plug in finding the slope of a line - matermiddlehigh.org lesson 5... · finding the slope of a...
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44 Lesson 5
Plug In
LESS
ON 5Finding the Slope of a Line
Slope
Theslopeofalineistheratioofthechangeinthey-valuestothechangeinthecorrespondingx-values .
x
y
1–1
–1 2 3 4 5 6 7 80
1
2
3
4
5
6
3
6
Slope5changeiny-values
______________changeinx-values
5522_____82253__651__2
They-interceptiswherealinecrossesthey-axis .
x
y
1–1
–1 2 3 4 5 6 7 80
1
2
3
4
5
6
(0, 1) y-intercept is 1.
Theequationofalinecanbewrittenintheformy5mx1b,wheretheslopeofthelineism,andthey-interceptisb .
x
y
1–1
–1 2 3 4 5 6 7 80
1
2
3
4
5
6
(0, 1)
3y �12x � 1
6
slope aratioofthechangeiny-coordinates(rise)ofagraphtothechangeincorrespondingx-coordinates(run)
y-interceptthey-coordinateofthepointatwhichalinecrossesthey-axis
Wordsto
Know
I see! I choose two coordinate pairs and then
divide the difference of the y-values by the difference of the corresponding x-values.
The y-intercept is the y-coordinate of the point where the line
intersects the y-axis.
I get it! The slope is m 5 1 _ 2 , and the y-intercept is b 5 1, so the equation of
the line is y 5 1 _ 2 x 1 1.
Willtheslopeofthelinethroughthepoints(1,6)and(7,22)bepositiveornegative?Howdoyouknow?
DISCUSS
A Youcanusetheslopeformulam5y22y1
_____x22x1tofindtheslopeofaline .
Findtheslopeofthelinethroughpoints(5,1)and(24,3) .
1 Choose(x1,y1)and(x2,y2) .
2 Writetheslopeformula .Substituteforx1,x2,y1,andy2 .
3 Subtract .Simplify,ifpossible .
(x1,y1)5(5,1) (x2,y2)5( , )
slope5y22y1
_____x22x1532
_________
25
5 ______
DO
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LC5 Slope
PRACTICEFind the slope of the line that contains the given points.
1 (2,1)and(4,2)
m522
_________
22
5 ______
2 (1,3)and(5,6)
m5 2
_____________
2
5 ______
Find the slope and y-intercept of each line.
3
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
slope5
y-intercept5
4
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
slope5
y-intercept5
5 y51__2x15
slope5
y-intercept5
6 y522x13
slope5
y-intercept5
B Youcanusetheequationofalinetofinditsslopeandy-intercept .
Findtheslopeandy-interceptofthelinewiththeequationy53x12 .
1 Comparetheequationofthegivenlinetoy5mx1b .
2 Identifytheslope,m,andthey-intercept,b .
y5 x1
y5m3x1b
m5
b5
Theslopeofthelineis .
They-interceptofthelineis .
DO
Whydoyouthinktheslopeofaverticallineisdescribedas“undefined”?Explainbycomputingtheslopeoftheverticallinethroughpoints(2,6)and(2,23) .
DISCUSS
I have to make sure the equation of a line is in the form
y 5 mx 1 b to determine the slope
and y-intercept.
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46 Lesson 5
POWER uPYoucanwritetheequationofalinefromitsgraphusingtheformy5mx1b .
1 Find the slope.Choosetwopointsontheline,andusetheslopeformulatocalculatetheslope .Usingthepoints(23,21)and(2,4):
m5changeiny-values
______________changeinx-values
5y22y1
_____x22x1
542(21)
_______22(23)
55__5
51
2 Find the y-intercept.Identifywherethelinecrossesthey-axis .
Thelinecrossesthey-axisat(0,2) .
They-interceptis2 .
3 Substitute the slope, m, and the y-intercept, b, in the equation y 5 mx 1 b.
y5mx1b
y5x12isanequationfortheline .
Thexandyintheequationrepresentallthepossiblecoordinatepairsofpointsontheline .
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
Doestheslopechange,dependingonwhichpointyouassignas(x1,y1)or(x2,y2)?Explain .
DISCUSS
A Youcanwritetheequationofalineifgiventheslopeandy-intercept .
Whatistheequationofalinethathasaslopeof8anday-interceptof22 .
1 Identifymandb .
2 Substitutetheslope,m,andthey-intercept,b,intheequationy5mx1b .
3 Writetheequation .
m5
b5
y5mx1b
y5 x1
y5
DO
Writing an Equation of a Line
I see! If I know the slope and y-intercept of a line, then I can
write an equation for the line.
I remember! m stands for slope and b represents
the y-intercept.
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LC5 Slope
Andrewsaysthathecanusetheformy5btowritetheequationforanyhorizontallinethatpassesthroughthepoint(0,b) .IsAndrewcorrect?Explain .
DISCUSS
B Youcanwritetheequationofalinethatpassesthroughtheorigin .
Writeanequationforthelineshowninthegraph .
1 Choosetwopointsontheline .
2 Findtheslope .m5y22y1
_____x22x1
3 Identifythey-intercept .
4 Writetheequation .Simplify,ifpossible .
(x1,y1)5(2, )
(x2,y2)5( ,22)
m5 2
_____________
2
5 ______
Thelinecrossesthey-axisattheorigin .
Sob5 .
y5 ______x1 5 ______x
DO
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
I remember! The coordinates of the
origin are represented by the point (0,0).
PRACTICEWrite the equation for the line.
1
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
2
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
I remember! All horizontal lines have
a slope of 0. That means m 5 0 for all
horizontal lines.
Write an equation of the line with the given slope and y-intercept.
3 slope53;y-intercept57 4 slope522___5 ;y-intercept523
Write an equation of the line that passes through the given points.
5 (0,4)and(25,2) 6 (0,2)and(4,6)
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48 Lesson 5
READY TO gOYoucanuseanytwopointsonalinetofindtheslopeoftheline .Youcancomparetheslopesoftwosegmentsofthelinetoshowthis .
1 Usethepoints(0,23)and(3,21)tofindtheslopeoftheline .
m5212(23)
________320
52__3
2 Usethepoints(3,21)and(9,3)tofindtheslopeoftheline .
m532(21)
_______923
54__6
52__3
3 Comparetheslopes .
2__352__3
Theslopesareequal .
x
y
1–1
–2
–3
–4
–5
–1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
44
22 66
33
Margaritalooksatatableofx-yvaluesandcalculatestheslopeusingonepairofpoints .Sheusesanotherpairofpointsandcalculatesadifferentslope .WhatcanMargaritaconcludeaboutagraphofthosepoints?
DISCUSS
Plug In POWER uP gO!lEssOn lInk
You can find the slope and y-intercept of a line given any two points on the line, its graph, or its equation.
x
y
1–1
–1–2 2 3 4 50
1
2
3
4
5
y �12x � 1
slope:m51__2
y-intercept:b51
You can use the slope and y-intercept of a line to write an equation for the line.
Slope:m55__8
y-intercept:b522
y5mx1b
y55__8x1(22)
55__8x22
Slope
I get it! Every segment of a
line will have the same slope!
I get it! I can investigate slope and y-intercept further
to better understand the characteristics
of lines!
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LC5 Slope
WORk TOgEThERUsetwopairsofpointstoconfirmtheslopeofaline .
• Thislineincludesthepoints(21,6),(1,2),(2,0),and(5,26) .
• Theslopebetween(21,6)and(1,2)is24__2,or22 .
• Theslopebetween(2,0)and(5,26)is26___3 522__1,or22 .
• Theslopeofthelineisthesamewitheitherpairofpoints .
x
y
1–1
–2
–3
–4
–5
–6
–1–2–3–4–5–6 2 3 4 5 60
1
2
3
4
5
6
4
66
2
3
A Youcanuseslopetocheckwhetherthreepointslieonthesameline .
Determineifthepoints(0,6),(3,3),and(6,21)areonthesameline .
1 Findtheslopebetweenanytwoofthethreepoints .
2 Nowfindtheslopebetweentwootherpoints .
3 Comparetheslopestodetermineifthepointsareonthesameline .
Use(0,6)and(3,3) .
m5 2
_____________
2
5 ______5
Use(3,3)and(6,21) .
m5 2
_____________
2
5 ______
Thethreepoints onthesameline
becausetheslopesare .
DO
I could also use any other two pairs of
points on this line to confirm the slope.
B Youcandetermineiftwosegmentsthatshareanendpointareonthesameline .
Determineiftwosegments,onewithendpoints(0,2)and(4,3)andonewithendpoints(0,2)and(8,4),liealongthesameline .
1 Findtheslopeofthefirstsegment .
2 Findtheslopeofthesecondsegment .
3 Comparetheslopestodetermineiftheyareonthesameline .
Use(0,2)and(4,3) .
m5 2
_____________
2
5 ______
Nowuse(0,2)and(8,4) .
m5 2
_____________
2
5 ______5 ______
Theslopesare .The3points onthesameline .
DO
Willalinesegmentthatispartofalargerlinesegmentalwayshavethesameslopeasthelargerlinesegment?Explain .
DISCUSS
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50 Lesson 5
READY TO gO
PRACTICESolve.
1 Alinesegmenthasendpointsat(22,5)and(1,4) .Anotherlinesegmenthasendpointsat(1,4)and(2,7) .Arethelinesegmentspartsofthesameline?
Findtheslopeofthefirstlinesegment .
m542
_________
2(22)
5 ______
Findtheslopeofthesecondlinesegment .
m5 24_______
22
5 ____5
Arethelinesegmentsonthesameline?Explain .
2 Choosetwopairsofpointsonthelinetoshowtheslopeofthelineisthesame .
Usepoints( , )and( , ) .
m5 2
_________
2
5 ______5
Nowusepoints( , )and( , ) .
m5 2
_________
2
5 ____5
Aretheslopesthesame?
3 Usetheendpointsoftwosegmentsalongthislinetoshowthattheslopesofthesegmentsarethesame .
Aretheslopesofthesegmentsthesame?
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
x
y
1–1
–2
–3
–4
–5
–1–2–3–4–5 2 3 4 50
1
2
3
4
5
REMEMBER Any two pairs of points on the line will show the same slope.
HINT You can check your answer by graphing the segments on a coordinate grid.
Coordinate Grid can be
found on p.213
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LC5 Slope
Solve.
4 Twolinesegmentsarepartoftheliney521__5x110 .Whatistheslopeofeachsegment?
5 Barbdrewalinethroughtheoriginandpoint(2,3) .Jamaldrewalinethroughtheoriginandpoint(4,6) .HowcanyoutellthatBarbandJamaldrewthesameline?
Use the slope formula to determine if the three points lie on a line.
6 (23,22),(21,3),(3,13)
Dothepointslieonthesameline?Explain .
7 (23,11),(2,7),(6,3)
Dothepointslieonthesameline?Explain .
Reasoning with Graphs
Doanysegmentswithineachofthesegraphsshareanendpoint?Explain .
x
y
1–1
–1 2 3 4 50
1
2
3
4
5
x
y
1–1
–1 2 3 4 50
1
2
3
4
5
x
y
1–1
–1 2 3 4 50
1
2
3
4
5
m � 14
m � 1 4m �
Whatdoyounoticeabouttheslopesasthelinesgetssteeper?
DISCUSS
I see! I can think of any two points as the endpoints
of a segment.
I get it! I can relate the steepness of a line with the value
of its slope.
CC13_Mth_G8_SE_PDF.indd 51 17/05/13 2:19 PM
52 Lesson 5
READY TO gO
PROblEm sOlvIng
sOlvE
READ
UnderWater PressUreTheunitsofmeasureforpressurearecalledatmospheres .Thepressureatsealevelis1atmosphere .Atadepthof20meters,thepressureis3atmospheres .Writeanequation,andfindthepressureatadepthof100meters .
• Youneedtofindthenumberofatmospheresatadepthof .
• Use(0,1)forthepressureat .
• Use forthepressureatadepthof20meters .
• Usethepointstofindthe .
• Usethemeasureofpressureat asthey-intercept .
• Usey5mx1bandsubstitute forx .
Writeanequation .
Theslope,m5 2
_____________
2
5 ______5 ______
They-interceptis .Theequationisy5 ______x1 .
Findthenumberofatmospheresatadepthof100meters:y51__10( )115
Writeanorderedpairfortheatmosphereatadepthof100meters .(100, )
Checkthatthispointandthegiventwopointslieonthesameline .
Theslopebetween(0,1)and(20,3)is .
Theslopebetween(20,3)and(100,11)is:m5 2
_____________
2
5 ______5 ______
Theslopesare ,sothepoints .
Atadepthof100meters,thepressureis atmospheres .
3 1
PlAn
52 Lesson 5
ChECk
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53
5 Slope
PRACTICEUse the problem-solving steps to help you.
1 Acomputerrepaircompanycharges$50foranappointment .For2hoursofrepairwork,itchargesatotalof$120 .Whatisthetotalchargefor5hoursofrepairwork?
2 Atdawn,thetemperatureis70°F .Thetemperaturefallsatthesameratefortherestoftheday .In4hours,thetemperatureis60°F .Whatisthetemperature6hoursafterdawn?
3 Jennieworksforacellphonecompany .Sheearns$250perweekpluscommissionsshemakesonsales .oneweek,shehadsalesof$1000,andherpaywas$275 .Howmuchwouldherpaybeifshehadsalesof$2,000?
CHECKLIST
READ
PLAN
SoLVE
CHECk
CHECKLIST
READ
PLAN
SoLVE
CHECk
CHECKLIST
READ
PLAN
SoLVE
CHECk
53
I see! For the company to perform any repair work, a customer is charged for making
an appointment. That amount is the
y-intercept!
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