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5/23/2018 Chapter 1 Coordinate Geometry
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Centre For Foundation Studies
Department of Sciences and Engineering
Chapter 1
FHMM1034Mathematics III
1
5/23/2018 Chapter 1 Coordinate Geometry
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Topics1.1 The Cartesian Coordinate
1.2 The Straight Line
1.3 Shortest Distance from a Point to aStraight Line
FHMM1034Mathematics III
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1.4 Circle
1.5 Intercepts and intersections
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1.6 Parabola
Topics
1.7 Ellipse
1.8 Hyperbola1.9 Shifted Conics
FHMM1034Mathematics III
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1.10 Parametric Equations
1.11 Loci
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.
The CartesianCoordinate
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Cartesian Coordinate
y
0x
a
bP(a, b)
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Distance between 2 points
lane.coordinate
on the),(and),(pointsLet 2211 yxQyxP
y
Q(x2,y2)
2 1y y
2y
FHMM1034Mathematics III
60 x
P(x1,y1)
6
2 1x x
2x1x
1yR(x2,y1)
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Distance between 2 points
The Pythagorean Theorem gives
DISTANCE FORMULA
2 2
2 1 2 1PQ x x y y= +
2 2
2 1 2 1( ) ( )x x y y= +
FHMM1034Mathematics III
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( ) ( )
1 1 2 2
2 2
2 1 2 1
The distance for ( , ) and ( , ) is
P x y Q x y
x x y y +
7
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Example 1
Which of the points P(1, 2) or Q(8, 9) is
closer the point A(5, 3) ?
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Example 2
pointsthreeare),3(and)1,4(,)5,2(25CBA
area.itsfindandtriangleangled-righta
isthatshowHence,.and,ofdistancetheFindplane.coordinateon the
ABCACBCAB
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Point Dividing Straight Line
pointjoininglinethedivides),(If yxR
then,:ratioin then erna y,po nan, 2211
yxyx
+=
+=
1212 , yy
yxx
x
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Point Dividing Straight Line
,externallylinethedivides),(If PQyxR
then,signsoppositehavewilland
=
=
1212 , yy
yxx
x
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Example 3
divideswhichpointtheofscoordinatetheFind R
internally(i)
2:5ratioin the)5,4(pointtheand)3,8(pointthejoininglinethe
Q
P
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externally(ii)
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Point Dividing Straight Line
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R divides PQ internally. R divides PQ externally.
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Mid-Point
and),(pointofpoint-midThe 11 yxP
are),(point 22 yxQ
++
2,
2
2121 yyxx
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Example
Show that the quadrilateral with vertices
(1, 2), (4, 4), (5, 9) and (2, 7) is a
paralelogram by proving that its two
diagonals bisect each other.
P Q R S
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1.2
The Straight Line
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Straight Line
passinglinestraightaofslopegradient /The m
is),(and),(pointsthrough 2211 yxQyxP
2112 , xx
xx
yym
=
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The Equation of Straight Line
throughpassinglinestraightaofequationThe
,,2211
121
xxyy
xxyy
=
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The Equation of Straight Line
gradientwithlinestraightaofequationThe m
s,po ntet roug tpass ngan a
)( axmby =
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Example 4
passinglinestraighttheofequationtheFind
.,an,po ntset roug t KH
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Example 5
withlinestraighttheofequationtheFind
).5,1(pointsthroughtpassesthat32gradient
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Parallel and Perpendicular Lines
linestheofgradientstherepresentandLet 21 mm
.tanandtanThen,
axis.positivethely torespective,
2211
2121
==
mm
x
parallel.areandlinestwothe,If(i) 2121 llmm =
FHMM1034Mathematics III
22
lar.perpendicu
areandlinestwothe,1If(ii) 2121 llmm =
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Example
(a) Find an equation of the line through
t e po nt , t at s para e to t e line 4 6 5 0.
x y+ + =
FHMM1034Mathematics III
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perpendicular to the line 4 6 5 0
an
x y+ + =
d passes through the origin.
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Angle Between 2 Lines
andlinestwoebetween thangleThe 21 ll
yv
21
12
1
tan
mm
mm
+
=
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angle.obtuseanis,0tanIf(ii)
angle.acuteanis,0tanIf(i)
1l2l
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Example 6
linesstraightebetween thangletheFind
.02035and0843 =+=+ yxyx
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1.3
Shortest Distance
from a Point to a
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The Shortest Distance
),(pointafromdistanceshortestThe
=
khPd
22 ba
cbkahd
+
++=
FHMM1034Mathematics III
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ve).orve(eithersignsamehaveall
sexpressionthe,0linestraight
,
+++
=++
cbkah
cbyax
ii
ii
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Example 7
fromdistancelarperpendicutheFind
.0543linestraighttheto)4,2(pointthe
=+
yxP
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Example 8
ifdeterminediagram,ausingWithout
.042
linestraighttheofsidesametheonlie)2,(and),2(pointsthe 2121
=+ yx
QP
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1.4
Circle
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Circle
),(centrewith thecircleaofequationThe baC
sra usan r
222 )()( rbyax =+
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,
222 ryx =+
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Circle
:iscircleaofequationgeneralThe
02222 =++++ cfygxyx
and),(iscentreitswhere fg
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.isradiusits cfg +
E l 9
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Example 9
circletheofequationanFind
).5,2(centerand3radiuswith
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E l 10
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Example 10
thehasthatcircletheofe uationanFind
diameter.theof
endpointstheas)6,5(and)8,1(point QP
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E l 11
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Example 11
ofgraphthat theShow
radius.andcentreitsfindandcircle,ais
09)34(2 22 =+++ xyyx
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Example 12
at thecentrewithcircletheofe uationtheFind
.632
linethefromcentretheofdistanceshortest
thetoequalradiushavingand)2,3(point
+= xy
36
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Example 13
.02042circleon the
22=++ yxyx
37
E l 14
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Example 14
)2,4(pointtheofdistancelarperpendicutheFind A
pointtheofscoordinatethealsoFind.42line
straightthetouching)2,4(centrewithcircleaof
equationthefindThen,.42linestraighttheto
=+
=+
yx
A
yx
38
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.Intercepts andintersections
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I
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Thex-coordinates of the points where a graph
Intercepts
intersects thex-axis are called thex-intercepts ofthe graph andy = 0.
They-coordinates of the points where a graph
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intersects they-axis are called they-intercepts ofthe graph andx = 0.
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Intercepts
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Sety =0 and solve forx Setx = 0 and solve fory
E l 15
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Find thexintercepts andyintercept of the
Example 15
equation .22 =xy
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P i t f I t ti
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In general, the coordinate of the points of
Points of Intersection
intersection of two equations can be found bysolving the two equations simultaneously.
Each real solution gives a point of intersection.
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T f P i t f I t ti
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(i) 2 distinct real roots
Types of Points of Intersection
(ii) 2 equal real roots(iii)No real roots
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(i) (ii) (iii)
Example 16
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Example 16
Find the coordinates of all points of intersection
xxxyxyyyxxy65,2(ii)
033,32(i)23
22
++===+=+
e ween e curves n eac o e o ow ngcases.
FHMM1034Mathematics III
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Example 17
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Example 17
and242ofra hsSketch the x =+
,andofscoordinatethefindingWithout
.andon,intersectiofpointsmark theand
diagram,sameon the106
2
QP
QP
xxy +=
FHMM1034Mathematics III
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.ofpoint-midtheofscoordinatethefind PQ
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1.6
Parabola
FHMM1034Mathematics III
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Parabola
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Parabola
A parabola is the set of points in the plane
equ s an rom a xe po n ca e e ocus and a fixed line l (called the directrix).
FHMM1034Mathematics III
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F(0,a)
y = - a
a
a
Parabola
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Parabola
isparabolaaofequationgeneralThe
ayx 42 =
directrix.thecalledislinefixedthe
andfocusthecalledis),0(pointfixedThe
= ay
aF
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.0foronlyexists
andaxisaboutsymmetricisgraphThe
y
y
Parabola
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If a>0 opens upward If a
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Parabola
Parabola with vertical Parabola with horizontal
Equation :
Properties: Vertex V (0,0)
Focus F (0, a)Directrix y = - a
Equation:
Properties: Vertex V (0,0)
Focus F (a, 0)Directrix x = - a
ayx 42 = axy 42 =
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Parabola
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Parabola
The line segment that
run t roug t e ocusperpendicular to the
axis with endpoints on
the parabola is called
aa
2a
FHMM1034Mathematics III
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e a us rec um an
its length is the focal
diameter.
,
x = -a
Example 18
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Example 18
a) Find the equation of the parabola with vertex
, an ocus , , an s etc ts grap .
b) Find the focus and directrix of the parabola
and sketch the graph.06 2 =+ yx
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,
the parabola and sketch its graph.221 xy =
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1.7
Ellipse
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Ellipse
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An ellipse is the set of all points in the plane the
Ellipse
sum o w ose s ances rom xe po n s.
These 2 fixed points are the foci of the ellipse.
y
P(x, y)
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x0F1(-c,0) F2(c,0)
Ellipse
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Ellipse
isellipseanofequationgeneralThe
12
2
2
2
=+b
y
a
x
axisma orthecalledis2andGenerall >
FHMM1034Mathematics III
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axis.minorthecalledis2and
b
Ellipse
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Ellipse
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Ellipse
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Ellipse
12
2
2
2
=+a
y
b
x1
2
2
2
2
=+b
y
a
xEquation: Equation:
0>> ba0>>
baa a
c 222 bac = c222 bac =
c c
Vertices: ( , 0)
Major axis: 2a
Minor axis: 2bFoci: ( ,0) ,
Vertices: ( 0, )
Major axis: 2a
Minor axis: 2bFoci: (0, ) ,
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a
e=
a
e=, ,
Ellipse
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Eccentricity of an ellipse, e
Ellipse
the ellipse is.
Eccentricity of an ellipse,
The eccentricit of ever elli se satisfies
.
a
ce=
.10
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a p e 9
representequationsfollowingtheofeachthatShow
84(ii)
42(i)
:e psean
22
22
=+
=+
yx
yx
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curves.sketch theandcase,eachpropertiestheallState
Example 20
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Example 20
a) Find the equation of an ellipse with its
vert ces are an t e oc are .
b) Find the equation of the ellipse with foci
and eccentricity .4
=e
, ,
)8,0(
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Example 21
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Example 21
Find the equations of the tangents with gradient 2
,63222
=+ yxto the ellipse with equation andfind their points of intersection.
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1.8
Hyperbola
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Hyperbola
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yp
A hyperbola is the set of all points in the plane,
points. These fixed points are the foci of thehyperbola.
y
P (x, y)
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x0 F2 (c, 0)F1 (-c, 0)
Hyperbola
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yp
ishyperbolaaofequationgeneralThe
12
2
2
2
=b
y
a
x
:thatNote
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.forexistnotdoescurveThe
axa
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veorveeitherofvaluelar eFor
:Notice
x +
:i.e.
2
2
22
b
x
a
by
x
FHMM1034Mathematics III
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.asymptotestheareHence, xa
by
a
=
Hyperbola
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yp
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12
2
2
2
=b
y
a
x
Hyperbola
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Equation : Equation:
yp
12
2
2
2
= yx
12
2
2
2
= xy
Vertices ( ,0)
Asymptotes
Foci ( ,0) ,
Vertices (0, )
Asymptotes
Foci (0, ) ,
a
xa
by =
a
xb
ay =
c 222 bac += c 222
bac +=
FHMM1034Mathematics III
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The Rectangular Hyperbola
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:If ba =
toreduceshyperbolaaofequationThe
222ayx =
h erbolarrectan ulathecalledish erbolaThis
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.areasymptoteswith the
xy =
The Rectangular Hyperbola
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The graph of and
222ayx =
2cxy =
are shown below222
ayx =y
2
cxy =
FHMM1034Mathematics III
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x0
Example 22
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:curvesfollowingeSketch th
5(ii)
149(i)
22=
=
yx
yx
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4)1((iii) =yx
Example 23
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a) State all the properties of the hyperbola and
b) Find the equation of the hyperbola withvertices and foci .
099(ii)144169(i) 2222 =+= yxyx
)0,4()0,3(
FHMM1034Mathematics III
72
c) Find the equation of the hyperbola with
vertices and asymptotes .)2,0( 2=y
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1.9
Shifted Conics
FHMM1034Mathematics III
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Shifted Conics
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In previous section, we studied parabolas with
vert ces at t e or g n an e pse an yper o as
with centers at origin.
In this section, we consider conics whose
FHMM1034Mathematics III
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origin, and we need to determine how this affectstheir equations.
Shifted Conics
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Given h and kare positive real numbers,
Replacement How the graph is shifted
1. x replaced byx h
2. x replaced byx + h
Right h units
Left h units
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(Page 776)
.
4. y replaced byy + k
Downward kunits
Shifted Ellipses
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General equation of an ellipse:
If we shift it so that its center is at the point (h, k)instead of at the origin, then its equation
122 =+by
ax
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becomes:1
)()(2
2
2
2
=
+
b
ky
a
hx
Shifted Ellipses
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(Page776)
Example 24
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Sketch the graph of the ellipse
and determine the center.
19
)2(
4
)1( 22=
+
+ yx
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Shifted Parabolas
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Applying shifts to parabolas leads to the
equations and graphs shown as followings:
FHMM1034Mathematics III
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0for
)(4)()i( 2
>
=
a
kyahx
0for
)(4)()ii( 2
=
a
hxaky
0for
)(4)()iv( 2