Chapter3–GraphingLinearEquationsRectangularCoordinateSystemCartesianCoordinateSystemOriginQuadrantsy-axisx-axisScaleCoordinates

Ex:Ploteachpoint:(0,0),(-1,3),(1,3),(1,-3),(-1,-3)

D location

Descartes} "GaPhig

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⇐aaxxiesgi.IE2 -

<

o•; > × Ordvetpair :( X. y )

.

v

I YnIi(x ,y )

z •

→• =\

,)= Graphing

2- Right 1

1- ( 0,0 ) ( 11

' 3)Down }

< 1 1 1 •.... 1 1 > X-3-2 - till23

( y,} ) Leftl ( -1

,

-3) left 1

-2 -

: Up 3 Down }• - •( i , } ) Right

'QuadrantsIT ✓ Up 3

-

I] 4

.

attendto Trig

Right Angle

Me't nd}#rectangleatstart

Ex:Givethecoordinatesofeachpoint:

Ex:(#22)Giventhefollowinggraphthatgivestheheartrateofawomanbefore,duringandafteranaerobicworkout.

a)Whatwasthewoman’sheartratehalfanhourafterbeginningtheworkout?b)Forhowlongdidthewomanworkoutathertrainingzone?

I

IAC 5,0 )

F¢5,-3.5 )

B( 1,4 )< :

: cf4,-3 )

:

# - : .

"D( 0,0 )

I ✓ ¥EC -5,5 )

P1015209530;

: :

"::#: :

.

Half an hour = 30 min

( 30,140 ) -→ 140 beatspermin .

10mi -→ 40mm 30 minutes

3.2GraphingLinearEquationsEx:Listpairsofnumbersthatsatisfytheequationy=x+1byconstructingatableofvalues:c)Does(4,4)satisfythatequation?Howabout(5,-5)?d)Plotthosepairsonarectangularcoordinatesystem.Doyouseeapattern?

if"

'Ff¥¥I26 } 22+1=3

66+1=7

I¥,÷¥t¥u "

•

.l¥I¥¥¥÷¥±i)

7- • ( 0,1 ) ( T,Tt D

6 - T5- ( 1,2 )

34µW( z ,} )

-43.44'D

i - • ( 6,7 )i•<viii.

t.tt ; > ×

• (5. ' 5). .

Ex:Graph2x–y=-3bysolvingforyfirstandthensettingupatableofvalues.

2×-4=-3 y= 2×+3

zxtI.IT#fEIEetfE5+ 3 +3-

2 2 # 3=4+3--72×+3 = Y 3 2C3) +3--6+3=9yY=21¥- 1 2ft )+3= -2+3=1

8

:.5 - •H•< I.

iiiz's ; > ×

.

3.3InterceptsEx:Graphtheliney=3x–6byfindingthex-andy-interceptsandusingonemorepointasacheck.

HorizontalandVerticalLinesEx:Graphy=3bysettingupatableofvalues.

← Math English

Running into something -- intersect t¥¥.

.←y.int#Y )

4=3×-6y.int :×=o

{•#y=3(o)-6

y =O - 6

Y=- 6 X - int :* 0.

Y ( o ,- 6)

n 0=3×-6

" ,×

:* IFby 64 - §= 3¥2 -

z=X

6 @

*< >-2 - ( 2,0 )-4 - Pick one point :x=4;

✓ ¥3,466 ( 4,6 )= 6

yY=3 Horizontal ^

( 2,3)

¥5line <••<> x

- it 3

Y48ow| }

v

Ex:Graphx=-3bysettingupatableofvalues.

Ex:(Ex8)Usinginformationabouta2010ToyotaPriusHybrid,wecansetuptheequation 12m+600g=7200wheremrepresentsthenumberofmilesdriven,andgrepresentsthenumberofgallonsofgasleftinthetank.Findtheinterceptsforthislineandexplainwhattheyrepresent.

*s

§fyf¥¥EFI< :p},

" ?}Eop.VerticalGm

§o±⇒a so

m=O 120+6009=7200 ToOt 6009=7200 §

s( MB )600g 7200 # °

=( 0,12) 60-0=60-0 or

g=R 3 ?

Before you drive anywhere ,

the car has 11 J12 gallons of gas .= size of the

gas tank . TWO

go.rmt.fm?ey=wnooo7#mI6Fo

.( m , g) = ( 600,0 )

3.4SlopeandRateofChangeEx:Pretendyouaregoingtogoskiingdownamountain.Drawexamplesofthefollowing:a)averysteepslopeb)aslightlysteepslopec)aflatslopeQuestion:Howcouldwedifferentiatebetweengoingupamountainandgoingdownamountain?Question:Howcouldwequantifyhowsteeptheslopesare?

YI.

¥f¥#Fit÷- 1

←.#tIf¥a¥ea

Slope=!"#$!%& =

)*+&,$"&-)*+&,$"&.

Ex:Graphy=2x–1andthen,usingtriangles,calculatetheslopebetween:

a)x=0andx=2 b)x=2andx=6 c)x=-5andx=1Whatdoyounoticeaboutallthreeslopes?

Rie=Y 's

Fun Fnx 's~

sin't

•µ)y=2x. I

I

;Y÷•l.ie#bnionionsii.t.IAY.imSEE'¥o¥'s 'm- 1 •516 z 2.2-1=4 - k ) ( 3,5)ai#¥¥II÷f¥¥E2i5)g-y5z=XRR¥n=z4=z

⇒× . '

iE÷da¥=zT¥¥¥a

-2 zfz ) - 1

= - 5a.al

2k¥ .

#2

UsingtheSlopeFormulaTheslopeofthelinepassingthrough(x1,y1)and(x2,y2)isgivenbytheformula:

slope=m=-/0-1./0.1

Ex:Usingtheslopeformula,findtheseslopes:1)Theslopeofthelinepassingthrough(-1,3)and(2,6)2)Theslopeoftheliney=33)Theslopeofthelinex=3

= ¥n¥÷fI÷¥¥E¥÷¥÷¥÷,

= ;5¥= IT -2

m=¥nII¥e ¥ 5*5=1or }I==÷= 1 #

m = II = YIXz - X

,

y =3

K¥3 m= #= 0=2--0

IT m=¥n¥¥=yg== f- oh no

!

Undefined

ParallelandPerpendicularLinesParallelLines:PerpendicularLines:Ex:Fillinthischart:

Slope ParallelSlope PerpendicularSlope

2

3

12

34

−56

-6

0

Same SlopeM

,= mz

¥ %\a

oppositeReciprocals"

<f€°°s #' • change the sign

M ,mz=- 1 or me #

" flip UpsideDown

2 - ±3 - st± - z

÷ - ¥- E + E

- 6 + I0 undefined

Ex:Determinewhetherthelinesthrougheachpairofpointsareparallel,perpendicular,orneither.1) (3,3)and(4,4) (3,3)and(2,4)2) (2,4)and(-1,-1) (8,0)and(11,5)

← Line I← Line 2

Line I m :

m.ir#=hxIx.I=3I4'- ¥=l Parallel ? No

" mm?Ir÷ey÷¥=¥e¥= . ,I ? YesDifferentSigns

• Flipped

mi¥n=k¥eIi¥F¥i÷ panel ?

me # =Ii¥=F÷F=¥ Yes=§

3.5Slope-InterceptFormIfyouhaveanequationofalineinthisform: 2x–y=5andyousolvefory:thenyouhaveSlope-InterceptForm,whereEx:Findtheslopeandthey-interceptofthelinewithequation 9x–3y=10Ex:Writeanequationofthelinewithslope=-2andy-intercept(0,5.4)

2×-4=5

ty+y×¥Iy¥fk2×'s

4=2×-5 ifmxtbp p in Tm y.int slope yint

( o ,. 5)

Solve try :

a*¥¥÷S¥E¥i¥÷¥fEEEI¥Coins )

Y= -2×+5.4

Ex:Useslope-interceptformtograph 2x+2y=-6

Ex:Determinewhetherthegraphsofy=4x+6andx+4y=-2areparallel,perpendicularorneither.

"

¥÷¥÷¥S*E±¥a⇒m= - 1

3= b= -30,-3 )

-3 -1-

< Toy:L ; > × m= - 1

%§i.ie:6#gg• •ig . ,

v ¥31 - 4Y 2 - 5

4 Same slope

1 Opposite reciprocal slope

4=4×+6 mi

4¥IiI¥j¥¥¥I, 'fP%¥y=-¥ . ÷ met- in

Ex:(#100)AnewPlaystation3costs$310.50andmembershipinanonlinevideogamemultiplayernetworkcosts$18.49permonth.a)Writealinearequationthatgivesthecostforsomeonetobuythemachineandbelongtotheonlinenetworkformmonths.b)Useyouranswerinpartatofindthecosttobuythemachineandbelongtothenetworkfor3years.

:8.49mi310.50- -

Variable Kxed

Y= 18.49 m + 310.50

m=# months

3 years = 36 months

Y= 18.49 ( 3 6) +310.50= $976.14

3.6Point-SlopeFormm=Point-SlopeForm:y–y1=m(x–x1)Ex:Findanequationofthelinethathasslope5andpassesthrough(-1,13).Writetheanswerinslope-interceptform.

You don't need to know this

Yz - Yl

Fx ,

Gzx ,)m=YIg¥4¥ )

( xz - × , )m= Yz - Yi

Yz - Y , = ( xz - ×, )m

yz - Y , = m ( xz - × , )

t

y . y , = m ( x - xD

m=5 C- I , 13 ) ← Not an

intercept

y - y ,=m ( x - x, )

y - 13=5 ( x - G) )

y- 13=5 ( xti ) Point . Slope

Form

Solve for y :

Y - 13=5×+5+ 13 + 13

y=5xt Slope - InterceptForm

Ex:Findanequationofthelinethathasslope0andpassesthrough(3,4).Ex:Findanequationofthelinethatpassesthrough(4,2)and(-1,12).Writetheanswerinslope-interceptform.

me 0 ( 3,4 ) ← Nfitvoinpt

y- Y ,=mCx - x

, )

y- 4=06 - 3) Point . Slope

for form

y- 4=0+4+4-4=4

Method I : Point Slope Form

Slept : Find the slope

m=YI¥=t}=¥= -2

Step 2 : Use Point . Slope form

Pick one point : ( 4 ,2) m= -

zY - Y ,=m( x - X )

y-

2=-2( X . 4)Step 3 : Solve Fry C if necessary )

a

y -2=-2 ( × - 4)

y . 2=-2×+8+ 2 + 2

y=2xt←s6pEIt

Method # 2 : Slope - Interceptform

( 4,2) t I

,1 2)

step 1 : find the Slope

m.IE#=Eg=Ef=Fo= -2

Step 2 : shove the points into

Slope . Intercept form

y=m×+b ( 4,2) ft,I 2)

y=- Zxtb T 9

Not intercepts

Pick One point : f bid in

y=- 2×+6

12=-2 fi ) + b12=2 + b4=-2*-2-210

= b