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S  0{ ^ `  >*  0t ^ `  @*:  <%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(a D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<* %(D2 ' =%(D' =%(D' =A@BBBB0B@L>%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 +I 0 G?p $(  ~  s *085P  8   ZA grid1Dgf 8.  <8"`<s#,$0 DLets think about the co-ordinates where y = 2. What could they be?  E 2D  < Q ,$  0 %Draw the line y=1  <8 7 ,$ 0 Or y=-4  048 ,h ,$ 0 (Now try x=3 and x=-2LB   c $D99   6oh / ,$D  02   6oVD,$D02   6oZ L ,$D02  6oG,$D02  6oR> ,$D0  <D"`C2,$ 0 vIf we connect these points we get a horizontal line and everywhere along the line y=2. So, the line equation is Y=2fwRl ? [,$D  0ZB  s *Do?  <  7Y=2l  p c9,$D  0ZB  s *Dop  <8  7Y=1l I 0 0 ,$D 0ZB  s *Do( (  <I w0 8Y=-4l @  v ! m,$D 0ZB  s *Do@ @ v  <@ /  7X=3l x  A~,$D 0ZB  s *Do  <L7x 8X=-2 " <(,r,$ 0 We have drawn simple equations , Lets look at equations that link x and y together & "VU # 0za,$ 0 T"(1,2), (3,2), (5,2) (-1,2),(-4, 2)# 2#2 $ 6oXG,$D0H  0޽h ? 33_4W4___PPT1074.}1 W+ZUD2' = @B D>2' = @BA?%,( < +O%,( < +D]' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<*%(D7' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D'  =-s6Bwipe(left)*<3<*#D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(|DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(dDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D' =%(XDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(MD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bwipe(up)*<3<*!D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D'  =-s6Bwipe(left)*<3<*"++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+"0 ++0+#0 +O 0 ;3%A (   ~  s *\G5     ZA grid1"   0I`,$0 GLets say we have an equation y=x. Give some co-ordinates on this line. H 2G  <P ,$  0 &Draw the line y=2x  <T  ,$ 0 Or y=-x  0X ~ ,$ 0 /Now try y=1/2 x Or y= -1/2xLB  c $D    6op` ,$D  02  6o{ ,$D02  6o = ,,$D02  6o 9J ,$D02  6o=1,$D02  6ol s ,$D0l 2 I % '2 ,$D  0ZB # s *Dod I $  <_2z a  7y=xl ice < 5 ie ,$D 0ZB , s *Doice 1 -  <dxU0< 8y=2xdl    8    D,$D 0TB / c $D   0  <i9   y=-xl    ;   ,$D 0`B 1 0DjJ(   2  Bl   :y=1/2xl KP  7 6KP ,$D 0ZB 3 s *DjJKP  4  < q@ c J  ;y=-1/2x 9 0tj=9,$0 Notice that as the number in front of x gets bigger the line gets steeper. This is the GRADIENT. For y=x, gradient =1, y=2x, gradient = 2^&0  : <L}"`,$ 0 f& (0,0),(1,1), (2,2),(-1,-1), (-2,-2) $'# < <,$  0 QDraw this line equation{l  A m,$D  0 =  08 -So, y=x is at 450 clockwise from the y scale :. @ mP @ mP`B > 0DomPTB ?  c $DPH  0޽h ? 3387___PPT107.}1 W+ЯɯD*6' = @B D5' = @BA?%,( < +O%,( < +D]' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<* %(D7' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*: %(D' =-s6Bwipe(left)*<3<*: D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*< %(D' =-s6Bwipe(left)*<3<*< D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*% %(D' =-s6Bwipe(down)*<3<*% D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A %(D'  =-s6Bwipe(left)*<3<*A D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5 %(D' =-s6Bwipe(down)*<3<*5 D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =-s6Bwipe(down)*<3<*8 D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*; %(D' =-s6Bwipe(down)*<3<*; D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7 %(D' =-s6Bwipe(down)*<3<*7 D]' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<*9 %(++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+9 0 ++0+: 0 ++0+< 0 +4 0 %C(PC(  (~ ( s *P5P    ( ZA grid11  ( 0$n ,$0 f*What are the gradients of the lines below  + 2*z 2 I ( '2 ,$D 0`B ( 0Dod I ( <2z a  7y=xl z  &(@" ,$D 0fB ( 6Doz  ( B9   8y=2xz    (    D,$D 0`B ( 0Do   ( <ė9   6y=-xl   '( ,$D 0`B ( 0DjJ'  ( BPȗ   :y=1/2xz KP  ( P ,$D 0ZB ( s *DjJKP  ( <̗@ c J  ;y=-1/2x  ( 0XїvH ,$ 0 Notice that y=-x has a negative gradient of 450 anticlockwise from the y scale Negative gradients point to the left, positive gradients point to the rightF k (( <חo[p,$D  0 Qy=x y=-x y=2x y=-2x y= x y=-x   *( <ܗop,$D  0 A1 -1 2 -2 -l SH -(HS,$D  0 +( <SH 8Equation   ,( <LH 8Gradient  z M 2( @+ ,$D 0`B 3( 0DoI 4( <f M 9y=-2x8 F)   A(   T mP :(# F) ) `B ;( 0DomPTB <( c $DP@ )   @()  fB >( 6Do)  ZB ?( s *D{   B( <` WC$ JPositive Negative Gradient^ C( 6o YIH ( 0޽h ? 33___PPT10.}1 W+D' = @B D<' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(a D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(down)*<3<*(D' =%(=D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(down)*<3<*(D' =%(1D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(down)*<3<*(D' =%(%D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'(%(D' =-o6Bwipe(up)*<3<*'(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&(%(D' =-s6Bwipe(down)*<3<*&(D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2(%(D' =-s6Bwipe(down)*<3<*2(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-(%(D' =-o6Bwipe(up)*<3<*-(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*((%(D' =-o6Bwipe(up)*<3<*((D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**(%(D' =-o6Bwipe(up)*<3<**(D]' =%(D' =%(D' =A@BBBB0B@>%(D' =1:Bvisible*o3>+B#style.visibility<* (%(++0+(0 ++0+ (0 ++0+((0 ++0+*(0 +g 0 43`aaX3(  Xx X c $ 5P   Hz @ p X @,$D 0b XB 6Sp (TB X c $D%%pTB X c $DpTB X c $DRRpTB X c $D@TB  X c $DITB  X c $DR  z 7p   X 7 ,$D 0R  X 6`7p ,  X T"`  73z  X ,$D 0b X 6= , X N"`(bO 73z e p X e ,$D  0`b XB 0z pTB X c $Dn n pTB X c $D@ @ pTB X c $D  pTB X c $De TB X c $Dn z 7p  X m  ,$D  0R X 6%7p , X T<("`!  73z  X ,$D  0b X 6&= , X N,0"`(a 72z  X h,$D 0b X 6,6= ,  X N8"`(aN 73z 7p  !X BV) ,$D 0R "X 6H=7p , #X T@"`  71Zz I} $X I",$D 0`b %XB 0VpTB &X c $DVV}TB 'X c $DITB (X c $DRTB )X c $DR z  D  *X  e ,$D 0N  J  +X  J N t J  ,X t J  -X <pH    change in y .X <Ltc J   change in xZB /X s *Do   0X <lP    Gradient =N  [  1X  ^ T  J  2X#  [  3X <@T" =  3 4X <Wc -J  3ZB 5X s *Do   6X <[    Gradient = 7X <x_] :D   Gradient = 1z   8X  ,$D0 9X <a 7y :X <e  7xz  ? ;X ? ,$D 0 X #%,$D0 ?X <rD# 7y @X <k% 7xz  D  AX  je ,$D 0N  J  BX  J N t J  CX t J  DX <X|    change in y EX <ttc J   change in xZB FX s *Do   GX <l    Gradient =N  [  HX  ^ T  J  IX#  [  JX <" =  2 KX <8c -J  3ZB LX s *Do   MX <    Gradient = NX <] D  "Gradient = 2/3z  D  OX  e ,$D 0N  J  PX  J N t J  QX t J  RX <    change in y SX <Xtc J   change in xZB TX s *Do   UX <,    Gradient =N  [  VX  ^ T  J  WX#  [  XX <" =  3 YX <c -J  1ZB ZX s *Do   [X <    Gradient = \X <8] :D   Gradient = 3 ]X 6o ,$D 0 ^X 0h6,$ 0 s=Gradient = change in vertical divided by change in horizontal> 2> _X` `,)Aj0212957/<$D0  `X@ s <A j0212957? ,$D 0 aX@ s <A j0212957t*,$D0H X 0޽h ? 3333___PPT10m3./2+#RJD 3' = @B D2' = @BA?%,( < +O%,( < +Dt' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8X%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_X%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-s6Bwipe(left)*<3<* XD' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]X%(D' =-s6Bwipe(left)*<3<*]XD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**X%(D' =-o6Bwipe(up)*<3<**XD ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;X%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`X%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(left)*<3<*XD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*AX%(D' =-o6Bwipe(up)*<3<*AXD ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$X%(D' =-s6Bwipe(down)*<3<*$XD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*aX%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>X%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!X%(D' =-s6Bwipe(left)*<3<*!XD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*OX%(D' =-o6Bwipe(up)*<3<*OXD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^X%(D' =-s6Bwipe(left)*<3<*^X+8+0+^X0 +r5 0 1|1pbb\1(  \x \ c $(5P   \0l  b\,$D0N 7p  \ pd R \ 67p , \ TD"`  73N 7p  \ 0 Cy R \ 67p , \ T"`!  73N  D   \  e N  J   \  J N t J   \ t J   \ <d    change in y  \ <tc J   change in xZB \ s *Do   \ <8    Gradient =N  [  \  ^ T  J  \#  [  \ <\" =  3 \ <c -J  3ZB \ s *Do   \ <    Gradient = \ <] jD  ! Gradient = -1N  D  \ j e N  J  \  J N t J  \ t J  \ <    change in y \ <tc J   change in xZB \ s *Do   \ <     Gradient =N  [  \  ^ T  J  \#  [   \ < " =  2 !\ < c -J  3ZB "\ s *Do   #\ <    Gradient = $\ <p] D  #Gradient = -2/3N  D  %\  e N  J  &\  J N t J  '\ t J  (\ <    change in y )\ <4tc J   change in xZB *\ s *Do   +\ <     Gradient =N  [  ,\  ^ T  J  -\#  [  .\ <(%" =  3 /\ <(c -J  1ZB 0\ s *Do   1\ <\,    Gradient = 2\ </] jD  ! Gradient = -3f 3\ 6o  4\ 0426 s=Gradient = change in vertical divided by change in horizontal> 2>"T @ p 5\C# b 6\B 607Sp (TB 7\ c $D%%pTB 8\ c $DpTB 9\ c $DRRpTB :\ c $D@TB ;\ c $DITB <\ c $DR  T  =\C# @b >\ 6(== , ?\ N?"`(bO 73T   @\C# .(  A\ <(D 7y B\ <B  7x C\b `hAj0212957"?./ T e p D\C# q `b E\B 0z pTB F\ c $Dn n pTB G\ c $D@ @ pTB H\ c $D  pTB I\ c $De TB J\ c $Dn T  K\C# e C b L\ 6O= , M\ NQ"`(a 72T  ? N\C# G ?d  O\ <U?_ 7y P\ <tZ   7xz Q\B s <ڞA j0212957"?D T  R\C# tRb S\ 6_= , T\ N]"`(aN 73T 7p  U\C# B R V\ 6$f7p , W\ Ti"`  714T I} X\C# "`b Y\B 0VpTB Z\ c $DVV}TB [\ c $DITB \\ c $DRTB ]\ c $DR T #% ^\C# % _\ <oD# 7y `\ <s% 7xz a\B s <hMA j0212957"?hH \ 0޽h ? 33___PPT10n./2+oDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b\%(+?Y 0 H@%O$ (  $~ $ s *d{5P    $ ZA grid1 u LB $ c $D--  $ 6oGp,$D 02