4.1 graphs in the coordinate plane
TRANSCRIPT
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 1
4.1 Graphs in the Coordinate Plane
Answers
1. (-2, -2)
2. (5, 6)
3. (2, -6)
4. (3, -4)
5. (-5, 5)
6. (-2, 3)
7. – 14.
15. – 22.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 2
23.
24.
25. (2, 0) is not in a quadrant, it is on the line between QI and QIV.
26. All coordinate pairs describe points by distance from (0, 0).
27. a) 𝑦 = 3𝑥
b) 𝑦 = {0, 3, 6, 9, 12,15
c)
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 3
28.
29.
The percentage of employed men trended downward between 1973 and 2009.
58
60
62
64
66
68
70
72
74
76
78
1973 1980 1986 1992 1997 2002 2005 2007 2009
Pe
rce
nta
ge
% of Men Employed in the U.S.
𝑥 0 2 4 6 8
𝑦 8 8.5 9 9.5 10
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 4
4.2 Graphs of Linear Equations
Answers
1. Solutions to an equation in two variables graph as a line. Solutions to an equation in one
variable are a single point.
2. 𝑦 = 3𝑥 − 7 :
3. 𝑦 = 2𝑥 + 7 :
4. 𝑦 = 0.7𝑥 − 4 :
𝑥 -1 0 1 2 3
𝑦 -10 -7 -4 -1 2
𝑥 -3 -2 -1 0 1
𝑦 1 3 5 7 9
𝑥 1 2 3 4 5 𝑦 -3.3 -2.6 -1.9 -1.2 -.5
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 5
5. 𝑦 = 6 − 1.25𝑥 :
𝑥 -1 0 1 2 3 𝑦 7.25 6 4.75 3.5 2.25
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 6
4.3 Horizontal and Vertical Line Graphs
Answers
1. 𝑦 = 0
2. 𝑥 = 0
3. 𝑥 = 6
4. 𝑦 = −2
5. 𝑦 = −7
6. 𝑦 = 5
7. 𝑥 = −4
8. – 10.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 7
4.4 Applications of Linear Graphs
Answers
1. $9.00
2.
a) 32°
b) ≈ −17
c) 100°
3. (𝑑 − 5)0.7 = 𝑒 : (50 − 5)0.7 = 𝑒 ∶ 31.50 = 𝑒
4. ≈ 9𝑙𝑏𝑠
5. ≈ 20𝑙𝑏𝑠
6. ≈ 5.5𝑘𝑔
7. ≈ 7.5𝑘𝑔
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 8
4.5 Intercepts by Substitution
Answers
1. An intercept is a coordinate describing a value on the 𝑥 or 𝑦-axis.
2. Any point on the 𝑥-axis is an 𝑥-intercept, so the point will have the form: (𝑥, 0)
3. (0, −6)(2, 0)
4. (0, 4)(2, 0)
5. (0, −21) (1.5, 0)
6. (0, 7) (21
3, 0)
7. A vertical line (𝑥 ≠ 0) will have only an 𝑥-intercept.
8. 𝑦 = 5 has only a 𝑦-intercept as it runs above and parallel to the 𝑥-axis.
9. 𝑥 = −4
10. An infinite number.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 9
4.6 Intercepts and the Cover-Up Method
Answers
1. The “Cover-Up” method involves solving the equation for the constant, then covering up each
variable and associated coefficient in turn and solving for the visible variable.
2. Answers will vary
3. (0, −2.5) (3, 0)
4. (0, 1.25) (−12
3, 0)
5. (0, −11
7) (−
11
2, 0)
6. (0, 2.5) (5, 0)
7. (0, 3) (−1.5, 0)
8. (2, 0) (0, −6)
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 10
9. (0, −5) (5, 0)
10. (0, 8) (8, 0)
11. (0, 0)
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 11
12. (24, 0) (0, 3)
13. (0 − 2) (4, 0)
14. (10
7, 0) (0, −2)
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 12
15. (0, 3) (−3
4, 0)
16. (0, 0)
17. (0, 5) (1, 0)
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 13
18. (0, 3) (−6
7, 0)
19. Distribute the 3 and the 2
20. 3𝑠 + 𝑏 = 10 : Where 𝑠 is the horizontal axis
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 14
21. 7.5𝑥 + 4.5𝑦 = 900 :
22. 6𝑡 + 3𝑓 = 36 : Where 𝑡 is the horizontal axis
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 15
4.7 Slope
Answers
1. Slope is the relationship between change in horizontal and vertical location between different
points on the graph of a linear equation.
2. Answers will vary
3. A vertical line has an undefined slope, since it represents division by 0.
4. A horizontal line has a slope of 0, since it represents zero divided by a constant.
5. a) (−1, −6) (3, 6) =12
4=
3
1= 3 b) (−6, −2) (0, 1) =
3
6=
1
2
6. c) (−1, 6) (5, −6) = −12
6= −2 d) (−2, −4) (4, 2) =
6
6= 1
7. d/e) (4, −6) (4,2) =8
0= undefined f) (−6, −2) (3, 1) =
3
7
8. −7
5
9. 16
6= 2
2
3
10. 14
−5= −2
4
5
11. 4
0 = undefined
12. −18
−18= 1
13. 2
−5= −
2
5
14. 5
1
4
−21
2
= −21
10= −2
1
10
15. 5
6
16. 0
21= 0
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 16
17. −21
0 = undefined
18. 2
3
19. −5
5= −1
20. 1
42
3
=3
8
21. Slope = 0
22. Slope = undefined
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 17
4.8 Rates of Change
Answers
1. Slope and Rate of Change are the same.
2. Section ‘B’ represents the stop light, and Section ‘E’ represents the tire change. All other
sections are motion, with steeper slopes representing shallower-angled hillsides.
3. 155
3= 51.66̅
4. 605
2
=24
1
5. Answers will vary but should describe a situation wherein Geoffrey is associated with an
increase in altitude of 10ft/sec
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 18
4.9 Slope-intercept Form
Answers
1. 𝑚 = 2, 𝑏 = 5
2. 𝑚 = −0.2, 𝑏 = 7
3. 𝑚 = 1, 𝑏 = 0
4. 𝑚 = 0, 𝑏 = 3.75
5. 𝑚 =2
3, 𝑏 = −9
6. 𝑚 = −0.01, 𝑏 = 10,000
7. 𝑚 =3
5, 𝑏 = 7
8. Not a line, equation describes the point 𝑥 = −8
5
9. 𝑚 = −2
4= −
1
2
10. 𝑚 = 0
11. 𝑚 = −2
1= −2
12. 𝑚 =4
1= 4
13. 𝑚 = −4
3
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 19
14. 𝑚 =2
5
15. 𝑚 = −2
8= −
1
4
16. 𝑚 = −1, 𝑏 = 0
17. 𝑚 = −2
3, 𝑏 = 1
1
3
18. 𝑚 = −1
5, 𝑏 = −1
19. 𝑚 = 3, 𝑏 = 1
20. 𝑚 = 0, 𝑏 = 3
21. 𝑚 =1
2, 𝑏 = −2
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 20
4.10 Graphs Using Slope-Intercept Form
Answers
1.
2.
3.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 21
4.
5.
6.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 22
7.
8.
9. 𝑚 = 2
10. 𝑚 = −0.2
11. 𝑚 = −1
12. 𝑚 = 0
13. 𝑚 = −1
5
14. 𝑚 = −5
15. 𝑚 = −3
16. 𝑚 = 3
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 23
4.11 Direct Variation
Answers
1. Direct variation indicates that both variables in a situation rise and fall at a constant relative
rate and when one variable is 0, the other is also.
2. 𝑦 = (𝑘)𝑥, 𝑘 is the constant of proportionality
3. 𝑚 = 𝑘ℎ
4. 𝑀 = 𝑘𝐸 𝑜𝑟 𝑊𝑀 = 𝑘𝑊𝐸
5. 𝑉 = 𝑘𝐾 𝑜𝑟 𝑉 = 𝑘𝑇 𝑜𝑟 𝑉𝑔 = 𝑘𝑇𝐾
6. 𝑝 = 𝑘𝑚
7. 𝑐 = 𝑘𝑝 𝑜𝑟 𝐴 = 𝑘𝑝
8. This is an inverse variation, 𝑦 decreases as 𝑥 increases.
9. There is only one variable (𝑦).
10. There is only one variable (𝑥).
11. The point (0, 0) is not a solution.
12. The point (0, 0) is not a solution.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 24
13.
14.
15.
16.
17. No, it is not direct since (0, 0) is not a solution.
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 25
4.12 Applications Using Direct Variation
Answers
1. Answers will vary: Cross Products and Isolating the Variable
2. False
3. 𝑘 = 12
4. 𝑘 = 47
5. 𝑘 = −7
6. 𝑘 = 5.15
7. 𝑘 = 8
8. $12.50
5=
$𝑥
2 ∶ 𝑥 = $5
9. 57.14 minutes
10. 12 minutes
11. a) R = 9Ω :
b) 585 V
12. ≈ 4.78 𝑖𝑛
13. Noon (14 hrs after start)
14. $51,853.45
15. a) 𝑘 = 1.2
b) 8.4 N
c) 19.167 cm
16. a) 𝑦 = 3𝑥
b) 𝑦 = −2𝑥
c) 𝑦 = −1
5𝑥
d) 𝑦 =2
9𝑥
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 26
4.13 Function Notation and Linear Functions
Answers
1. 𝑓(𝑥) is read as “f of x”
2. Answers will vary: Function notation allows the same equation to be used with different
values, particularly useful in the sciences.
3. A function is an equation with two variables where each input relates to one and only one
output.
4. Function, each 𝑥 relates to a unique 𝑦.
5. Not a function, each 𝑥 (aside from the vertex) relates to two different 𝑦’s.
6. Not a function, each 𝑥 (aside from the vertices) relates to two different 𝑦’s.
7. Function, each 𝑥 relates to a unique 𝑦.
8. 𝑓(𝑥) = 7𝑥 − 21
9. 𝑓(𝑥) = −2
3𝑥 + 4
1
2
10. 𝑓(𝑥) =1
9𝑥 −
1
3
11. 𝑓(𝑥) = 6
12. 𝑓(𝑡) = 65𝑡 + 100
13. 𝑓(𝐶) = 1.8𝐶 + 32
14. 𝑓(𝑚) = 0.10𝑚 + 25,000
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 27
15. a) 9
b) -11
c) 3
d) −2𝑧 + 3
16. a) 1.1
b) 8.1
c) 3.2
d) 0.7𝑧 + 3.2
17. a) 25
11
b) −25
11
c) 0
d) 5(2−𝑧)
11
18. a) 8.5
b) 28.5
c) 4
d) 1
2𝑧2 + 4
19. a) 4.5
b) −1
2
c) 3
d) 3 −1
2𝑧
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 28
4.14 Graphs of Linear Functions
Answers
1. a) 𝑓(𝑥) = 8𝑥 + 100 b) 180 min c) 316 min d) 21.25 lbs
2. 212 ℉ is the boiling point of water in degrees Fahrenheit.
3. Answers may vary: 𝑚(𝑑) =𝑑
0.16 : 𝑚(20) =
20
0.16= 125 𝑚𝑖𝑛
4. 𝑏(ℎ) = 330ℎ : 𝑏(. 75) = 247.50 𝑐𝑎𝑙𝑜𝑟𝑖𝑒𝑠 : The number of calories burned in 45 mins
5. a) 𝑏(𝑤) = 𝑏 − 55𝑤 :
𝑏) 𝑏(10) = 100 : She will run out of money at 12 weeks
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Chapter 4 – Graphs of Linear Equations and Functions Answer Key
CK-12 Basic Algebra Concepts 29
4.15 Problem Solving with Linear Graphs
Answers
1. a) 40 ℎ𝑟𝑠 ≈ $350
b) 30 hrs
c) 𝑚 =200
30=
20
3= 6
2
3 Aatif earns $6.75 per hr
d) 𝑏 = $50 : Aatif makes a flat rate of $50 per job before his $6.75 per hr
2. 8 inches (𝑦 = 2𝑥 + 8)
3. $668
4. 52
3𝑖𝑛 (𝑙 = 5
2
3−
1
3𝑚)
5. 0.0023𝑖𝑛
6. 56 glasses (technically 55.55̅)
7. $2.53
8. 3.375 mi @ 45 mins : 4.3 mi @ 55 mins