linear relationships math notes
Post on 18-Jul-2016
35 Views
Preview:
DESCRIPTION
TRANSCRIPT
5ISERIES TOPIC
1Linear RelationshipsMathletics Passport © 3P Learning
Q Twosnailshavespottedthebestlettuceinthegardenandaretravellinginstraightlinestowardsitfordinner.SlidySamisfollowingthepath: 2y x2= - andSlipperyShellyisfollowingthepath:2 2y x= - - Whichlettucearetheybothmovingtowardsifitisatthepointwheretheirpathscross?
Work through the book for a great way to solve this
Give this a go!
Writeasinglesentencethatdescribesthemathematicalmeaningfortheseterms:Youcanuseasmalldiagram/exampletosupportyourdescription.
Relationship LinearContinuous
Thisbookletextendsonstraightlineconceptsbyinvestigatingtherelationshipbetweenpointsonaline
Jointhesedefinitionstogethertodescribeinaonesentencewhatyouthinkismeantbyalinear relationship.
x-axis
y-axis
1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
A BC
E
I
J
H
F
D
G
LettuceSamandShellyaremovingtowards:
2 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
Linear RelationshipsHow does it work?
Fortheequationy x3 4= -
Gradient-intercept form
Alinearrelationshipisanalgebraicrulethatformsacontinuous,straightlinewhengraphed.Let’sreviewthebasiclinear equation.
(i) Writethegradientandy-interceptforthegraphofy x3 4= -
gradient m 313= =
++^ h y-intercept b 4= -^ h
(ii) Describewhatthesevaluesmeanforthegraphofy x3 4= -
y mx b= +
coefficientofx (m)isthegradient theconstantterm(b)isthey-intercept
Beforegraphingitisimportanttobeabletoextractthecorrectinformationfromalinearequation.
thepowerofxandyisalways1
gradient-interceptform
y x3 4= -
:m 3= graphmoves 3+ vertically(slopesup)forevery 1+ horizontally
:b 4= - graphpassesthroughthepoint ,0 4-^ hor
thegraphcrossesthey-axisat 4-
Fortheequationy x 2= +
(i) Writethegradientandy-interceptforthegraphofy x 2= +
gradient 1m11= =
++^ h y-intercept b 2= +^ h
The 1infrontofthex ishidden
(ii) Describewhatthesevaluesmeanforthegraphofy x 2= +
Sometimestheinformationneededcanappeartobe‘hidden’.
:m 1= graphmoves 1+ vertically(slopesup)forevery 1+ horizontally
:b 2= + graphpassesthroughthepoint ,0 2^ hor
thegraphcrossesthey-axisat2
y x 2= +
Equationisanother wordforrule
5ISERIES TOPIC
3Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear Relationships
y x31
21= - +y x
3 21= - +
y x32 1= +
Fortheequationy x32 1= +
Herearesomemoreexampleswithdifferentpointstolookoutfor.
(i) Writethegradientandy-interceptforthegraphofy x32 1= +
gradient m32
32= =
++^ h y-intercept b 1= +^ h
(ii) Describewhatthesevaluesmeanforthegraphofy x32 1= +
:m32= graphmoves 2+ vertically(slopesup)forevery 3+ horizontally
:b 1= + graphpassesthroughthepoint ,0 1^ h or thegraphcrossesthey-axisat 1
Fortheequationy x4 5= - +
(i) Writethegradientandy-interceptforthegraphofy x4 5= - +
gradient m 414= - =
+-^ h y-intercept b 5= +^ h
(ii) Describewhatthesevaluesmeanforthegraphofy x4 5= - +
Becarefultocorrectlyinterpretthemeaningofnegativesigns.
:m 4= graphmoves 4- vertically(slopesdown)forevery 1+ horizontally
:b 5= + graphpassesthroughthepoint ,0 5^ h orthegraphcrossesthey-axisat5
y x4 5= - +
Writethegradientandy-interceptforthegraphofy x3 2
1= - +
Whenthecoefficientofxisafraction,itcanappearinaslightlydifferentform.
gradient m31
31= - =
+-^ h y-intercept b
21=^ h
Writethegradientandy-interceptforthegraphof 3 2y x= -
Don’tletexpressionswritteninadifferentordertrickyou.
y-intercept b 3= +^ h gradient m 212= - =
+-^ h
3 2y x= -
or
Remember:Thecoefficientisthenumberinfrontofthevariable
sameas 2 3y x= - +
4 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear RelationshipsYour Turn
Gradient-intercept form
1 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Describewhatthesevaluesmeanforthegraphofeachlinearequation
GRADIENT-INTERCEPT FORM * GRADIENT-IN
TERCEPT FORM
*
..../...../20...
a y x2 1= - b 6y x= -
Gradient(m) =
y-intercept(b)=
(i)
(ii) Thegraphofy x2 1= - moves:
verticallyforeveryhorizontally
Gradient(m) =
y-intercept(b)=
(i)
(ii) Thegraphofy x 6= - moves:
c y x3 2= - + d y x5 2= +
Gradient(m) =
y-intercept(b)=
(i) Gradient(m) =
y-intercept(b)=
(i)
e y x 6= - + f y x341= - +
Gradient(m) =
y-intercept(b)=
(i) Gradient(m) =
y-intercept(b)=
(i)
verticallyforeveryhorizontally
(ii) Thegraphofy x3 2= - + moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x5 2= + moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x 6= - + moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x341= - + moves:
verticallyforeveryhorizontally
y mx b=
+
5ISERIES TOPIC
5Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear RelationshipsYour Turn
Gradient-intercept form
1 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Describewhatthesevaluesmeanforthegraphofeachlinearequation
a 2y x= + b y x21
32= +
Gradient(m) =
y-intercept(b)=
(i)
(ii) Thegraphof 2y x= + moves:
verticallyforeveryhorizontally
Gradient(m) =
y-intercept(b)=
(i)
(ii) Thegraphofy x21
32= + moves:
c 3y x= - d y x2
8= - +
Gradient(m) =
y-intercept(b)=
(i) Gradient(m) =
y-intercept(b)=
(i)
e y x31= - - f y x
54
23= -
Gradient(m) =
y-intercept(b)=
(i) Gradient(m) =
y-intercept(b)=
(i)
verticallyforeveryhorizontally
(ii) Thegraphof 3y x= - moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x2
8= - + moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x31= - - moves:
verticallyforeveryhorizontally
(ii) Thegraphofy x54
23= - moves:
verticallyforeveryhorizontally
6 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear Relationships
Writethegradientandy-interceptforthegraphof2 6 12y x- =
Rearranging linear relationships
Somelinearrelationshipsneedtobere-arrangedbeforethegradientandy-interceptcanberead.
move6xovertotherighthandside
gradient m 313= =
++^ h y-intercept b 6= +^ h
Usingasimilarmethodtosolvingequations,makeythesubjectoftheformula.
2 6 12y x- =
3 6y x= +
2 6 12y x= +
x x6 6+ +
2 2 2' ' '
Writethegradientandy-interceptforthegraphof2 3 9x y+ =
gradient m32
32= - =
+-^ h y-intercept b 3= +^ h
Leavevaluesassimplifiedfractionsafterrearranging.
y x
y x
332
32 3
= -
= - +
3 9 2y x= -
x x2 2- -
3 3 3' ' '
Writethegradientandy-interceptforthegraphof 2y x9
3 -=
gradient m31
31= =
++^ h y-intercept b 6= +^ h
Hereisanotherexamplewherea1ishiddenattheend.
6y x3
= +
1y x3 8= +
x x+ +
2 3 9x y+ =
divideeverytermby2toisolatey
linearequationisnowintheform y mx b= +
move2xovertotherighthandside
divideeverytermby3toisolatey
linearequationisnowintheform y mx b= +
9 2 9y x9
3##
-=
1y x3 8- =
3 3 3' ' '
multiplybothsideby9
movexovertotherighthandside
divideeverytermby3toisolatey
5ISERIES TOPIC
7Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear RelationshipsYour Turn
Rearranging linear relationships
1 (i) Rearrangeeachoftheselinearrelationshipsintogradient-interceptform y mx b= +^ h(ii) Writethegradientandy-interceptforeachoftheselinearequations
a 4 8 12y x= - b 2 14 6y x= +
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii)
c 10 10 25y x- = d 4 3 12y x+ =
e 6 2 1x y+ = f 8 4 16x y- =
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(ii)
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii)
hint:becarefulwithnegativevalueshere
..../...../20...REARRA
NGING LINEAR RELATI
ONSHIPS * REARRANGING LINEAR RELATIONSHIPS *
ymx b
=+
8 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear RelationshipsYour Turn
Rearranging linear relationships
2 (i) Rearrangeeachoftheselinearrelationshipsintogradient-interceptform y mx b= +^ h(ii) Writethegradientandy-interceptforeachoftheselinearequations
a 1y
x3
= - b 2 3y x21 = +
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii)
c 2y x3-
= d 6y x5
2 6+=
e 1y x2
5 4+= f 5 3y x
35+ =
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii)
Gradient(m) =
y-intercept(b)=
(i)
(ii) Gradient(m) =
y-intercept(b)=
(i)
(ii)
hint:becarefulwiththesignofthegradient
5ISERIES TOPIC
9Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear Relationships
Fortheequation 2y x= +
Graphing using the intercept and gradient
Afterfirstplottingthey-intercept,thegradientcanthenbeusedtofindasecondpointtohelpdrawthegraph.
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1:Plotthey-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
(i) Writethegradientandy-interceptforthegraphof 2y x= +
7
6
5
4
3
2
1
0
-1
-2
-3
Step 3:Ruleadoublearrowedline throughthepoints
1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
y-axis
x-axis
y-axis
gradient m 111= =
++^ h y-intercept b 2= +^ h
` Graphoftherulemoves 1+ vertically(slopesup)forevery 1+ horizontallyandpassesthroughthey-axisat(0 , 2)
(ii) Describewhatthesevaluesmeanforthegraphofy x 2= +
1+
Step 2:Usethegradienttoplotasecondpoint
Step 4:Writetherulealongtheline
1+
2y x= +y
x2
=
+
10 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear Relationships
Fortheequation y x3 2 12+ =
Rearrangelinearrelationshipsfirsttofindthey-interceptandgradient.
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1:Plotthey-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
(i) Writethegradientandy-interceptforthegraphof y x3 2 12+ =
7
6
5
4
3
2
1
0
-1
-2
-3
Step 3:Ruleadoublearrowedline throughthepoints
1 2 3 4 5 6 7 -5 -4 -3 -2 -1 x-axis
y-axis
x-axis
y-axis
(ii) Usethisinformationtographthelinearequation y x3 2 12+ =
2-
Step 2:Usethegradienttoplotasecondpoint
Step 4:Writetherulealongtheline
3+
move2xovertotherighthandside
gradient m32
32= - =
+-^ h y-intercept b 4= +^ h
y x
y x
432
32 4
= -
= - +
y x3 12 2= -
x x2 2- -
3 3 3' ' 'divideeverytermby3toisolatey
linearequationisnowintheform y mx b= +
y x3 2 12+ =
yx
32
12
+
=
Rememberthegradientis:
Howfarup(+)ordown(-)
Howfaracross(+)
5ISERIES TOPIC
11Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear RelationshipsYour Turn
Graphing using the intercept and gradient
1 Sketcheachoftheselinearequationsusingthey-interceptandgradient
a y x2 3= + b 3 1y x= - +
Gradient(m) =
y-intercept(b)=
Gradient(m) =
y-intercept(b)=
c y x21 3= + d y x
52 2= +
GRAPHING USING THE INTERCEPT A
ND GRADIENT
*
..../...../20...
Gradient(m) =
y-intercept(b)=
Gradient(m) =
y-intercept(b)=
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
12 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear RelationshipsYour Turn
Graphing using the intercept and gradient
2 Sketcheachoftheselinearequationsusingthey-interceptandgradient
a 2y x 4= - - b y x 5= - -
Gradient(m) =
y-intercept(b)=
Gradient(m) =
y-intercept(b)=
c 3y x53= - d y x
35= -
Gradient(m) =
y-intercept(b)=
Gradient(m) =
y-intercept(b)=
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
y-axis
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
y-axis
5ISERIES TOPIC
13Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear RelationshipsYour Turn
Graphing using the intercept and gradient
3 (i) Writethegradientandy-interceptforeachoftheselinearequations(ii) Usethisinformationtographeachlinearequation
a y x4 3 4- = -
Gradient(m) =
y-intercept(b)=
(i) (ii)
b y x3
3 21
+=
(i) (ii)
c x y2 329+ =
(i) (ii)
Gradient(m) =
y-intercept(b)=
Gradient(m) =
y-intercept(b)=
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
14 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
How does it work? Linear Relationships
The linear equation from the graph
Byworkinginreversetobefore,wecanfindthey-interceptandgradientoftheline.Thesevaluesarethensubstitutedintothegradient-interceptformula.
Findtheequationofthelineforthelineargraphbelow:
Findtheequationofthelineforthislineargraph
Forgraphswithonlyonevisibleintercept,searchforanothereasy-to-readpoint.
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
Step 1:Readthey-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
2+
Step 2:Determinethegradientinsimplestform
y mx b= +
y-intercept b 0=^ h Gradient m53
53=
+- = -^ h
y-intercept b 2= +^ h Gradient m32
32=
++ =^ h
` y x53= -
y mx b= +
y-intercept 0=
5+
3+
2y x32` = +
3-
5ISERIES TOPIC
15Linear RelationshipsMathletics Passport © 3P Learning
How does it work? Linear RelationshipsYour Turn
The linear equation from the graph
Matchtheletterofeachgraphwithitslinearrelationship.
A
x-axis
y-axis
1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
0
N P
G
C
H
S
I C R
E
I
5 7 35x y+ =-5x= y x8
4 8= +2 3y x= +3 8 0x y+ =y x=1y
x2= -4 20y x= -
y x3 1= - y x6+ =y x7= +2y x= THE
LINEAR EQUAT
ION FROM THE GRAPH
..../...../20...
16 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
Linear Relationships
Comparing graphs
Bygraphingmorethanonelinearequationontothesamesetofaxes,wecanseeifthereisaspecialrelationshipbetweenthem.
Let’sseewhathappenswhentwolinearequationswiththesamegradientareplottedtogether.
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
Allthreegraphsgothroughthesamepoint:(2 , 2)
` ifyousubstitutex = 2intoanyoftheseequations,ywillbe2
1 2 3 4 5 -5 -4 -3 -2 -1
For:y 2=
For:y x21 1= +
For:y x4= -
Graphtheseequationsonthesamesetofaxes: , y x221 1= = + andy x4= -
Samepoint =
Concurrentlines
m 0= andb 2= +
1m2
= andb 1= +
x-axis
y-axis7
6
5
4
3
2
1
0
Parallellinesnevercrosseachother
1 2 3 4 5 -5 -4 -3 -2 -1
For:y x 1= +
For:y x 3= -
Graphthelinearequations 1y x= + andy x 3= - onthesamesetofaxes
Samegradient =
Parallellines
m 1= andb 1= +
m 1= andb 3= -
-1
-2
-3
-4
-5
1
y
x=
+
y
x3
=
-
Theyallcrosshere
1m = - andb 4= +
Let’sseewhathappenswhenthesethreeequationsaregraphedonthesamesetofaxes.
Where does it work?
y
x4
=
-
(horizontalline)
.1
y
x0 5
=
+
y 2=
5ISERIES TOPIC
17Linear RelationshipsMathletics Passport © 3P Learning
Where does it work? Linear RelationshipsYour Turn
Comparing graphs
1 Comparethegradientsforeachofthefollowingpairsoflinearequations andtickwhethertheyareparallelornot parallel.
a y x2 3= + and 1 2y x= + b x y 4+ = andy x 4= -
Parallel
c y x3
= andy x131= + d y x2 2- = andy x2 2= -
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
y-axis
2 Ineachofthese,onelinearequationhasalreadybeengraphedbutnotlabelled. Comparethegradientofthegraphedequationwiththeun-graphedequation. Tickwhethertheywouldbeparallelornot parallelwhengraphedonthesamesetofaxes.
psst!:Youcangraphtheotherequationifithelps!
a y x 4= +
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
y-axis
NotParallel Parallel NotParallel
Parallel NotParallel Parallel NotParallel
Remember:Parallellineshavethesamegradient
Parallel NotParallel
x-axis
COMPAR
ING GR
APHSCOMPARING GRAPHSCOMPARING GRAPHSCO
MPARING GRAPHS
..../...../20...
18 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
Where does it work? Linear RelationshipsYour Turn
Comparing graphs
b y x32 2= -
Parallel NotParallel
c y x3 9= Parallel NotParallel
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
y-axisd y x2 2 3+ = -
7
6
5
4
3
2
1
Parallel NotParallel
x-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
y-axis
7
6
5
4
3
2
1
x-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
y-axis
7
6
5
4
3
2
1
x-axis
5ISERIES TOPIC
19Linear RelationshipsMathletics Passport © 3P Learning
Where does it work? Linear RelationshipsYour Turn
a , 4 , 4 2 4y x x x= = = -
Concurrent NotConcurrent
b , ,y x y y x21 1 1= = = -
Comparing graphs
3 (i) Sketcheachofthethreelinearequationsonthesamesetofaxes(ii) Tickwhethertheywouldbeconcurrentornot concurrent
Remember:Concurrentlinesallpassthroughthesamepoint
Concurrent NotConcurrent
c 3 6 , 2 , 2 6y x x y x6 8+ = = + =
Concurrent NotConcurrent
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
7 6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
y-axis
0 1 2 3 4 5 -5 -4 -3 -2 -1 x-axis
20 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
Linear RelationshipsWhat else can you do?
Intersection of two linear graphs
Wherethegraphsoftwolinearrelationshipscrosseachotheriscalledtheirintersection point. Atthispoint,thesamexandyvaluesworkinbothlinearequations.
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
For: 2 1y x= +
For: 4y x= -
Fortheequations 1y x2= + andy x4= -
m 1= andb 1= +
1m = - andb 4= +
y
x4
=
-
1
yx2
=+
(i) Graphthelines 2 1y x= + and 4y x= - onthesamenumberplane
Drawverticalandhorizontallinesfromtheintersectionpointtotheaxestofindthecoordinates(ii) Writedowntheirpointofintersection
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
y
x4
=
-
1
yx2
=+
Pointofintersection (1 , 3)
Writedownthepointofintersectionforthelinesx 4= andy 1=
Theintersectionofhorizontalandverticalgraphsiseasilyfoundbycombiningtheirequations.
x 4= y 1=
` Pointofintersectionissimply:(4 , 1)
5ISERIES TOPIC
21Linear RelationshipsMathletics Passport © 3P Learning
What else can you do? Linear Relationships
Hereisasimilarexampleinadifferentcontext.
Twotreasurehuntersarewalkinginstraightlinestowardhiddentreasurealongtwodifferentpaths. Theirpathscrossattheexactlocationofthetreasure.
(i) “One-eyedPi”iswalkingalong2 4y x= +
“Two-eyedSquinty”iswalkingalong 6x =
Graphtheirpathstodeterminethelocationofthetreasure
Thesegraphsintersecteachotherat (6 , 5)
y x2
2` = +
2m b21` = =
` thetreasureisatcoordinates(6 ,5)
(ii) Athirdtreasurehunterbelievesthetreasureliesonthepath 3y x= - .Ifheiswalkingfromthethirdquadrantalongthispath,showwhetherhewillnotfindthetreasureornot.
One-eyedPi:2 4y x= + Two-eyedSquinty: 6x =
` when ,x y6 6 3= = -
= 3
Treasureisat(6 , 5)
` whenx 6= ,hewillbeatcoordinates(6 , 3), 2 unitsawayfromthetreasure ` hewillnotfindthetreasureonthispath.
x-axis
y-axis
1 2 3 4 5 6 7 8
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1 0
substitute 6x = intotheequation
6x=
2
4
yx
=+
22 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
What else can you do? Linear RelationshipsYour Turn
a y x3= - andy x2= b y x2 2= - andy x 4= - +
Pointofintersection: ( , )
c y 3= andx 1= - d x 3= - andy x2 1= - -
Intersection of two linear graphs
1 (i) Grapheachpairofequationsbelowonthesamenumberplane(ii) Writethecoordinatesoftheirpointofintersection
Pointofintersection: ( , )
Pointofintersection: ( , ) Pointofintersection: ( , )
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
INT
ERSECTION
OF TWO LINEAR GRAPHS * * *
..../...../20...
5ISERIES TOPIC
23Linear RelationshipsMathletics Passport © 3P Learning
What else can you do? Linear RelationshipsYour Turn
a y x 2= + andy x53= b 2y x
21= + andy x
23 6= +
Pointofintersection: ( , )
c y x 2= - andy x42
= - d y x= - and 2 1y x4 1 6= +
Intersection of two linear graphs
2 (i) Grapheachpairofequationsbelowonthesamenumberplane(ii) Writethecoordinatesoftheirpointofintersection
Pointofintersection: ( , )
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
Pointofintersection: ( , ) Pointofintersection: ( , )
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
24 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
What else can you do? Linear RelationshipsYour Turn
Remember me?
Intersection of two linear graphs
3 Twosnailshavespottedthebestlettuceinthegardenandaretravellinginstraightlinestowardsit.
*
AWESOM
E *
..../...../20...
* AWESOME *
(i) Graphthepathofeachsnail.
(ii) Whichlettucearetheymovingtowardsifitisatthepointwheretheirpathscross?
SamandShellyaremovingtowardslettuce:
(iii) SluggySteveisalsointhegardenslidingtowardsalettucealongthepath2 3 12y x= - . ShowthatSluggySteveisnotmovingtowardsthesamelettuceasthetwosnailsbygraphinghispath. WhichlettuceisSluggyStevemovingtowards?
SluggySteveismovingtowardslettuce:
(iv)Writedownthelinearequationsfortwodifferentpathsthatpassthroughthecentreofthreelettuces. hint:findthepossiblepathsfirstusingaruler
Firstpathpassingthroughthecentreofthreelettuces:
Secondpathpassingthroughthecentreofthreelettuces:
SlidySamisfollowingthepathy x2 2= -
SlipperyShellyisfollowingthepath y x2 2= - -
x-axis
y-axis
1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
A BC
E
I
J
H
F
D
G
5ISERIES TOPIC
25Linear RelationshipsMathletics Passport © 3P Learning
What else can you do? Linear RelationshipsYour Turn
Intersection of two linear graphs
4 DaniandReecebothrodetheirbicyclesthroughaparktwiceatdifferenttimesoftheday.
(i) Graphthepathofeachridethroughthepark.
(ii) HowmanytimesdidDani’sridingpathscrosswithReece’swithinthepark?
DaniandReececrossedpaths timeswithinthepark.
(iii)WritedownallthecoordinateswhereDani’spathscrossedwithReece’s.
(iv)Writetwoequationsfordifferentpathsthroughtheparkthat only crossthroughoneofReece’spaths.
Firstpaththrough:
Secondpaththrough:
Danirodealongthefollowingpaths: 2 6y x= - and 3y x+ = -
Reecerodealongthefollowingpaths:6 12y x= - and3 9 21y x= -
x-axis
y-axis
-8 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
26 Linear RelationshipsMathletics Passport © 3P Learning
5ISERIES TOPIC
What else can you do? Linear RelationshipsYour Turn
Reflection Time
Reflectingontheworkcoveredwithinthisbooklet:
1 Whatusefulskillshaveyougainedbylearningaboutlinearrelationships?
2 Writeaboutoneortwowaysyouthinkyoucouldapplylinearrelationshipstoareallifesituation.
Ifyoudiscoveredorlearntaboutanyshortcutstohelpwithlinearrelationshipsorsomeothercoolfacts,jotthemdownhere:
3
5ISERIES TOPIC
27Linear RelationshipsMathletics Passport © 3P Learning
Cheat Sheet Linear Relationships
Here is a summary of the important things to remember for linear relationships
Gradient-intercept form
Graphing using the intercept and gradient
Afterfirstplottingthey-intercept,thegradientcanthenbeusedtofindasecondpointtohelpdrawthegraph.
The linear equation from the graph
Byworkinginreversetobefore,youcanfindthey-interceptanddeterminethegradientoftheline.Thesevaluesarethensubstitutedintothegradient-interceptformula.
Comparing graphs
• Graphswiththesamegradientareparalleltoeachother(theynevercross) • Ifanumberofgraphspassthroughthesamepoint,thesegraphsarecalledconcurrent graphs
Intersection of two linear graphs
Wherethegraphsoftwolinearrelationshipscrosseachotheriscalledtheirintersection point. Atthispoint,thesamexandyvaluesworkinbothlinearequations.
Picture Summary
y mx b= +
theconstantterm(b)isthey-intercept
thepowerofxandyisalways1
gradient-interceptform
coefficientofx (m)isthegradient
x-axis
y-axis
7
6
5
4
3
2
1
0
Parallellines(samegradient)
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
y
x3
22
=
+
Concurrentlines(allintersectatthesamepoint)
Intersectionpoints(wherelinescross)
Negativegradient(slopingdown)
Positivegradient(slopingup)
y-intercept
x-intercept
2
y
x3
2=
-
top related