Name:____________________________ Block:_________ Teacher: Miss Zukowski Date Submitted: / / 2018
Unit #6 Part II: Slope, Intercepts & Linear Relations Submission Checklist: (make sure you have included all components for full marks)
โ Cover page & Assignment Logโ Class Notesโ Homework (attached any extra pages to back)โ Quizzes (attached original quiz + corrections made on separate page)โ Practice Test/ Review Assignment
Assignment Rubric: Marking Criteria
Excellent (5) - Good (4) - Satisfactory (3) - Needs Improvement (2) - Incomplete (1) - NHI (0) Self
Assessment Teacher
Assessment
Notebook โ All teacher notes completeโ Daily homework assignments have been recorded &
completed (front page)โ Booklet is neat, organized & well presented (ie: name on,
no rips/stains, all pages, no scribbles/doodles, etc)
/5 /5
Homework โ All questions attempted/completedโ All questions marked (use answer key, correct if needed)
/5 /5
Quiz (1mark/dot point)
โ Corrections have been made accuratelyโ Corrections made in a different colour pen/pencil
(+ยฝ mark for each correction on the quiz)/2 /2
Practice Test (1mark/dot point)
โ Student has completed all questionsโ Mathematical working out leading to an answer is shownโ Questions are marked (answer key online)
/3 /3
Punctuality โ All checklist items were submitted, and completed on theday of the unit test. (-1 each day late)
/5 /5
Comments: /20 /20
5) Introduction to Linear Relations
โข Linear relations are ______________________________ relationships
โข Linear relations are always ______________________
o One exception:
Part 1: Algebra Review
Example #1: Solve the following equations for y.
a) 4x + 6y = 24 b) ๐ฅ๐ฅ2
+ ๐ฆ๐ฆ3
= 1
Part 2: Graphing Using Slope and Y-Intercept
We know two ways of graphing equations:
1. _____________________________________________________________________________________
Best used when: _______________________________________________________________________
2. _____________________________________________________________________________________
Best used when: _______________________________________________________________________
A new third way! Graphing from the equation ๐๐ = ๐๐๐๐ + ๐๐
m = ________________________________ b = ________________________________
Example #2: ๐ฆ๐ฆ = 12๐ฅ๐ฅ + 3
STEPS
1. Solve for y (ifnecessary)
2. Plot the y-intercept3. Use the slope ๏ฟฝ๐๐๐๐๐๐๐๐
๐๐๐๐๐๐๏ฟฝ to
plot 2nd point4. Keep plotting more
points using the sameslop until you have atleast 4 points
ASSIGNMENT # 5pages #3-10 questions #1-25
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KeyTermsTerm Definition Example
LinearRelation
LinearFunction
Orderedpair
Slope
y-intercept
x-intercept
Slope-interceptformofalinearequation
Point-slopeformofalinearequation
Generalformofalinearequation
ParallelLines
PerpendicularLines
DependentVariable
IndependentVariable
LinearFunction
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IntroductiontoLinearRelationsWehaveexaminedrelationsbetweentwoquantitiesearlierinthiscourse.Nowwewillnarrowourfocustoexamineonlylinearrelations.Linearrelationsarestraightlinerelationships.Eachoutputvalueisproportionatetotheinputvalue.Thatis,thechangeoccursataconstantrate.Eg.Anemployeethatworksforanhourlywage($10perhour).
Thisisalinearrelationshipbecausetheemployeesearningsincreaseataconstantrate.TheequationthatrelatestheEarningsandthehoursworkedis๐ฌ = ๐๐๐.
1. Plottherelationshipdescribedaboveifthedomainis{0,1,2,3,4,5,6,7}.
2. Whatistheshapeofthegraphyoujustplotted?
3. Istherelation๐ธ = 10โafunction? 4. Whichvariableintherelation๐ธ = 10โisthedependentvariable
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๐ฆ = 3๐ฅ x y โ2
โ6
โ1
0
1
2
5. Challenge#1:If๐ฆ = 3๐ฅ,findthemissingvaluesofy.
6. Whatnamedowegivethepairsofnumbersineachrow?
7. Does(โ8,โ24)satisfytheequationabove.
8. Howmanypairsofnumbersaretherethatsatisfythatequation?
9. Whatshapedoyouseeifyouploteachofthepairsofnumbersinthetableabove?
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๐ฆ = 3๐ฅ x y -2 -6-1 -30 0 1 3 2 6
๐ฆ = 3๐ฅ x y
๐ฆ = 3๐ฅ
x y
โ2 โ6
โ1
0
1
2
Findingcoordinatesfromanequation:
ATableofValuesisatoolusedtofindorderedpairsfromanequation.
Thisisatableofvaluessetuptofind5orderedpairsfortheequation๐ฆ = 3๐ฅ.
Step1: Select5inputvaluesofxandwritetheminthexcolumn.Eg.-2,-1,0,1,2
Step2: Substitutethemintotheequationandsolvetofindtheyvalue.
**Ichosetoinputvaluesofx,butIcouldhaveselectedvaluesofyandsolvedforx(althoughIfindthatmoredifficultinthiscase).
10. Challenge#2: Usingthetableofvalues,graphtheequation๐ฆ = 3๐ฅonthegraphprovided.
This is the completed table from above.
This gives us 5 ordered pairs: (โ2, โ6), (โ1, โ3)(0,0)(1,3)(2,6)
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SomeAlgebraReview:
Whenworkingwithatableofvaluesandlinearequations,itismostusefultohaveโyโisolatedontheleft.Example: 2๐ฅ + 3๐ฆ = 12
3๐ฆ = โ2๐ฅ + 12
๐ฆ = โ67๐ฅ + 4
11. Isolatey.
2๐ฅ โ 4๐ฆ = 1612. Isolatey.
4๐ฆ โ 8๐ฅ + 12 = 013. Isolatey.
20 + 3๐ฅ = 5๐ฆ
14. Isolatey.13๐ฅ โ
32๐ฆ = 1
15. Isolatey.
๐ฅ โ3๐ฆ4+ 9 = 0
16. Isolatey.2๐ฅ3+๐ฆ2โ35= 1
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๐ฆ = 2๐ฅ + 1 x y -2 -1 0 1 2
๐ฆ = 4๐ฅ โ 1 x
-2 -1 0 1 2
3๐ฅ + ๐ฆ = โ2 x y
-2 -1 0 1 2
๐ฆ = 3๐ฅ
x y โ2 โ6 โ1 โ3 0 0 1 3 2 6
GraphingfromaTableofValues.Usingthetableofvalues,graphtheequation๐ฆ = 3๐ฅonthegraphprovided.
Step1: Fromthetableofvalueswegetthe followingorderedpairs. (โ2, โ6), (โ1, โ3), (0,0), (1,3), (2,6)Step2:Ploteachoftheorderedpairs.Step3:Drawalinethroughthepoints witharrowsoneachend.
Usethetableofvalues,ifnecessary,tographeachofthefollowingequations.
17. ๐ฆ = 2๐ฅ + 1
18. ๐ฆ = 4๐ฅ โ 1
19. 3๐ฅ + ๐ฆ = โ2
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๐ฆ =๐ฅ2 + 1
x y -2 -1 0 1 2
๐ฆ =43๐ฅ โ 2
x y -2 -1 0 1 2
๐ฆ + 3 = ๐ฅ x y -2 -1 0 1 2
2๐ฅ + 3๐ฆ = 12 x y
๐ฆ = x y
๐ฆ = x y
Usethetableofvalues,ifnecessary,tographeachofthefollowingequations.
20. ๐ฆ = :6+ 1
21. ๐ฆ = ;7๐ฅ โ 2
22. ๐ฆ + 3 = ๐ฅ
23. 2๐ฅ + 3๐ฆ = 12
24. <7๐ฆ โ ๐ฅ = 1
25. 6=>โ 2 = ๐ฅ
6) Linear & Non-Linear Graphing
Linear Equations Non-Linear Equations
Shape of Graph
Variables Present
Exponents on Variables
Examples
Example #1: Graph the following relations (using any method you choose: table of values, intercepts, or slope),
and circle whether they are linear or non-linear.
a) ๐ฆ๐ฆ = 2๐ฅ๐ฅ + 4
Linear: YES or NO
b) ๐ฆ๐ฆ = โ๐ฅ๐ฅ2
Linear: YES or NO
c) ๐ฆ๐ฆ = 2โ๐ฅ๐ฅ + 5
Linear: YES or NO
Example #2:Graph the line ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐
x-intercept:
y-intercept:
Example #3: Graph the line โ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐ = ๐๐
x-intercept:
y-intercept:
Example #4: Determine the intercepts of the following relations. As well, circle whether the relation is linear.
a) ๐ฅ๐ฅ3
+ ๐ฆ๐ฆ5
= 2
Linear: YES or NO
b) ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 9
Linear: YES or NO
-10 -5 5 10
-10
-5
5
10
-10 -5 5 10
-10
-5
5
10
ASSIGNMENT # 6pages #11-17 questions #26-63
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GraphingEquations:Areviewfromabove.UsingaTableofValues:Step1: Chooseappropriatevaluesofโxโtoputinthetable.Step2: Inputeachโxโintotheequationtofindthecorrespondingโyโ.Step3: Plotthenew-foundโorderedpairsโ.Step4: Drawalinethroughthepoints.(becarefuloftheshapeโฆnotallarelines)
Inthisunit,wewillbestudyinggraphsofstraightlinesandtheirequations.WecalltheseLINEAREQUATIONS.Anequationissaidtobelinearifitformsastraightlinewhengraphed.
EquationofaLineProperty:
Thecoordinatesofeverypointonthelinewillsatisfythe
equationoftheline.
26. Howmanypointsdoyouneedtographaline?
27. Tobesafe,atleasthowmanyshouldyouhave?
Graphtheseequationsโฆ28. ๐ฆ = โ3๐ฅ โ 1
x y
29. ๐ฆ = 5 + ๐ฅ
x y
You should REALLY memorize this!
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๐ฆ = โ2๐ฅ โ 4 x y -2 -1 0 1 2
๐ฆ = ๐ฅ6 x y -2 -1 0 1 2
๐ฆ = 5๐ฅ x y -2 0 1 4 9
๐ฆ = ๐ฅ7 x y -2 -1 0 1 2
๐ฆ = โ2๐ฅ6 + 6 x y -2 -1 0 1 2
๐ฆ = โ๐ฅ x y -2 -1 0 1 2
Graphthefollowingequations,thendetermineiftheyarelinearornot.30. ๐ฆ = โ2๐ฅ โ 4
Linear:YESorNO
31. ๐ฆ = ๐ฅ6Linear:YESorNO
32. ๐ฆ =5xLinear:YESorNO
33. ๐ฆ = ๐ฅ7Linear:YESorNO
34. ๐ฆ = โ2๐ฅ6 + 6Linear:YESorNO
35. ๐ฆ = โ๐ฅLinear:YESorNO
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36. Canyoudescribeaโruleofthumbโthatwillenableyoutotellifanequationrepresentsalinearequationornot?
Challenge#3: Theequation2๐ฅ + 4๐ฆ = 16isalinearequation.
37. Findthecoordinatesofthepointwherethelinecrossesthey-axis.(Thinkโฆwhatwouldbethevalueofโxโhere?)
38. Whatisthevalueofโxโwherethelinecrossesthey-axis?
39. Findthecoordinatesofthepointwherethelinecrossesthex-axis.
40. Whatisthevalueofโyโโwherethelinecrossesthex-axis?
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InterceptsThelocationwherealinepassesthroughthex-axisiscalledthex-intercept.Thispointwillhavethecoordinates(๐ฅ, 0)Thelocationwherealinepassesthroughthey-axisiscalledthey-intercept.Thispointwillhavethecoordinates(0, ๐ฆ)Consider:2๐ฅ + 4๐ฆ = 16
Thislinehasanx-interceptat(8,0).Anday-interceptat(0,4).Youmayseethiswrittenas:x-interceptis8y-interceptis4
Calculatinginterceptsfromanequation:Thex-interceptwillhavecoordinates(x,0).Thismeanswecansubstitute0inforyandsolvetofindthex-intercept.They-interceptwillhavecoordinates(0,y).Eg.Findthex-interceptfor 2๐ฅ + 4๐ฆ = 16 Findthey-intercept: 2๐ฅ + 4๐ฆ = 16 2๐ฅ + 4(0) = 16 2(0) + 4๐ฆ = 16 2๐ฅ = 16 4๐ฆ = 16 ๐ฅ = 8 ๐ฆ = 4
Interceptscanbeexpressedasorderedpairsorsimplyasvalues.Fortheexampleabove,thex-interceptis8orthex-interceptis(8,0).
Somenoteshereโฆ
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Calculate the intercepts and graph each equation using them. Fractions can be estimated on the grid.
41. 2๐ฅ + 3๐ฆ = 12
42. 3๐ฅ + 5๐ฆ = 30
43. 3๐ฅ โ 4๐ฆ + 24 = 0
44. 4๐ฅ + 5๐ฆ = 20
45. 6๐ฅ โ 3๐ฆ โ 18 = 0
46. 3๐ฅ โ 7๐ฆ = 21
47. 4๐ฅ + 5๐ฆ = 10
48. 9๐ฅ + 3๐ฆ โ 18 = 0
49. 3๐ฅ โ 2๐ฆ = 9
50. When do you think it would be appropriate (or the best scenario) to graph a line using the intercepts as opposed to using some other technique?
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Answer the following questions about intercepts and linear relations. For these questions the domain is all real numbers. 51. Draw a graph of a linear
relation that has two intercepts.
52. Draw a graph of a linear relation that has two positive intercepts.
53. Draw a graph of a linear relation that has two negative intercepts.
54. Draw a graph of a linear relation that has one negative and one positive intercept.
55. Draw a graph of a linear relation that has an infinite number of intercepts.
56. Draw a graph of a linear relation that has only one intercept.
57. Consider your answer to the previous question. What other type of line could you draw that would satisfy the problem?
58. Find the intercepts of the following linear equation.
๐ฅ2 +
๐ฆ3 = 1
59. Find the intercepts of the following non-linear relation.
๐ฆ = ๐ฅ6 โ 4
7) Slope & Points on Lines Warm-Up:
1. Determine the slopes of the following lines:
2. On the grid, draw a line through the point (-4, 2) with the following
slope:
a) Parallel to โ23
b) Perpendicular to โ23
3. Compute the slopes of the following line segments, using the
coordinates provided:
a) AB b) CD
Line Segment Slope
AB
CD
EF
GH
A (1, 8) C (2, 5)
B (7, 11) D (6, 2)
4. A line has a slope of 97. It passes through (-1, -4) and (x, 5). Find the value of x.
5. The slope of a line is โ75. The line passes through the point (-1, 3). Find the coordinates of another point
on the line.
Part 1: Linear Relations
Linear Relations have a _________________ rate of change/slope.
Example #1: Determine whether the following relations are linear.
a)
Linear: YES or NO
Slope:
b)
Linear: YES or NO
Slope:
c)
Linear: YES or NO
Slope:
x y
4 10
6 16
8 22
10 28
x y
4 10
6 16
8 22
10 28
x y
5 8
10 7
15 6
x y
4 10
6 16
8 22
10 28
x y
1 -9
2 -6
3 -1
4 6
Part 2: Finding the Slope from an Equation
To find the slope from an equation, always change the equation to be in the form ๐ฆ๐ฆ = ๐๐๐๐ + ๐๐.
Example #2: Find the slope of the following lines (without graphing).
a) ๐ฆ๐ฆ = โ4๐๐ + 7 b) 3๐๐ โ 2๐ฆ๐ฆ = โ18 c) ๐ฆ๐ฆ โ 2 = 14
(๐๐ + 3)
Part 3: Determining if Points are on a Line
If a point falls on a line, it must โsatisfyโ the equation of the line (ie: it must fit into the equation and remain equal)
Example #3: Do the following points fall on the line 2๐๐ + 5๐ฆ๐ฆ = 20?
a) (5, 2) b) (-5, 6) c) (4, 0) d) (0, 4)
Example #4: Do the following points fall on the line ๐ฆ๐ฆ = โ๐๐2?
a) (2, -4) b) (-3, 9) c) (-9, -81)
ASSIGNMENT # 7pages #18-24-questions #64-97
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Slope of a Line Challenge #4:
60. Plot the following points: (โ1, โ5), (2, โ4), (5, โ3), (8,โ2)
61. Draw a line through the points you plotted.
62. Choose three sections of the line you just plotted and find their slopes. Slope of section 1: Slope of section 2: Slope of section 3:
63. What do you notice?
Some notes hereโฆ
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Slope of a Line Recall from our discussion of line segments that slope can be calculated using: ๐ = ๐๐D๐๐
๐๐D๐๐ or FGHI
FJK
For a straight line, the slopes of all segments on the line are equal. That is, if you find the slope of any two parts of the line, they will be equal.
Pick any three segments of the line and calculate the slope. Slope will always be <
7.
The equations discussed earlier in this booklet result in lines that continue in two directions. Working with slope allows us to extend the line if we need to. Remember:
โข Parallel lines have equal slopes. โข Perpendicular lines have slopes that are negative reciprocals.
64. Find the slope of the line represented by
the equation ๐ฆ = 3๐ฅ โ 5. 65. Find the slope of the line represented by
the equation 2๐ฅ + 5๐ฆ = 20.
66. Find the slope of the line represented by the equation ๐ฆ โ 4 = 3(๐ฅ โ 5).
67. Find the slope of the line represented by the equation <
7(๐ฅ + 2) = ๐ฆ โ 1.
68. Find the slope of the line below.
Slope is _____
69. Find the slope of the line below.
Slope is _____
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70. Draw a line through T(5,7) with slope 6>.
71. Draw a line through U(2, -2) parallel to the line in the previous question.
72. Draw a line through U(2, -2)
perpendicular to the line in the question above.
73. If you were given a triangle with its vertices drawn as coordinates on an x-y coordinate plane, how could you determine if the triangle was a right triangle? Do you know another way?
74. The slope of a line is 76. If the line
passes through point ๐ต(5,2), find the coordinates of another point.
75. The slope of a line is โ2.5. If the line passes through point ๐ถ(โ1,2), find the coordinates of another point.
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76. Julanyaโs internet provider charges a flat fee for the first 8 hr of access per month, plus an hourly rate for additional access. One month, 15 hr of usage cost her $25.88. The next month, 27 hr of access cost her $49.76. a) Graph the data.
77. Find the hourly rate for access above 8 hr/month.
78. What word is synonymous with rate in this unit?
79. Find the flat fee for the first 8 hours. (Where will you find this value on the graph?)
cost
Hours of usage
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Find the slope of the line passing through the points: 80. (2,1) and (6,6) 81. (โ5,2) and (4,2) 82. (โ3,0) and (3, โ4)
83. The slope of a line is -2. The line passesthrough (0,0) and (-3,y). Find the valueof y.
84. A line has a slope of 1.5. It passesthrough (-2,1) and (x,7). Find the valueof x.
85. Challenge#5: Show that (7, -1) is on the line ๐ฆ = 2๐ฅ โ 15
Algebraically: Graphically:
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The Equation of a Line As you have seen, equations such as 2๐ฅ + 3๐ฆ = 12 or 3๐ฆ = ๐ฅ + 9 or ๐ฆ = >
O๐ฅ โ 4 produce straight lines
when graphed. They are linear equations. Linear Equations may be written in several forms:
Slope-Intercept Form: ๐ฆ = ๐๐ฅ + ๐ ๐ฆ = 3๐ฅ + 2 Point-Slope Form: ๐ฆ6 โ ๐ฆ< = ๐(๐ฅ6 โ ๐ฅ<) (๐ฆ โ 2) = 3(๐ฅ โ 0) General Form: ๐ด๐ฅ + ๐ต๐ฆ+ ๐ถ = 0 3๐ฅ โ ๐ฆ + 2 = 0 Recall the Equation of a Line Property: The coordinates of every point on the line will satisfy the equation of the line. Eg.1. Show that (7, -1) is on the line ๐ฆ = 2๐ฅ โ 15 ๐ฆ = 2๐ฅ โ 15 If (7, -1) is on the line, it will satisfy the equation. (โ1) = 2(7) โ 15 Substitue the ordered pair into the equation. โ1 = 14 โ 15 Does the left side = right side? โ1 = โ1 Yes. The point IS on the line.
Determine if the following points lie on the line ๐ฆ = 2๐ฅ + 4 (HINT:substitution!)
86. (-10, 24) 87. (5,14) 88. (-7,-10)
Determine if the following points lie on the line 3๐ฅ โ 2๐ฆ + 6 = 0 89. (10, 18) 90. (0,-3) 91. (-6,-6)
Do you recall the โtext boxโ like this on page 10?
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92. Determine if the point (2,โ3) is onthe line ๐ฆ = 3๐ฅ โ 9.
-3=3(2)-9
-3=6-9
-3=-3
Explain why or why not:
Yes, it is on the line because when
the coordinates 2,-3 are substituted
into the equation, left side and right
side are equal. 93. Determine if the point (โ1, โ4) is
on the line 3๐ฅ โ 2๐ฆ โ 11 = 0.Explain why or why not:
94. Determine if the point (2,โ3) is onthe line ๐ฆ + 1 = 7:
6.
Explain why or why not:
95. Determine if the set of orderedpairs represents a linear relation.
(2,3), (3,4), (4,5), (5,6)
Explain why or why not:
96. Determine if the set of orderedpairs represents a linear relation.
(1,1), (1,2), (1,3), (1,4)
Explain why or why not:
97. Determine if the set of orderedpairs represents a linear relation.
(2,1), (3,0), (4, โ1), (5, โ2)
Explain why or why not:
8) Slope-Intercept Form
Warm-Up:
1. Determine the slope, y-intercept, and equation of the
following lines:
Part 1: Using Slope Intercept Form
Example #1: Identify the slope and y-intercept for each of the following linear equations.
Linear Relation Slope y-intercept
a) ๐ฆ๐ฆ = 42๐ฅ๐ฅ + 15
b) ๐ฆ๐ฆ = โ 9100
๐ฅ๐ฅ โ 72
c) ๐ฆ๐ฆ = 5
d) ๐ฆ๐ฆ = 22๐ฅ๐ฅ + 13
e) ๐ฆ๐ฆ = ๐ฅ๐ฅ
Line # Slope y-intercept Equation
1
2
3
4
Example #2: Write the equation of a line based on the following slopes and y-intercepts.
Linear Relation Slope y-intercept
a) โ23
7
b) 3 12
c) 16
-2
Example #3: Sketch a graph of the following equations. Make sure you have at least points on your graph!
a) ๐ฆ๐ฆ = 12๐ฅ๐ฅ + 3 b) ๐ฆ๐ฆ = โ4๐ฅ๐ฅ
c) 2๐ฅ๐ฅ + 4๐ฆ๐ฆ = 8 d) ๐ฅ๐ฅ3โ 2๐ฆ๐ฆ
6= 2
Part 2: Finding Missing Parts in the Equation of the Line
Example #4: Write the equation of a line where the slope is 10, and it passes through the following coordinates.
a) (3, 2) b) (-6, 6) c) (-1, 8)
Example #5: Write the equation of a line where the y-intercept is 3, and it passes through the following
coordinates.
a) (7, 2) b) (-4, 11) c) (9, 3)
ASSIGNMENT # 8pages #24-29 questions #98-137
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Equation of a Line: Slope-Intercept Form
98. Graph the line ๐ฆ = 6
7๐ฅ โ 5 using a table of
values.
99. Graph the line ๐ฆ = โ3๐ฅ + 5 using a table
of values.
100. What is the slope of the line above? 101. What is the slope of the line above?
102. What is the y-intercept of the line above?
103. What is the y-intercept of the line above?
104. Compare these values to the equation. What do you notice?
105. Compare these values to the equation. What do you notice?
We say the equations above are written in slope-intercept form. A general formula for an equation in slope intercept form is ๐ฆ = ๐๐ฅ + ๐ Remember, x and y are the coordinates of ANY point on the line. When substituted, they will satisfy the equation. See your work on the previous page!
The slope is the coefficient of x.
The y-intercept. (Make note of the sign)
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State the slope and y-intercept for the line represented by each equation. 106. ๐ฆ = โ3๐ฅ + 2
107. ๐ฆ = โ7>๐ฅ โ 7 108. ๐ฆ = a
6๐ฅ โ 7
6
Write the equation of each line given the slope and y-intercept. 109. m =2 ,b =-5
110. m = b7 ,b = 6
7 111. m =-3 ,b =-2
For each line below, state the slope, y-intercept, and equation. 112.
slope_____ y-intercept_____ equation:
113.
slope_____ y-intercept_____ equation:
114.
slope_____ y-intercept_____ equation:
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For each line below, state the slope, y-intercept, and equation. 115.
slope_____ y-intercept_____ equation:
116.
slope_____ y-intercept_____ equation:
117.
slope_____ y-intercept_____ equation:
118. What do you notice about the equation of the lines passing through the origin?
119. When is b positive? 120. When is b negative?
Graph the equations below by finding the slope and y-intercept from the equation. 121.
๐ฆ = โ3๐ฅ
122. ๐ฆ = >
6๐ฅ
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Graph the equations below by finding the slope and y-intercept from the equation. 123.
๐ฆ = โ๐ฅ + 3
124. 2๐ฆ = โ10๐ฅ + 12
125.
๐ฆ โ 5 =13๐ฅ โ 3
126. 2๐ฅ โ 5๐ฆ + 20 = 0
127. ๐ฅ3 โ
๐ฆ4 = 1
128. 2๐ฅ3 +
3๐ฆ4 = โ6
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Determine the value of b for the equation ๐ = ๐๐ + ๐ if the line passes through the following points. Then write the equation in slope-intercept form.
129. ๐ (2,1)
y=3x+b
1=3(2)+b
1=6+b
-5=b
Therefore:
y=3x-5
130. ๐พ(โ1,4) 131. ๐ด(3, โ2)
132. ๐ฝ(2,1) 133. ๐ iโ2, <6j 134. ๐ฟ i6
7, 1j
Determine the value of m for the equation ๐ = ๐๐+ ๐ if the line passes through the following points. Then write the equation in slope-intercept form.
135. ๐ (12,5) 136. ๐พ(1,โ3) 137. ๐ด(โ5,1)
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What you just did above is one way that you will be able to find the equation of a line. IF you have
the ____________ or the ______________, you can input the _____________ of a point on the
line to solve for the unknown part of the equation.
Then you will write the full equation with __________ and _____________ in place of m and b.
The following is another method.
9) Equations of Lines in Three Forms
The Three Forms of Writing Equations of Lines
1. Point-Slope Form:
2. Slope-Intercept Form:
3. General Form
Part 1: Writing the Equation of a Line in General Form
Example #1: Write the following equations in general form.
a) ๐ฆ๐ฆ = 4๐ฅ๐ฅ โ 10 b) 34๐ฆ๐ฆ โ 4 = 5๐ฅ๐ฅ c) 1 = โ2
5๐ฅ๐ฅ + 1
2๐ฆ๐ฆ
Part 2: Writing the Equation of a Line in Three Forms
Example #2: Use the following slope and point on the line to write the equation of the line in all three forms.
a) ๐๐ = 2 (4, 7) b) ๐๐ = โ34 (12, 4)
When you have a slope and a point, ALWAYS come up with your equations in this order:
1. Slope-Point Form2. Slope-Intercept Form3. General Form4. Check: if you plug your point back into all three equations, does it work?
c) ๐๐ = โ13 (-6. 2) d) ๐๐ = โ2
5 (4, -3)
ASSIGNMENT # 9pages #30-35 questions #138-156
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The Equation of a LineThe three forms
Slope-Intercept Form Point-Slope Form General Form
๐ฆ = ๐๐ฅ + ๐
๐ is the slope ๐ is the y-intercept
๐ฆ โ ๐ฆ< = ๐(๐ฅ โ ๐ฅ<)
Derived from ๐ = =lD=m:lD:m
Cross multiply to get point-slope form. Need one point and slope
๐ด๐ฅ + ๐ต๐ฆ + ๐ถ = 0
A must be positive. A,B,C are integers.
Write in general form.
138. ๐ฆ = 3๐ฅ โ 5 139. ๐ฆ โ 5 = ๐ฅ + 7 140. 5 โ 2๐ฅ = โ4๐ฆ + 2
141. โ<7๐ฅ โ 4๐ฆ = 2 142. ๐ฆ โ 5 = 6
7๐ฅ + 7 143. 5 = 6
7๐ฆ + 7
;๐ฅ
144. Challenge #6Write the equation of the line that passes through A(2,5) and has slope 3. Express your answer in general from and in slope intercept form.
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The Equation of a Line IMPORTANT!!! There is only one line that passes through a given point with a given slope.
Given the slope and a point: Eg.1. A line passes through A(2,5) and has slope 3. Write the equation of the line. Use the slope formula : ๐ = ๐๐D๐๐
๐๐D๐๐ Cross-Multiply. This creates the Point-Slope form of an equation.
๐(๐๐ โ ๐๐) = ๐๐ โ ๐๐ Fill in what you know. m = 3. Substitute the given point in for x1 and y1. 3(๐ฅ โ 2) = (๐ฆ โ 5) This is our equation in point-slope form. We no longer need the subscripts on x and y 3๐ฅ โ 6 = ๐ฆ โ 5 Expanded. 3๐ฅ โ ๐ฆ โ 1 = 0 Collecting the terms to the left side is called writing the equation in general form. Or ๐ฆ = 3๐ฅ โ 1 Isolate for โyโ to get the equation in slope-intercept form.
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Write the equation of the line that passes through the given point and has the given slope. Express the equation in a) point-slope form b) general form c) slope-intercept form.
145. (-2,3), -2
y-3=-2(x- -
2)
y-3 = -
2(x+2) point-slope
y-3=-2x-4
y=-2x-1 slope-
intercept
2x+y+1=0 general
146. (-5,2),2 147. (-5,-1),-2
a) y-3=- a) a)
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2(x+2)
b) 2x+y+1=0 b) b)
c) y=-2x-1 c) c)
148. (-3,4),โ๐๐
149. (2,4),๐
๐
150. (0,7), -1
a) a) a) b) b) b) c) c) c) Write the equation of the line that passes through the given point and has the given slope. Express the equation in a) point-slope form b) slope-intercept form c) general form.
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151. (3, โ6),๐ = โ3
Start with Point-
Slope formula:
y2-y1 = m(x2-x1)
y - -6 = -3(x-
3)
y + 6 = -3(x-
3)
y + 6 = -3x+9
y = -3x + 3
3x + y - 3 = 0
152. (4,6), ๐ = 5 153. (โ2, โ1),๐ = <6
a) y+6 = -3(x-3) a) a)
b) y=-3x+3 b) b)
c) 3x + y - 3 = c) c)
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0
154. (5, โ6),๐ = โ 7;
155. i<
6, 6j , ๐ = ;
7
156. (โ2,1), ๐ = 1.5
a) a) a) b) b) b)
c) c) c)
10) Equations of Lines in Three Forms...Continued...
Remember! What are the three different equations of lines?
Example 1: Write the following equations in general form.
Remember: General form means NO ___________________ or ___________________,
The ____________________________ ALWAYS has to be ___________________.
Order matters! Always write in this order: _________________________.
a) โ9๐ฅ๐ฅ + 2 = โ2๐ฆ๐ฆ + 5 b) 34๐ฆ๐ฆ โ 4 = 5๐ฅ๐ฅ โ 10 c) 1 = โ2
5๐ฅ๐ฅ + 1
2๐ฆ๐ฆ
Example 2: Write the equation of the line in all three forms.
a) ๐๐ = โ3 (5, 2) b) ๐๐ = โ25 (6,โ1)
Example 3: Write the equation of the line, in general form, that has the points (8, -2) and (6, 3).
Example 4: Write the equation of the line, in slope intercept form, that has the points (9, 5) and (-6, 4).
Example 5: Write the equation of the line, in general form, that has the points (-0.9, 0.2) and (-0.1, -0.8).
ASSIGNMENT # 10pages #36-38 questions #157-165
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157. Challenge #7:Write the equation of a line in general form given that the line passes through (3,4) and (4,6).
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Given two points:
When given two points we must first find the slope of the line. Then we will follow the same process as above.
Write the equation of the line that passes through (3,4) and (4,6).
๐ = =lD=m:lD:m
Find the slope.
๐ = OD;;D7
= 6<= 2 The slope is 2.
2 = =D;:D7
Substitute slope and ONE of the points.
2(๐ฅ โ 3) = ๐ฆ โ 4 Cross-multiply. Point-slope form
2๐ฅ โ 6 = ๐ฆ โ 4 Expand and simplify.
2๐ฅ โ ๐ฆ โ 2 = 0 Write in general form.
๐ฆ = 2๐ฅ โ 2 And in slope-intercept form if necessary.
Write the equation of the line that passes through the following two points in general form. 158. (3,4) and (4,6) Explain your reasoning
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159. (-2, -4) and (0, 6) Explain your reasoning
Write the equation of the line that passes through the following two points in general form. 160. (-5, -8) and (-7, -9) 161. (-1, -2) and (3,0) 162. (0,4) and (5, 0)
163. (8, -7) and (-6, -7) 164. i67, <;j ๐๐๐ i<
7, <7j 165. (0.3, 0.4) and (0.5, 0.7)
11) Working with Linear Relations
Part 1: Graphing a Line from an Equation
Recall the THREE ways we have to graph a line from an equation:
1. _________________________________________
2. _________________________________________
3. _________________________________________
Example 1: Graph the lines represented by the equation. Use any method you wish.
a) 3๐ฅ๐ฅ โ 2๐ฆ๐ฆ = 12 b) ๐ฆ๐ฆ โ ๐ฅ๐ฅ = 4 c) ๐ฆ๐ฆ = ๐ฅ๐ฅ2 โ 2
Part 2: Describing Equations
Example 2: Represent the following in either equations or words.
a) Represent the followingstatement with an equation:
Each element of the range is 4 less than triple the domain.
b) T or F: Each element in therange is 2 more than one-fifth ofthe domain is represented by thefollowing equation:
๐๐๐๐ โ ๐๐ = ๐๐๐๐
c) Describe the following, inwords, as a function:
๐๐๐๐ โ ๐๐๐๐ = ๐๐๐๐
Remember:
DOMAIN =
RANGE =
Part 3: Writing the Equation of a Line from a Graph
Example 3: Write the equation of the following line in slope-intercept form.
Example 4: Write the equations of the following lines in general form.
a) b)
Hereโs How You Do It:
1. Pick out two points on the line
2. Find the _________________
3. Plug the slope and ONE point into
_______________________________ OR
plug the slope and the y-intercept (if itโs a nice
point) into ______________________________
4. Transform your equation into the form asked.
ASSIGNMENT # 11pages #40-43 questions #166-188
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Working With Linear Equations:
โข Be able to convert equations between general form and slope-intercept form.โข Be able to graph equations given to you in either form.โข Be able to make comparisons based on parallel and perpendicular lines.
Eg.1. Graph the line 3๐ฅ + 2๐ฆ โ 6 = 0.
Your Options: 1) use intercepts 2)make a table of values 3)convert to slope-intercept form
I chose option 1 because this equation allows for easy calculations to find both intercepts.
3(0) + 2๐ฆ โ 6 = 0 2๐ฆ โ 6 = 0 2๐ฆ = 6 ๐ฆ = 3 The y-intercept is 3.
3๐ฅ + 2(0) โ 6 = 0 3๐ฅ โ 6 = 0 3๐ฅ = 6 ๐ฅ = 2 The x-intercept is 2.
Plot the two points & draw the line through them.
My second choice would have been option 3, conversion to slope-intercept form. 3๐ฅ + 2๐ฆ โ 6 = 0 2๐ฆ = โ3๐ฅ + 6 ๐ฆ = D7
6๐ฅ + 3
Plot the y-intercept then use the slope to plot another point, draw a line through the two points.
Graph the lines represented by each of the following equations. Use any method. 166. 3๐ฅ + 2๐ฆ + 6 = 0 167. 5๐ฅ + 2๐ฆ โ 10 = 0 168. ๐ฅ โ ๐ฆ = 10
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Graph the lines represented by each of the following equations. Use any method. 169. 3๐ฅ + 2๐ฆ โ 4 = 0
170. Explain your strategy:
171. ๐ฅ โ 4๐ฆ = 0
Explain your strategy:
172. 2(๐ฅ โ 3) = ๐ฆ โ 3
Explain your strategy:
Match the following graphs to their corresponding equations. Choose the best match. 173.
a) ๐ฅ โ 3๐ฆ + 3 = 0
b) 3๐ฅ โ ๐ฆ โ 12 = 0
c) 3๐ฅ + ๐ฆ โ 12 = 0
d) None of the above
174.
a) 4๐ฅ โ 3๐ฆ + 9 = 0
b) 3๐ฅ โ 4๐ฆ + 9 = 0
c) 3๐ฅ + 4๐ฆ โ 9 = 0
d) None of the above
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175. Which equation on the right represents the graph below?
a) 2๐ฅ โ 3๐ฆ + 6 = 0
b) 3๐ฅ โ 2๐ฆ + 6 = 0
c) 3๐ฅ + 2๐ฆ + 6 = 0
d) None of the above
176. Which of the following equations represents the word statement โeach element of the range is equal to one less than double an element in the domain.โ
a. 2๐ฅ โ ๐ฆ โ 1 = 0 b. ๐ฅ โ 2๐ฆ = โ1 c. 2๐ฅ + ๐ฆ + 1 = 0
177. Which of the following equations represents the word statement โeach element of the range is equal to two more than one third an element in the domain.โ
a. 3๐ฅ โ ๐ฆ = 6 b. ๐ฅ โ 3๐ฆ = โ6 c. ๐ฅ + 3๐ฆ + 6 = 0
178. Which of the following equations represents the word statement โtriple each element of the range is equal to one less than double an element in the domain.โ
a. 2๐ฅ โ 3๐ฆ = โ1 b. 2๐ฅ โ 3๐ฆ = 1 c. 2๐ฅ + 3๐ฆ = 1
179. Write a โword statementโ to describe the following equation.
๐ฆ = 3๐ฅ โ 2
180. Write a โword statementโ to describe the following equation. 2๐ฅ + 4๐ฆ โ 8 = 0
181. Write a โword statementโ to describe the following equation. 3๐ฅ โ 5๐ฆ = 20
182. Which of the following equations represent the same line as ๐ฆ = 3๐ฅ โ 2?
Circle all that apply.
a. 3๐ฅ = ๐ฆ + 2 b. 3๐ฅ โ ๐ฆ โ 2 = 0 c. ๐ฆ โ 3๐ฅ = โ2 d. none
183. Which of the following equations represent the same line as 5๐ฅ โ 2๐ฆ + 10 = 0?
Circle all that apply.
e. ๐ฆ = >6๐ฅ + 5
f. 6>(๐ฅ โ 4) = ๐ฆ โ 15
g. ๐ฅ = 6>๐ฆ โ 2
h. none
184. Which of the following equations represent the same line as ๐ฆ โ 4 = 2(๐ฅ + 1)?
Circle all that apply.
i. 2๐ฅ โ ๐ฆ + 6 = 0 j. ๐ฆ = 2๐ฅ + 6 k. 2๐ฅ + ๐ฆ = 6 l. none
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Find the slope and y-intercept, write the equation in slope-intercept form, then in general form.
185.
m_____ b______
slope-intercept form___________
general form____________
186.
m_____ b______
slope-intercept form___________
general form____________
187.
m_____ b______
slope-intercept form___________
general form____________
188.
m_____ b______
slope-intercept form___________
general form____________
12) Parallel & Perpendicular Equations
We Always Need Two Pieces of Information to Write Equations:
1. _________________________________________________
2. _________________________________________________
Example 1: Write the equation of a line parallel to 2๐ฅ๐ฅ โ 4๐ฆ๐ฆ + 3 = 0 with the point (5, 1) in general form.
Example 2: Write the equation of a line perpendicular to 5๐ฆ๐ฆ โ 15๐ฅ๐ฅ = 1 with the point (7,โ3) in general form.
REMEMBER:
Parallel:___________________________
Perpendicular: ______________________
Example 3: Write the equation of a line perpendicular to the line through (7, 5) & (10, 9) passing through the point (โ1, 8) in general form.
Example 4: Two perpendicular lines intersect on ๐ฅ๐ฅ axis. One line is ๐ฆ๐ฆ = โ12๐ฅ๐ฅ + 5 in general form.
Example 5: Write the equation of a line parallel to โ2๐ฆ๐ฆ + ๐ฅ๐ฅ = 8 with the same y-intercept as 10๐ฆ๐ฆ + 32๐ฅ๐ฅ = 100 in general form.
ASSIGNMENT # 12pages #43-48 questions #189-200
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ParallelandPerpendicularLinesRecall:
โข Parallellineshaveequalslopes.โข Perpendicularlineshaveslopesthatarenegativereciprocals.
For each line below, state the slope of a line that would be (a) parallel (b) perpendicular. 189. ๐ฆ = 3๐ฅ โ 5
a)
b)
190. ๐ฆ โ 5 = โ67๐ฅ
a)
b)
191. 5๐ฅ โ 3๐ฆ = 14
a)
b)
192. CHALLENGE.
Writetheequationofthelineparallelto5๐ฅ โ 8๐ฆ + 12 = 0andthroughthepoint(-2,3).
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SOLUTIONtoQ.192.Writetheequationofthelineparallelto5๐ฅ โ 8๐ฆ + 12 = 0andthroughthepoint(-2,3).
**Parallelmeanssameslope.Soweneedtofindslopeof5๐ฅ โ 8๐ฆ + 12 = 0.
5๐ฅ โ 8๐ฆ + 12 = 0 Converttoslopeinterceptform.โ8๐ฆ = โ5๐ฅ โ 12๐ฆ = >
q๐ฅ + <6
q Thisgivesustheslope.๐ = >
q
Usetheslope,๐ = >qandthepoint(-2,3)towritetheequation.
๐ = ๐๐D๐๐๐๐D๐๐
Fillinwhatyouknow.๐ = >q.Substitutepoint(-2,3)
๐๐= ๐D๐
๐DD๐Cross-Multiply.
5(๐ฅ + 2) = 8(๐ฆ โ 3) Simplify.5๐ฅ + 10 = 8๐ฆ โ 24
5๐ฅ โ 8๐ฆ + 34 = 0 GeneralForm
๐ฆ = >q๐ฅ + <b
; Slope-InterceptForm
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193. Write the equation of the line parallel to 4๐ฅ โ 6๐ฆ + 12 = 0 and through the point (5,7).
Explain your reasoning
Eg.2. Write the equation of the line perpendicular to ๐๐ + ๐๐ โ ๐ = ๐ and through the point (2,3).
Perpendicular means slopes are negative reciprocals. Step 1: Find the slope of 3๐ฅ + 2๐ฆ โ 4 = 0.
3๐ฅ + 2๐ฆ โ 4 = 0 Convert to slope-intercept form.
2๐ฆ = โ3๐ฅ + 4
๐ฆ = D76๐ฅ โ ;
6 This line has a slope, ๐ = D7
6.
The perpendicular line will have a slope of ๐ = ๐๐
Use: ๐ = ๐๐D๐๐๐๐D๐๐
๐๐= ๐D๐
๐D๐ Fill in what you know. ๐ = 6
7. Substitute point (2,3)
2(๐ฅ โ 2) = 3(๐ฆ โ 3) Cross-Multiply.
2๐ฅ โ 4 = 3๐ฆ โ 9 Simplify.
2๐ฅ โ 3๐ฆ + 5 = 0 General Form
๐ฆ = 67๐ฅ + >
7 Slope-Intercept Form
Negative reciprocal!
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194. Write the equation of the line perpendicular to 4๐ฅ + 3๐ฆ โ 24 = 0 and through the point (1,4).
Eg.3.Write an equation for the line through C(2,4) that is perpendicular to the line through A(1,2) and B(4,8).
First find slope AB. ๐ = qD6;D<
= O7= 2 Therefore, the perpendicular line has slope, ๐ = D<
6.
๐ = ๐๐D๐๐๐๐D๐๐
Fill in what you know: ๐ = D<6
. & substitute point (2,4)
D๐๐= ๐D๐
๐D๐ Cross-Multiply.
โ1(๐ฅ โ 2) = 2(๐ฆ โ 4) Simplify.
โ๐ฅ + 2 = 2๐ฆ โ 8
๐ฅ + 2๐ฆ โ 10 = 0 General Form
๐ฆ = โ<6๐ฅ + 5 Slope-Intercept Form
Know which of these forms you are being asked to answer in. If it is not specified, you can choose.
Both describe the same line.
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195. Write an equation for the line through C(1,2) that is perpendicular to the line throughA(2,4) and B(5,5).
Explain your reasoning
196. Write an equation for the line through Q(1,2) that is perpendicular to the line throughR(-2,0) and S(3,5).
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Determine the equation of the following lines. Answer in general form. 197. The line parallel to 2๐ฅ โ 3๐ฆ + 1 = 0 and
passing through the point (1, 2).198. The line perpendicular to ๐ฅ โ 5๐ฆ + 2 = 0
and passing through the point (-2, 5).
199. The line perpendicular to3๐ฅ โ 12๐ฆ + 16 = 0 and having the samey-intercept as 14๐ฅ โ 13๐ฆ โ 52 = 0.
200. Two perpendicular lines intersect on thex-axis. An equation of one line is ๐ฆ =3๐ฅ + 9 Find the equation of the otherline.
13) Horizontal & Vertical Lines + Mixed Practice
Part 1: Horizontal & Vertical Lines
Equations: ___________________________ ___________________________
___________________________ ___________________________
Example 1: Graph the following:
a) 3๐ฆ๐ฆ โ 6 = 0 b) 5๐ฅ๐ฅ + 10 = 0
Part 2: Word Problems
Example 2: The slope of a line represented by 6๐ฅ๐ฅ โ ๐๐๐ฆ๐ฆ + 1 = 0 is 23. Determine the value of k.
Example 3: Determine the equation of the line (in general form) of the diameter of a circle if the center is (6,โ2) and a point on the diameter is (15, 3).
Example 4: The slope of a roof is 612
, and its height is 20 m. Calculate the total horizontal span of the roof.
Example 5: Anya is building a picnic table for her backyard. The slope of the table legs is 4 and the table height is 100 cm. Find the length of a table leg to the nearest cm.
ASSIGNMENT # 13pages #49- 56 questions #201-244 (skip #241-242)
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Horizontal & Vertical Lines:
The equation of a horizontal line that is 3 units above the x-axis will be ๐ = ๐ or ๐ โ ๐ = ๐. The equation of a horizontal line that is 12 units below the x-axis will be ๐ฆ = โ12 or ๐ฆ + 12 = 0.
The equation of a vertical line 7 units to the right of the y-axis will be ๐ = ๐ or ๐ โ ๐ = ๐. The equation of a vertical line 2 units to the left of the y-axis will be ๐ฅ = โ2 or ๐ฅ + 2 = 0.
The equation of this line is ๐ = ๐.
Write the equation of the following lines. 201. 202. 203.
Think to yourselfโฆ Every point on this line has an x-coordinate of 5. It makes sense for the equation to be x=5.
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Write the equation of the following lines. 204. 205. 206.
207. Graph the linerepresented by theequation 2๐ฆ โ 4 = 0.
208. Graph the linerepresented by theequation 3๐ฅ โ 2 = 0.
209. Graph the linerepresented by theequation
๐ฆ โ 4 = 2๐ฆ โ 10.
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Mixed Practice: 210. Which of the following equations
represents the steepest line?
a. 5๐ฅ + 4๐ฆ โ 12 = 0b. 6๐ฅ + 2๐ฆ = 14c. โ3๐ฅ โ 7๐ฆ โ 21 = 0d. 12๐ฅ + 24๐ฆ + 64 = 0
211. Which of the following passes through(9, โ8) and has an x-intercept of โ3?
a. 3๐ฅ + 2๐ฆ + 9 = 0b.5๐ฅ + 9๐ฆ + 27 = 0c. 2๐ฅ + 3๐ฆ + 6 = 0d. 4๐ฅ + 3๐ฆ + 12 = 0
212. What is unique about lines that arewritten in the form ๐ฅ = ๐.
213. What is unique about lines that arewritten in the form ๐ฆ = ๐.
214. What is the equation, in general form, ofthe line that passes through the point(6, -3) and is parallel to ๐ฆ = 6
7๐ฅ + 4.
215. Determine the slope of the lineperpendicular to ๐ฅ โ 2๐ฆ โ 3 = 0.
216. Determine the equation of the line thatcontains the diameter of the followingcircle.Centre (โ4,3)Point on circumference (2,โ1)
Answer in general form.
217. The slope of the line represented by theequation 8๐ฅ โ ๐๐ฆ + 2 = 0 is 6
7.
Determine the value of k.
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218. What is the equation of a line withundefined slope and an x-intercept of 5.Write your answer in general form.
219. Write the equation ๐ฆ = <>๐ฅ โ 4 in the
form ๐ด๐ฅ + ๐ต๐ฆ + ๐ถ = 0 where A is positiveand all coefficients are rationalnumbers?
220. Find the value of k if 2๐ฅ + ๐๐ฆ + 7 = 0 isparallel to 3๐ฅ โ 6๐ฆ + 12 = 0.
221. Find all of the following points that areon the line 3๐ฅ = 2๐ฆ + 24?
a. (8,0)b. (6,-3)c. (4,6)d. (-2,9)e. (0,-12)
222. The slope of the roof on Mr. Jโs hiddensurf shack is;
7. If the roof is 14m tall, how
wide is it?
223. Anya is building a picnic table for herbackyard. The slope of the table legsis 2 and the table height is 90cm.Find the length of a table leg to thenearest cm.
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224. Write an equation that represents thegraph below.
225. What is a possible relationship forthe graph (and equation) above?
226. Challenge#8
The equation ๐ฆ = 75๐ฅ + 1500 represents the cost of a wedding reception. The total costconsists of $1500 fee to rent the hall plus $75 per guest. Express the equation of thisrelation using function notation.
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Linear Functions
Function notation is used to show the relationship between two quantities. The use of function notation allows the reader to identify the dependent and independent variable. Also, the letters chosen often identify what the variables represent.
Eg. The equation ๐ฆ = 75๐ฅ + 1500 represents the cost of a wedding reception. The total cost consists of $1500 fee to rent the hall plus $75 per guest. Express the equation of this relation using function notation.
C(n) = 75n+1500 Cost is a function of the number of guests.
227. The cost of a taxi ridein Victoria is $5.25 plus$0.35 per kilometer.Write an equation usingfunction notation forthis relation.
228. J-Tees Pedi-Cabsprovide tours forvisitors to Victoria. Thecost is 25 cents perminute. Write anequation using functionnotation for thisrelation (in dollars).
229. JLA-Skuterz rent gas-powered scooters. Thecost is $40 per day plus25 cents per kilometreridden. Write anequation using functionnotation for thisrelation.
230. The skating rink at therecreation centrecharges students $5.00admission. Write anequation for the cost(C) as a function of thenumber of students (s).
231. The skating rink will leta group of studentsbook the entire rink for$500. Write anequation for the cost(C) as a function of thenumber of students (s).
232. At the same skatingrink, another option isto reserve the rink for$200 and then pay $4per student. Write anequation for the cost(C) as a function of thenumber of students (s).
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Find the range value for each of the following. 233. ๐ถ(๐) = 25๐ + 12
Find ๐ถ(12).234. ๐(๐ฅ) = <
6๐ฅ โ 3
Find ๐(โ3).
235. โ(๐ก) = โ250๐ก + 1200Find โ(20).
Find the domain value for each of the following. 236. ๐ถ(๐) = 25๐ + 12
Find n if ๐ถ(๐) = 24.237. ๐(๐ฅ) = <
6๐ฅ โ 3
Find x if ๐(๐ฅ) = 12.
238. โ(๐ก) = โ250๐ก + 1200Find t if โ(๐ก) = 1000.
239. Below is a graph of C(n).
Find C(4).
240. Below is a graph of f(x).
Find x if f(x) = 7.
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Extended Practice 241. The centre of a circle is located at
(0,-3). Draw a tangent at (5,-3). What isthe equation of the tangent?
242. The centre of a circle is located at(1,-1). Draw a tangent at (2,5). What isthe equation of the tangent?
243. Are the lines below parallel?
Explain how you know___________________
___________________________________
___________________________________
244. Draw a line through A(1,2) and B(-3,-7).Now draw a perpendicular line throughC(9,-3).
What is the equation of the perpendicular line?
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Part II Answers:
1.
2. Aseriesofdotsarrangedinaline.3. Yes.4. E,earnings.5.
๐ฆ = 3๐ฅx y-2 -6-1 -30 01 32 6
6. Orderedpairs.7. Yes,itmakestheleftsideoftheequationequalthe
right.8. Infinitenumber.9. Youcandrawastraightlinethroughthem.10.
11. ๐ฆ = <6๐ฅ โ 4
12. ๐ฆ = 2๐ฅ โ 313. ๐ฆ = 7
>๐ฅ + 4
14. ๐ฆ = 6a๐ฅ โ 6
7
15. ๐ฆ = ;7๐ฅ + 12
16. ๐ฆ = โ ;7๐ฅ + <O
>
17.
18.
19.
20.
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21.
22.
23.
24.
25.
26. Atleasttwo.27. Threewouldbesaferbecauseitwouldpointout
anyerrors.28.
29.
30. Linear.
31. Non-linear.
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32. Linear.
33. Non-linear.
34. Non-linear.
35. Non-linear.
36. Theexponentonboththexandtheymustbe1.37. (0,4)38. 039. (8,0)40. 0
41. x-int:6y-int:4
42. x-int:10y-int:6
43. x-int:โ8y-int:6
44. x-int:5y-int:4
45. x-int:3y-int:โ6
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46. x-int:7y-int:โ3
47. x-int:>
6or2.5y-int:2
48. x-int:2y-int:6
49. x-int:3y-int:โ a
6orโ4.5
50. Whenthecoefficientsarefactorsoftheconstant
term.Thiswillproduceinterceptsthatcanbegraphedeasily.
51. Answerswillvary.
52. Answerswillvary.
53. Answerswillvary.
54. Answerswillvary.Twooptionsbelow.
55. Therelationwillbethesamelineasoneofthe
axes.Eg.
56. Therelationwillbeparalleltooneoftheaxis.Two
optionsareshownbelow.
57. Eitheralinewithundefinedslopeorzeroslope.58. x-int:2y-int:3
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59. x-int:ยฑ2y-int:โ4
60.
61. Ongraphabove.62. Slopeofallthreesectionsis<
7.
63. Differentsectionsofthesamelineorlinesegmetwillhaveequalslopes.
64. ๐ = 365. ๐ = โ6
>
66. ๐ = 367. ๐ = <
7
68. ๐ = <6
69. ๐ = โ270.
71.
72.
73. Findtheslopesofeachside(segment)and
determineifanypairofsegmentshadperpendicularslopes(negativereciprocals).
74. Manyanswers.Eg.(1, โ4), (3,โ1), (7,5), (9,8)75. Manyanswers.Eg.(โ3,7), (1,โ3), (3,โ8)76.
77. Rate=Slope.๐ = $<.aa
๏ฟฝFor$1.99perhour.
78. Slope(costperhour)79. $11.9580. ๐ = >
;
81. ๐ = 082. ๐ = โ6
7
83. 684. 285. Algebraically:
โ1 = 2(7) โ 15โ1 = 14โ 15โ1 = โ1Leftsideequalsrightside,therefore(7, โ1)satisfiestheequationandisontheline.Graphically:Thelinepassesthrough(7,โ1)
86. No87. Yes88. Yes89. Yes90. No
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91. Yes92. Answeredinbooklet.93. No.Itdoesnotsatisfytheequation.94. No.Itdoesnotsatisfytheequation.95. Yes.Asthex-valuesincreaseby1,they-valuesalso
increaseby1.Constantrateofchange(slope)meanstherelationislinear.
96. Yes.Thesepointsformaverticalline.(Thisisalinearrelationbutnotafunction.)
97. Yes.Asthex-valuesincreaseby1,they-valuesdecreaseby1.Constantrateofchange(slope)meanstherelationislinear.
98.
99.
100. ๐ = 6
7
101. ๐ = โ3102. โ5103. 5104. Theslopeisthecoefficentonthexandthey-
interceptistheconstantterm.105. Theslopeisthecoefficentonthexandthey-
interceptistheconstantterm.106. ๐ = โ3,๐ = 2107. ๐ = โ7
>,๐ = โ7
108. ๐ = a6,๐ = โ7
6
109. ๐ฆ = 2๐ฅ โ 5110. ๐ฆ = b
7๐ฅ + 6
7
111. ๐ฆ = โ3๐ฅ โ 2112. ๐ = 7
;,๐ = 3,equation:๐ฆ = 7
;๐ฅ + 3
113. ๐ = โ1,๐ = โ2,equation:๐ฆ = โ๐ฅ โ 2114. ๐ = โ4,๐ = 0,equation:๐ฆ = โ4๐ฅ115. ๐ = โ<
7,๐ = โ3,equation:๐ฆ = โ<
7๐ฅ โ 3
116. ๐ = 76,๐ = โ4,equation:๐ฆ = 7
6๐ฅ โ 4
117. ๐ = 3,๐ = 0,equation:๐ฆ = 3๐ฅ118. Theequationwillnothaveaconstantterm,itis
zero.119. Ifthey-interceptispositive,thevalueofbwillbe
positive.
120. Ifthey-interceptisnegative,thevalueofbwillbenegative.
121.
122.
123.
124.
125.
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126.
127.
128.
129. Answeredinbooklet.130. ๐ฆ = 3๐ฅ + 7131. ๐ฆ = 3๐ฅ โ 11132. ๐ฆ = 3๐ฅ โ 5133. ๐ฆ = 3๐ฅ + <7
6
134. ๐ฆ = 3๐ฅ โ 1135. ๐ฆ = <
;๐ฅ + 2
136. ๐ฆ = โ5๐ฅ + 2137. ๐ฆ = <
>๐ฅ + 2
Whatyoujustdidaboveisonewaythatyouwillbeable
tofindtheequationofaline.IFyouhavetheslopeorthey-intercept,youcaninputthecoordinatesofapointonthelinetosolvefortheunknownpartofthe
equation.
Thenyouwillwritethefullequationwithslopeandy-interceptinplaceofmandb.
138. 3๐ฅ โ ๐ฆ โ 5 = 0139. ๐ฅ โ ๐ฆ + 12 = 0140. 2๐ฅ โ 4๐ฆ โ 3 = 0141. ๐ฅ + 12๐ฆ + 6 = 0142. 2๐ฅ โ 3๐ฆ + 36 = 0143. 9๐ฅ + 8๐ฆ โ 60 = 0
144. ๐ฆ = 3๐ฅ โ 1or3๐ฅ โ ๐ฆ โ 1 = 0145. Answeredinbooklet.146. a)๐ฆ โ 2 = 2(๐ฅ + 5)
b)๐ฆ = 2๐ฅ + 12c)2๐ฅ โ ๐ฆ + 12 = 0
147. a)๐ฆ + 1 = โ2(๐ฅ + 5)b)๐ฆ = โ2๐ฅ โ 11c)2๐ฅ + ๐ฆ + 11 = 0
148. a)๐ฆ โ 4 = โ <7(๐ฅ + 3)
b)๐ฆ = โ<7๐ฅ + 3
c)๐ฅ + 3๐ฆ โ 9 = 0149. a)๐ฆ โ 4 = <
6(๐ฅ โ 2)
b)๐ฆ = <6๐ฅ + 3
c)๐ฅ โ 2๐ฆ + 6 = 0150. a)๐ฆ โ 7 = โ(๐ฅ โ 0)
b)๐ฆ = โ๐ฅ + 7c)๐ฅ + ๐ฆ โ 7 = 0
151. Answeredonpage.152. a)๐ฆ โ 6 = 5(๐ฅ โ 4)
b)๐ฆ = 5๐ฅ โ 14c)5๐ฅ โ ๐ฆ โ 14 = 0
153. a)๐ฆ + 1 = <6(๐ฅ + 2)
b)๐ฆ = <6๐ฅ
c)๐ฅ โ 2๐ฆ = 0154. a)๐ฆ + 6 = โ 7
;(๐ฅ โ 5)
b)๐ฆ = โ7;๐ฅ โ a
;
c)3๐ฅ + 4๐ฆ + 9 = 0155. a)๐ฆ โ 6 = ;
7i๐ฅ โ <
6j
b)๐ฆ = ;7๐ฅ + <O
7
c)4๐ฅ โ 3๐ฆ + 16 = 0156. a)๐ฆ โ 1 = 7
6(๐ฅ + 2)
b)๐ฆ = 76๐ฅ + 4
c)3๐ฅ โ 2๐ฆ + 8 = 0157. 2๐ฅ โ ๐ฆ โ 2 = 0158. ๐ฆ = 2๐ฅ โ 2,2๐ฅ โ ๐ฆ โ 2 = 0159. ๐ฆ = 5๐ฅ + 6,5๐ฅ โ ๐ฆ + 6 = 0160. ๐ฅ โ 2๐ฆ โ 11 = 0161. ๐ฅ โ 2๐ฆ โ 3 = 0162. 4๐ฅ + 5๐ฆ โ 20 = 0163. ๐ฆ + 7 = 0164. 3๐ฅ + 12๐ฆ โ 5 = 0165. 30๐ฅ โ 20๐ฆ โ 1 = 0166.
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167.
168.
169.
170. Answerswillvary.Perhaps:Isolateโyโandplot
usingslopeandy-intercept.171.
Isolateโyโandplotusingslopeandy-intercept.
172.
Isolateโyโandplotusingslopeandy-intercept.
173. C174. D
175. C176. a177. b178. b179. Eachelementintherangeistwolessthanthree
timesthecorrespondingelementinthedomain.180. Addinganelementinthedomaintotwiceits
elementintherangegivesasumof4.181. Eachelementintherangeisfourlessthanthree-
fifthsthecorrespondingelementinthedomain.182. a,b,c183. e,g184. i,j185. ๐ = 6
7, ๐ = 3,
๐ฆ = 67๐ฅ + 3, 2๐ฅ โ 3๐ฆ + 9 = 0
186. ๐ = โ>7, ๐ = 4,
๐ฆ = โ>7๐ฅ + 4, 5๐ฅ + 3๐ฆ โ 12 = 0
187. ๐ = โ67, ๐ = 2,
๐ฆ = โ67๐ฅ + 2, 2๐ฅ + 3๐ฆ โ 6 = 0
188. ๐ = 0, ๐ = โ6,๐ฆ = โ6,๐ฆ + 6 = 0
189. a)3b)โ <7
190. a)โ 67b)7
6
191. a)>7b)โ 7
>
192. Answeredonpage.193. ๐ฆ = 6
7๐ฅ + <<
7or2๐ฅ โ 3๐ฆ + 11 = 0
194. ๐ฆ = 7;๐ฅ + <7
;or3๐ฅ โ 4๐ฆ + 13 = 0
195. ๐ฆ = โ3๐ฅ + 5or3๐ฅ + ๐ฆ โ 5 = 0196. ๐ฆ = โ๐ฅ + 3or๐ฅ + ๐ฆ โ 3 = 0197. 2๐ฅ โ 3๐ฆ + 4 = 0198. 5๐ฅ + ๐ฆ + 5 = 0199. 4๐ฅ + ๐ฆ + 4 = 0200. ๐ฅ + 3๐ฆ + 3 = 0201. ๐ฆ = 6๐๐๐ฆ โ 6 = 0202. ๐ฅ = 8, ๐๐๐ฅ โ 8 = 0203. ๐ฆ = 8๐๐๐ฆ โ 8 = 0204. ๐ฅ = โ1๐๐๐ฅ + 1 = 0205. ๐ฆ = โ3๐๐๐ฆ + 3 = 0206. ๐ฅ = โ6๐๐๐ฅ + 6 = 0207. ๐ฆ = 2
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208. ๐ฅ = 67or3๐ฅ โ 2 = 0
209. ๐ฆ = 6
210. b211. c212. Theyareverticallines(undefinedslope).213. Theyarehorizontallines(zeroslope).214. 2๐ฅ โ 3๐ฆ โ 21 = 0215. ๐ = โ2216. 2๐ฅ + 3๐ฆ โ 1 = 0217. ๐ = 12218. ๐ฅ โ 5 = 0219. ๐ฅ โ 5๐ฆ โ 20 = 0220. ๐ = โ4221. a,b,e222. 21mwide.223. 101cm.224. ๐ฆ = 50๐ฅ + 400225. Answerswillvary.Therelationshiphasaconstant
rateandainitialfixedcost.Thisgraphcouldrepresentthecostofrentingabanquethallwithafixedcostandanadditionalfeeperpersonattending.
226. ๐ถ(๐) = 75๐ + 1500227. ๐ถ(๐) = 0.35๐ + 5.25228. ๐ถ(๐ก) = 0.25๐ก229. ๐ถ(๐) = 0.25๐ + 40230. ๐ถ(๐ ) = 5๐ 231. ๐ถ(๐ ) = 500232. ๐ถ(๐ ) = 4๐ + 200233. ๐ถ(12) = 312234. ๐(โ3) = โ a
6
235. โ(20) = โ3800236. ๐ = <6
6>
237. ๐ฅ = 30238. ๐ก = ;
>
239. ๐ถ(4) = 40240. ๐ฅ = 2241. ๐ฅ = 5242. ๐ฅ + 6๐ฆ โ 32 = 0or๐ฆ = โ <
O๐ฅ + <O
7
243. Yes,bothlineshaveaslopeofโ 76.
244. ๐ฆ = โ;a๐ฅ + 1or4๐ฅ + 9๐ฆ โ 9 = 0
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Quizzes & Tests:
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Quiz 1
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Unit/ Chapter test