= + − ⋅= + −= +
4. 1 ( 1)10 ( 1) 210 2 22 8
nnnn
a a n da na na n
= + −= + − ⋅= + −= +
5. 188
( 1)21 (8 1) 984na a n daa
= + −= + − ⋅=
6. 12727( 1)5 (27 1) 383na a n d
aa
= + −= + − ⋅=
7. Find d by calculating the slope using (6,10) and (21,55) as two points: 55 10 45 321 6 15d −= = =
− Solve for 1a :
6 111(6 1)10 5(3)5a a d
aa
= + −= +
− = Write the rule:
1 ( 1)5 ( 1)35 3 38 3nnnn
a a n da na na n
= + −= − + −= − + −= − +
8. Find d by calculating the slope using (4,–23) and (22,49) as two points: 49 ( 23) 72 422 4 18d − −= = =−
Solve for 1a : 4 111
(4 1)23 3(4)35a a daa
= + −− = +− =
Write the rule: 1 ( 1)35 ( 1)435 4 439 4
nnnn
a a n da na na n
= + −= − + −= − + −= − +
12.2 Geometric Sequences 1. 12 2. –2 3. –4 4.
( )
11 14 2.5nnn
n
a a r
a
−
−=
=
61
5. 11 1 1( 1)( 2) ( 2)nn
n nn
a a ra
−
− −== − − = − −
6. ( ) ( )
( )
11 15 1 1415 5 2 5 25 16,384 81,920n
na a r
a
−
−=
= − = −= =
7. ( ) ( )
( )
11 7 1 61 17 2 21646 66 0.09375
nna a r
a
−
−=
= − = −
= =
8. 𝑎10 = 𝑎1𝑟 = 512 𝑎15 = 𝑎1𝑟 = 16384 141 91 16384512a r
a r= , so 𝑟 = 32, or 𝑟 = 2.
Using 𝑎 = 𝑎1(2) = 512, so 𝑎 = 1. 𝑎 = 1(2) = 536,870,912 12.3 Recursively Defined Sequences 1. 12 13 24 3
3 3 2 55 3 88 4 12aa a na a na a n
== + = + == + = + == + = + =
neither; there is no common difference nor common ratio
2. 12 13 24 3
4 2 4 2 2 82 8 2 3 142 14 2 4 22aa a na a na a n
= −= − = − − ⋅ = −= − = − − ⋅ = −= − = − − ⋅ = −
3. 12 13 2
32 1 2(3) 1 52 1 2(5) 1 9aa aa a
== − = − == − = − =
4. (a) 𝑎 = 40 𝑎 = 8 2 12n n
na aa − −+= for 𝑛 > 2 (b) 𝑎7 = 19
62
Chapter 13. Factoring
13.1 Factor Out the Greatest Common Factor 1. 24 6 2 2 2 32 (2 3)x x x x xx x
− = ⋅ ⋅ ⋅ − ⋅ ⋅= −
2. 25 10 5 2 55 ( 2)a a a a aa a
− = ⋅ ⋅ − ⋅ ⋅= −
3. 27 (2 1)x x + 4. 2( 1)x x x+ − 5. 26 (2 3 )xy x y+ 6. 22 ( 2 1)y y y− + 7. 23 ( 2 2)x x x− + 8. 2( )x y− + 9. 3𝑚𝑛(𝑚 + 4𝑛) 10. 2𝑥 𝑦 (3𝑥𝑧 − 2) 13.2 Factor a Trinomial 1. ( 7)( 2)x x+ + 2. ( 9)( 2)x x− − 3. ( 9)( 3)x x− + 4. ( 15)( 14)a a− + 5. (𝑥 + 8)(𝑥 − 3) 6. (𝑥 + 5)(𝑥 − 3) 7. (𝑥 − 12)(𝑥 + 2) 8. (𝑥 − 3)(𝑥 − 2) 9. yes, it is prime 10. 2( 2)( 5) 7 10x x x x+ + = + + , so 7b = 11. 2 2 2( 3 2) (4 3 10) 4 12 ( 6)( 2)x x x x x x x x− + − + + − = + − = + − 13.3 Factor the Difference of Perfect Squares 1. ( 6)( 6)x x+ − 2. (2 3)(2 3)x x+ − 3. (3 )(3 )x x+ − 4. ( 1)( 1)a a+ − 5. (7 )(7 )x y x y+ − 6. (2 3 )(2 3 )a b a b+ − 7. ( 4)( 4)xy xy+ − 8. 5 5( 10)( 10)x x+ − 9. (10𝑛 + 1)(10𝑛 − 1) 10. (11 + 𝑥)(11 − 𝑥) 11. (3𝑎 + 8𝑏)(3𝑎 − 8𝑏) 12. 10
63
13.4 Factor Trinomials with a≠1 by Grouping 1. 22 6 26 3 4 23 (2 1) 2(2 1)(3 2)(2 1)
x xx x x
x x xx x
+ − =− + − =
− + − =+ −
2. 2212 5 212 3 8 23 (4 1) 2(4 1)(3 2)(4 1)
x xx x x
x x xx x
+ − =− + − =
− + − =+ −
3. 2212 29 1512 9 20 153 (4 3) 5(4 3)(3 5)(4 3)
x xx x x
x x xx x
− + =− − + =− − − =
− −
4. 226 11 46 3 8 43 (2 1) 4(2 1)(3 4)(2 1)
x xx x x
x x xx x
− + =− − + =
− − − =− −
5. 2215 14 815 20 6 85 (3 4) 2(3 4)(5 2)(3 4)
x xx x x
x x xx x
+ − =+ − − =
+ − + =− +
6. 2210 29 1010 25 4 105 (2 5) 2(2 5)( 5 2)(2 5)
x xx x x
x x xx x
− − − =− − − − =− + − + =
− − +
7. 224 12 94 6 6 92 (2 3) 3(2 3)(2 3)(2 3)
x xx x x
x x xx x
+ + =+ + + =
+ + + =+ +
The square root is 2 3x + .
8. First, write in standard form: 2210 11 610 15 4 65 (2 3) 2(2 3)(5 2)(2 3)
x xx x x
x x xx x
+ − =+ − − =
+ − + =− +
9. 22 4 3 62 ( 2) 3( 2)(2 3)( 2)x x x
x x xx x
+ − − =+ − + =
− + So, (2 3)x −
13.5 Factor Completely 1. 222 12 542( 6 27)2( 9)( 3)y y
y yy y
+ − =+ − =
+ −
2. 223 15 423( 5 14)3( 7)( 2)x x
x xx x
+ − =+ − =+ −
64
3. 223 273( 9)3( 3)( 3)x
xx x
− =− =
+ −
4. 222 502( 25)2( 5)( 5)x
xx x
− =− =
+ −
5. 222 10 282( 5 14)2( 7)( 2)a a
a aa a
− − =− − =− +
6. 3 22 8 7( 8 7)( 7)( 1)x x x
x x xx x x
+ + =+ + =+ +
7. 8 7 66 26
2 16 322 ( 8 16)2 ( 4)( 4)x x xx x x
x x x
+ + =+ + =+ +
8. 2 23 273 ( 9)3 ( 3)( 3)ax a
a xa x x
− =− =
+ −
9. 2 32 25 1805 ( 36)5 ( 6)( 6)x y y
y x yy xy xy
− =− =
+ −
10. 542 22
2 322 ( 16)2 ( 4)( 4)2 ( 4)( 2)( 2)x x
x xx x x
x x x x
− =− =
+ − =+ + −
11. ( )
( )( )
2 22 10 122 5 62 6 1x x
x x
x x
+ − =
+ − =
+ −
12. ( )
( )( )
32 442 2a a
a a
a a a
− =
− =
+ −
13. ( )
( )( )
3 223 33 903 11 303 6 5x x x
x x x
x x x
− + =
− + =
− −
14. ( )
( )( )
2 62 63 3
36 1004 9 254 3 5 3 5x y
x y
x y x y
− =
− =
+ −
15. ( )
( )( )
3 32 24 364 94 3 3x y xy
xy x y
xy xy xy
− =
− =
+ −
16. ( )
( )( )
3 22 663 2x x x
x x x
x x x
− − + =
− + − =
− + −
65
Chapter 14. Quadratic Functions
14.1 Solve Simple Quadratic Equations 1. 81 9x = =± ± {9,–9} 2. 20 2 5y = =± ± { }2 5, 2 5− 3.
{ }
223 7525 255, 5x
x
x
=
=
= ±−
4.
{ }
2 224 36 04 369 93, 3
x
x
x
x
− =
=
=
= ±−
5.
{ }
222 126 66, 6xx
x
===
−±
6.
{ }
22 49 4 29 32 23 3
9 4,
xx
x
==
= =
−
± ±
7.
2 225 5 05 51 1 1{1, 1}
xxx
x
− ==== =
−± ±
8. 223 000{0}
mm
m
===
9. 2 22h s
h s=
=± 10. 2 2 22 2a c b
a c b
= −
= −
14.2 Solve Quadratic Equations by Factoring 1.
{ }
2 5 0( 5) 00 5 00,5x xx x
x or x
− =− =
= − =
2.
{ }
2 3 18 0( 6)( 3) 06 0 3 06,3x xx x
x or x
+ − =+ − =
+ = − =−
66
3.
{ }
224 36 04( 9) 04( 3)( 3) 03 0 3 03,3xx
x xx or x
− =− =
+ − =+ = − =
−
4.
{ }
2 4 32 0( 8)( 4) 08 0 4 08, 4x xx x
x or x
− − =− + =
− = + =−
5.
{ }
22 5 65 6 0( 6)( 1) 06 0 1 06, 1x x
x xx x
x or x
− =− − =
− + =− = + =
−
6.
{ }
22 3 22 3 0( 3)( 1) 03 0 1 03, 1x x
x xx x
x or x
− =− − =
− + =− = + =
−
7.
22 66 0( 3)( 2) 03 0 | 2 0{3, 2}x x
x xx x
x x
− =− − =
− + =− = + =
−
8.
22 30 1313 30 0( 15)( 2) 015 0 | 2 0{ 15,2}x x
x xx x
x x
= −+ − =
+ − =+ = − =
−
9.
{ }
22 4 245 24 0( 8)( 3) 08 0 3 08, 3x x xx xx x
x or x
− = +− − =
− + =− = + =
−
10.
{ }
2222 10 122 10 12 02( 5 6) 02( 6)( 1) 06 0 1 06,1
x xx x
x xx x
x or x
+ =+ − =
+ − =+ − =
+ = − =−
11.
{ }
22( 2) 32 32 3 0( 3)( 1) 03 0 1 03,1
x xx x
x xx x
x or x
+ =+ =
+ − =+ − =
+ = − =−
12.
{ }
22( 2)( 3) 125 6 125 6 0( 6)( 1) 06 0 1 06,1
x xx xx xx x
x or x
+ + =+ + =+ − =
+ − =+ = − =
−
67
14.3 Find Quadratic Equations from Given Roots 1. 2
10 210 0 2 0( 10)( 2) 08 20 0x or x
x or xx xx x
= = −− = + =
− + =− − =
2. 2
0 30 3 0( 3) 03 0x or x
x or xx xx x
= == − =
− =− =
3. ( )( )2
12 212 0 2 012 2 010 24 0x or x
x or xx x
x x
= − =+ = − =
+ − =
+ − =
4. ( )( )2
3 53 0 5 03 5 02 15 0x or x
x or xx x
x x
= − =+ = − =
+ − =
− − =
5. ( )( )2
5 25 0 2 05 2 03 10 0x or x
x or xx x
x x
= − =+ = − =
+ − =
+ − =
6. ( )( )2
1 31 0 3 01 3 04 3 0x or x
x or xx x
x x
= =− = − =
− − =
− + =
7. multiply both sides by 2 32
20 2 02 3 0 2 0(2 3)( 2) 02 7 6 0
x or xx or x
x xx x
− = − =− = − =
− − =− + =
8. 22( 1) 02 1 0x
x x− =
− + =
9. 2( 4)( 4) 016 0x xx− + =
− = 10.
230 1 1( 1)( 1) 0( 1) 00
x or x or xx x x
x xx x
= = = −− + =
− =− =
14.4 Equations with the Square of a Binomial 1. 5 16 45 4{ 1, 9}x
x+ = =
= −− −
± ±± 2.
{ }
4 104 104 10,4 10xx− ==
+ −
±±
68
3.
{ }
2( 1) 81 8 2 21 2 21 2 2,1 2 2b
bb
− =− = =
=± ±±
+ -
4.
{ }
22( 1) 30( 1) 301 301 301 30, 1 30mm
mm
− + = −+ =
+ == −
− −
±±
+ -
5. 2( 2) 02 02{2}
xx
x
− =− =
=
6.
2 222( 5) 50 02( 5) 50( 5) 255 25 55 5{0, 10}
xx
xx
x
+ − =+ =+ =
+ = == −
−
± ±±
14.5 Complete the Square 1. 2 210 252 2b = =
2 22 2
10 11 010 1110 25 11 25( 5) 365 365 65 6
x xx x
x xx
xx
x
+ − =+ =
+ + = ++ =
+ =+ =
= −±±
Solution: {1, –11}
2. 2 28 162 2b − = =
2 22 28 16 08 168 16 16 16( 4) 04 0
x xx x
x xx
x
− + =− = −
− + = − +− =
− =
Solution: {4}
69
3. 2 24 42 2b = =
2 22 24 2 04 24 4 2 4( 2) 22 22 2
x xx x
x xx
xx
+ + =+ = −
+ + = − ++ =
+ == −±±
Solution: { }2 2, 2 2− + − −
4. 2 24 42 2b − = =
2 22 24 8 04 84 4 8 4( 2) 122 12 2 32 2 3
x xx x
x xx
xx
− − =− =
− + = +− =
− = ± = ±= ±
Solution: { }2 2 3,2 2 3+ − 5.
222( 6 2) 06 2 0x xx x
− + =− + = 2 26 92 2b − = =
22 26 26 9 2 9( 3) 73 7
x xx x
xx
− = −− + = − +
− =− =±
Solution: { }3 7 ,3 7+ −
6. 2 22 12 2b − = =
2 22 22 3 02 32 1 3 1( 1) 21 2
x xx x
x xx
x
− + =− = −
− + = − +− = −
− = −±
No real solutions, since the square root of a negative number is not a real number. 7. 2( 10) 88010 880w w
w w+ =
+ = 2 210 252 2b = =
2 210 25 880 25( 5) 9055 9055 905 5 30.125.1 ( )
w ww
www reject negative w
+ + = ++ =
+ == − ± ≈ −=
±±
width ≈ 25.1 ft. and length ≈ 35.1 ft. 14.6 Quadratic Formula and the Discriminant 1. (2) 2 2. 𝑏2 − 4𝑎𝑐 = 42 − 4(1)(7) = −12, so the answer is (a) not real 3. 𝑏2 − 4𝑎𝑐 = 32 − 4(9)(−4) = 153, so (4) real, irrational, and unequal 4. 𝑏2 − 4𝑎𝑐 = (−9)2 − 4(2)(4) = 49, so (2) real, rational, and unequal
70
5.
2 22
6 2 3 0422 ( 2) 4(6)( 3)2(6)2 76 2 2 1912 121 196
x x
b b acxa
− − =
− ± −= =
± − − −=
= =± ±
±
Solution: 1 19 1 19,6 6 + −
6.
2 22
2 7 3 0427 (7) 4(2)( 3)2(2) 7 734
x x
b b acxa
+ − =
− ± −= =
− ± − −=
− ±
Solution: 7 73 7 73,4 4 − + − −
7.
2 22
7 8 042(7) (7) 4(1)(8)2(1) 7 172
x x
b b aca
+ + =
− ± − =
− ± −=
− ±
Solution: 7 17 7 17,2 2 − + − −
8.
2 22
2 8 3 042( 8) ( 8) 4(2)(3)2(2)8 40 8 2 104 4 102 2
x x
b b aca
− + =
− ± − =
− − ± − −=
= =± ±
±
Solution: 10 102 ,22 2 + −
14.7 Solve Quadratics for Trinomials with 𝒂 ≠ 𝟏 1. 10𝑥 + 9𝑥 + 2 = 0 𝑛 + 9𝑛 + 20 = 0 (𝑛 + 5)(𝑛 + 4) = 0 𝑛 = −5 or 𝑛 = −4 𝑥 = − or 𝑥 = − − ,−
2. 2𝑥 − 3𝑥 − 2 = 0 𝑛 − 3𝑛 − 4 = 0 (𝑛 + 1)(𝑛 − 4) = 0 𝑛 = −1 or 𝑛 = 4 𝑥 = − or 𝑥 = − , 2
71
3. 12𝑥 + 29𝑥 + 15 = 0 𝑛 + 29𝑛 + 180 = 0 (𝑛 + 20)(𝑛 + 9) = 0 𝑛 = −20 or 𝑛 = −9 𝑥 = − or 𝑥 = − − ,−
4. 4𝑥 + 109𝑥 + 225 = 0 𝑛 + 109𝑛 + 900 = 0 (𝑛 + 100)(𝑛 + 9) = 0 𝑛 = −100 or 𝑛 = −9 𝑥 = − or 𝑥 = − −25,− 5. 20ac = and 9b = , so use 5 and 4 210 5 4 2 05 (2 1) 2(2 1) 0(5 2)(2 1) 0x x x
x x xx x
+ + + =+ + + =
+ + =
255 2 0xx
+ == − 122 1 0x
x+ =
= − { }1 22 5,− −
6. 24ac = and 14b = − , so use –12 and –2 23 12 2 8 03 ( 4) 2( 4) 0(3 2)( 4) 0x x x
x x xx x
− − + =− − − =
− − =
233 2 0xx
− == 4 04x
x− =
=
{ }23 ,4 7. 4ac = − and 3b = − , so use –4 and 1 22 4 2 02 ( 2) 1( 2) 0(2 1)( 2) 0x x x
x x xx x
− + − =− + − =
+ − =
122 1 0xx
+ == − 2 02x
x− =
=
{ }12 ,2−
8. 22
4 ( 1) 154 4 154 4 15 0x xx x
x x
− =− =
− − =
60ac = − and 4b = − , so use 6 and –10 24 6 10 15 02 (2 3) 5(2 3) 0(2 5)(2 3) 0x x x
x x xx x
+ − − =+ − + =
− + =
522 5 0xx
− == 322 3 0x
x+ =
= − { }3 52 2,−
72
9. ( ) ( )
22 4542 222 22 45 44 42 494 49 74 27 2 72 2 2
4 8 452 12 1( 1)1 1
b
x xx x
x x
x
x
x
+ =+ =
= =
+ + = +
+ =
+ = =
= − = −
± ±
± ±
{ }5 92 2,−
10.
( ) ( )
( )
22 2 13 32 2 13 32 21 12 3 92 2 1 1 13 9 3 921 43 91 4 23 9 31 23 3
3 2 1 00b
x xx x
x x
x x
x
x
x
− − =− − =
− =
= − =
− + = +
− =
− = =
=
± ±
±
{ }131,− 14.8 Word Problems – Quadratic Equations 1.
22 2 482 48 0( 8)( 6) 08 0 6 08 ( 6)x x
x xx x
x or xx reject x
− =− − =
− + =− = + == = −
The number is 8.
2.
2 22 2 22( 8) 10416 64 1042 16 40 02( 8 20) 02( 10)( 2) 02 ( 10)
x xx x x
x xx xx x
x reject x
+ + =+ + + =
+ − =+ − =
+ − == = −
Numbers are 2 and 10. 3.
( )( )
22 36 55 36 09 4 09 0 4 09 ( 4)x x
x xx x
x or xx reject x
− =
− − =− + =
− = + == = −
Positive solution is 9.
4. ( )( )
22 5 245 24 08 3 08 0 3 08 ( 3)x x
x xx x
x or xx reject x
= +
− − =− + =
− = + == = −
The positive number is 8. 5.
2 ( 5) 5005 500 0( 25)( 20) 020 ( 25)w w
w ww w
w reject w
+ =
+ − =+ − =
= = −
Width is 20 and length is 25.
6. 22
( 7)( 3) 244 21 244 45 0( 9)( 5) 05 ( 9)t tt t
t tt t
t reject t
+ − =+ − =
+ − =+ − =
= = −
Tamara is 5 years old.
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7. 2 ( 2) 632 63 0( 9)( 7) 09 ( 7)
x xx xx x
x reject x
+ =+ − =
+ − == − =
Numbers are –9 and –7.
8. 2 2
( 2) ( 4) 82 1212 0( 4)( 3) 03 ( 4)x x x
x x xx x
x xx reject x
+ = + ++ = +
+ − =+ − =
= = −
Numbers are 3, 5, and 7. 9.
2 2( 1)( 10) 9011 10 9011 80 0( 16)( 5) 05 ( 16)
x xx x
x xx x
x reject x
+ + =+ + =
+ − =+ − =
= = −
Numbers are 5 and 6.
10. 2 2
( 4) 2( 2) 204 2 4 202 24 0( 6)( 4) 04 ( 6)x x x
x x xx xx x
x reject x
+ = + ++ = + +
+ − =+ − =
= = −
Ages are 4, 6, and 8 11.
2 2 22 22
( 1) 62 1 362 2 35 0x x
x x xx x
+ + =+ + + =
+ − = 2 284 { 4.714x −= ≈ −± ,3.71} 3.71 and 4.71
12. ( )
216 32 016 2 016 0 2 02 ( 0)x xx x
x or xx reject x
− + =− − =
− = − == =
2 seconds. 13. (a) 𝑤(0) = 120 gallons (b) 8 2464 { 5.810t = ≈ −
−± ,4.2} 4.2 mins.
14. (a) ( 40)( 60)x x+ + (b) 2400, 60x , 2x , 40x (c) Both equal 2 100 2400x x+ +
74
Chapter 15. Parabolas
15.1 Find Roots Given a Parabolic Graph 1. (c) 2. (a) 3. 2 and 4 4. 1 and 5 5. –2 and 3 6. –4 and 2 7. –1 and 4 8. –1 and 5 9. –6 and 3 10. 0 (only) 15.2 Find Vertex and Axis Graphically 1. Vertex is (3,–1); axis of symmetry is x = 3. 2. Vertex is (1,–5); axis of symmetry is x = 1. 3. Vertex is (–1,7); axis of symmetry is x = –1. 4. Vertex is (–2,–3); axis of symmetry is x = –2. 5. Vertex is (3,23); axis of symmetry is x = 3. 6. Vertex is (3,8); axis of symmetry is x = 3. 7. Vertex is (0,0); axis of symmetry is x = 0. 8. (3) (3,0) and (1,0) 15.3 Finding Vertex and Axis Algebraically 1. 4 22 2bx
a− −= = =
− 2 2 4 8(2) 4(2) 8 4y x x
y= − + −= − + − = −
Vertex is (2,–4); axis of symmetry is x = 2.
2. 6 32 2bxa
−= = = 2 2 6 10(3) 6(3) 10 1y x xy
= − += − + =
Vertex is (3,1); axis of symmetry is x = 3. 3. 6 12 6bxa
− −= = = − 23( 1) 6( 1) 1 4y = − + − − = − Vertex is (–1,–4). 4. 8 22 4bx
a− −= = = − 22( 2) 8( 2) 9 1y = − + − + = Vertex (minimum) is (–2,1).
75
5. 2 12 2bxa
− −= = = − 2( 1) 2( 1) 1y = − + − = − Vertex is (–1,–1); axis of symmetry is x = –1. 6. 0 02 6bx
a−= = = 23(0) 1 1y = + = Vertex is (0,1); axis of symmetry is x = 0. 7. ( 8) 22( 2)x − −= = −
− 22( 2) 8( 2) 3 11y = − − − − + = 2x = − and (–2,11)
8. 2
( 2) 12( 1)( 1) 2( 1) 1 2x
y
− −= = −−
= − − − − + =
1x = − and ( 1,2)− 9. 8 162 0.5bxa
−= = = 20.25(16) 8(16) 800 736y = − + = Vertex is (16,736). So, producing 16 units will result in the minimum cost of $736. 10. ( )
160 202 2 4bwa
− −= = =−
workers
15.4 Graph Parabolas 1.
2.
76
3.
4.
5.
6.
7.
8.
77
9.
Roots are –1 and 3.
10.
Roots are –4 and 2. 11. a)
6 32 2( 1)bxa
− −= = =−
seconds
12. a)
40 2.52 2( 8)bta
− −= = =−
seconds
78
13. (a)
(b) 12 32 2( 2)bxa
− −= = =−
22(3) 12(3) 18y = − + = feet
14. (a)
(b) 20 102 2( 1)bxa
− −= = =−
210 20(10) 100y = − + = feet 15.5 Vertex Form 1. 2 26 92 2b = =
2 22 22
6 1010 61 6 91 ( 3)( 3) 1y x xy x xy x xy xy x
= + +− = +− = + +− = += + +
vertex: (–3,1)
2. 2 210 252 2b = =
2 22 2210 2121 104 10 254 ( 5)( 5) 4
y x xy x xy x xy xy x
= + +− = ++ = + ++ = += + −
vertex: (–5,–4)
79
3.
4.
15.6 Vertex Form with a≠1 1. 2 242 22 2 2ba
a = = ⋅
2 222 22
2 4 66 2 44 2 4 24 2( 2 1)4 2( 1)2( 1) 4
y x xy x xy x xy x xy xy x
= + +− = +− = + +− = + +− = += + +
vertex: (–1,4)
2. 2 2183 272 2 3baa
= = ⋅
2 222 22
3 18 2626 3 181 3 18 271 3( 6 9)1 3( 3)3( 3) 1
y x xy x xy x xy x xy xy x
= + +− = ++ = + ++ = + ++ = += + −
vertex: (–3,–1) 3. ( , ) (60,150)h k = is the vertex. So, the maximum profit is $150. 4. 22 601 9002 2( 1)ba
a = − = − −
2 222 22
( ) 60 400( ) 400 60( ) 500 60 900( ) 500 ( 60 900)( ) 500 ( 30)( ) ( 30) 500
P n n nP n n nP n n nP n n nP n nP n n
= − + −+ = − +− = − + −− = − − +− = − −= − − +
vertex: (30,500) maximum daily profit: $500
80
5.
6.
81
Chapter 16. Quadratic-Linear Systems
16.1 Solve Quadratic-Linear Systems Algebraically 1. 22 5 44 5 0( 5)( 1) 0{ 5,1}
x xx x
x xx
− = −+ − =
+ − == −
When 5, 4( 5) 20x y= − = − − = When 1, 4(1) 4x y= = − = − Solutions: (–5,20) and (1,–4)
2. 2 24 1 5 32 0( 2)( 1) 0{2, 1}
x x xx x
x xx
+ + = +− − =
− + == −
When 2, 5(2) 3 13x y= = + = When 1, 5( 1) 3 2x y= − = − + = − Solutions are (2,13) and (–1,–2) 3.
2 22 1 3 56 0( 3)( 2) 0{3, 2}x x x
x xx x
x
+ − = +− − =
− + == −
When 3, 3(3) 5 14x y= = + = When 2, 3( 2) 5 1x y= − = − + = − Solutions: (3,14) and (–2,–1)
4. 2 24 2 2 12 3 0( 3)( 1) 0{ 3,1}
x x xx x
x xx
+ − = ++ − =
+ − == −
When 3, 2( 3) 1 5x y= − = − + = − When 1, 2(1) 1 3x y= = + = Solutions: (–3,–5) and (1,3) 5. 3 1 3 1y x y x+ = → = − +
22 7 22 3 110 21 0( 7)( 3) 0{ 7, 3}x x xx xx x
x
+ + = − ++ + =+ + =
= − −
When 7, 3( 7) 1 22x y= − = − − + = When 3, 3( 3) 1 10x y= − = − − + = Solutions: (–7,22) and (–3,10)
6. 2 23 6 3 62 6 2 6y x y xx y x y x x
+ = → = − += + + → = − −
2 22 6 3 612 0( 4)( 3) 0{ 4,3}
x x xx x
x xx
− − = − ++ − =
+ − == −
When 4, 3( 4) 6 18x y= − = − − + = When 3, 3(3) 6 3x y= = − + = − Solutions: (–4,18) and (3,–3) 7.
2 22 8 2 193x x x
xx
+ − = +==± 2(3) 1 72( 3) 1 5y
y= + == − + = −
(3,7) and (–3,–5)
8.
2 26 9 9 193 10 0( 5)( 2) 0{ 5,2}x x x
x xx x
− + = − ++ − =
+ − =− 9( 5) 19 649(2) 19 1y
y= − − + == − + =
(–5,64) and (2,1)
82
9.
2 2 5 17 54 12 0( 6)( 2) 0{ 6,2}x x x
x xx x
+ − = −+ − =
+ − =− ( 6) 5 11(2) 5 3y
y= − − = −= − = −
(–6,–11) and (2,–3)
10.
( ){ }
2 2 6 3 64 04 00,4x x x
x xx x
− − = −
− =− =
( )
( )3 0 6 63 4 6 6y
y
= − = −
= − = (0,–6) and (4,6)
16.2 Solve Quadratic-Linear Systems Graphically 1. (2) (–3, 5) 2. (1) (8, 9) 3.
Solutions: (1,3) and (–3,–5)
4.
Solutions: (3,14) and (–2,–1) 5.
Solutions: (2,13) and (–1,–2)
6.
Solutions: (–1,–4) and (–2,–5)
83
7.
Solutions: (0,5) and (4,–3)
8.
Solutions: (–4,–5) and (1,0) 9.
Solutions: (–1,8) and (5,–4)
10.
Solutions: (2,5) and (5,2)
84
11.
Solutions: (–4,12) and (2,0)
85
Chapter 17. Cubic and Radical Functions
17.1 Cubic Functions 1. The function has one root. x y –2 –5 –1 2 0 3 1 4 2 11
2. The function has three roots. x y –3 5 –2 15 –1 13 0 5 1 –3 2 –5 3 5
17.2 Square Root and Cube Root Functions 1. (A) 2. (A)
86
3.
(a) (b) (c) 670 (d) 12
87
Chapter 18. Transformations of Functions
18.1 Translations 1. (4) 2. (2) 3. (3) 4. 1y x= − 5. 4y x= + 18.2 Reflections 1. (1) 2. (4) 3. (1) 4. 2( 1)y x= − − 5. 𝑦 = √−𝑥 + 3 6. 𝑦 = (−𝑥 − 1) − 2 18.3 Dilations 1. (2) 2. (2) 3. (4) 4. (3) 5. 212y x= ; wider 6.
becomes narrower (horizontally shrinks)
7.
becomes wider (horizontally stretches)
88
8.
becomes narrower (vertically stretches)
89
Chapter 19. Discontinuous Functions
19.1 Piecewise-Defined Functions 1. ( 3) 3, (0) 1, (2) 3f f f− = = = 2.
Not a continuous function. 3.
4.
This is a continuous function. 5.
1( ) 2 1x xf x x x− ≤ −= + > −
6. 4 0 2( ) 2( 2) 8 2 616 6 8t tc t t t
t
< ≤= − + < ≤ < ≤
90
19.2 Step Functions 1. (6.25) 3 6.25 5 3(7) 5 26f = + = + = 2.
