Example1: Sketchandannotate
the graph
o fy = x ' - 2 x - 1 5
Note: whenquicklysketchinga quadraticgraph,the rootsand shapeC'happy"or "sad" face)are enough.
If thegraphof a quadratic
functionhasrootsat x = -L
guessat itsequation
andx5 5, a reasonable
wouldbe
=
y
(x
+
1)(x- 5).
)/ x' 4x 5, i.e.from
However,
asthediagram
shows,therearemany
parabolas
whichpassthroughthesepoints,allof which
belongto thefamily of functions
f = k (x + 1) (x- 5).
we needthe
To findtheequation
of the originalfunction,
rootsandoneotherpointon thecurue(to allowusto
determine
thevalueof $.
Exampfe2: Statethe equation
of thegraphbelowin the formy = axz+ bx + c .
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Thediagramshowsthegraphsof two quadratic
functions.
If the graphof y = x2 is shiftedg unitsto the right,
foflowed
by runitsup,thenthegraph
of y= (x-q)'+ r is
obtained.
Astheturningpointof f = x2 is (0, 0), it followsthatthe
newcuruehasa turningpointat (q, r).
'
,'y= (x-q)z + r
ta'
Notethat asthe dottedgraphdoesnotcut the x- axis,we
wouldnotbeableto findtheturningpointviafindingthe
roots.
A quadratic
equation
writtendsy = p (x + q)' + r is said
to be in the completedsquareform.
Example3:
(i) Writethe following
valueof
in the formy- (x + q)2 + r andfindthe minimum
(ii) Hencestatethe minimum
valueof x.
valueof yand thecorresponding
a )Y = x ' + 6 x + L 0
b )Y = x 2 ' 8 x + 3
c )Y = x z ' 5 x
d )Y = x ' ' 3 x + L
Example4: Writey = 3x' + IZx + 5 in the
f o r my = p ( x + q ) ' + r .
Example5: Write)/ = 5 + IZx - x2 in the form
y=p-(x+g)r.
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Example6:
a) Write)/= x2 - Llx+ 28 in the form
y = ( x + p ) 2+ g ,
valueof
b) Hencefindthe maximum
18
Llx
x2
+28
the square,or
equations
whichdo noteasilyfactorisecanbesolvedin two ways:(i) completing
Quadratic
(ii) usingthe quadraticformula.In fact,bothmethodsareessentially
the same,asthe quadraticformulais
obtainedby solving)/ = axz+ bx + c via completing
the square(seep L49for proof!).
Example7: Solve5x2- 3Ox- 18= 0 by:
a) usingthequadratic
formula
b) completing
the square
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are easilysolvedby makinga sketchof the equivalentquadraticfunction,and
Quadraticinequations
determiningthe regionsaboveor belowthe x- axis.
Example 8: Findthe valuesof x for which:
a) 2x2- 7x + 6 > 0
b)2x'-7x+6<0
First, sketch y = 2x' - 7x + 6
The Discriminant
Roots of
Usingthe quadraticformula,find the rootsof eachof the threequadraticequationsbelow.
(i)Y=x'-4x+3
(ii)Y=x2-4x+4
(iii)Y=x2-4x+5
yetthe rootsarealldifferent.
Thenature of the rootsis
In eachequation,
onlythevalueof cchanges,
purelybythe partof theformulainsidethe squarerootsign,i.e.bz '4ac .
determined
(i), b' - 4ac> Q,andtheequation
In equation
hadtwo unequal(i.e.differentor disctincflroots.
(iD,b2^-4ac= 0, andthe equation
hadtwoequal roots (asxz - 4x + 4 = (x- 2) (x- 2)).
In equation
(iii),
<
In equation b' 4ac 0, andtheequation
hadno real roots.
Fory- axz + bx+ c, b2 -4acis knownas
the discriminant.
,i:.;:t
.
.
.
b'- 4ac > o givesreal, unequalroots
bz - 4ac = o gives real, qual roots
bz - 4ac< o iives no reairoots
Wheneveryou are asked about the nature of the roots of a guadratic eguation. you should
alwavs usethe discriminant!
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the natureof the roots
Example9: Determine
44x- 3) = 9
of the equation
Example1O:Findthevalue(s)of p giventhat
2x2+ 4x+ p hasrealroots'
of rgiventhatx2 + (r- 3)x + r = 0 hasno realroots.
Exampfe11: Findthevalue(s)
equal
a lineanda culve,we (i) makecurveandlineequations
To findthe pointsof contactbetween
=
points
of
of the
to findthex- coordinates
to makeRHS 0; and(iii)factorise
eachother;(ii) rearrange
equation).
contact(i.e.findthe rootsof the rearranged
by part(ii) above,we couldworkout
obtained
equation
of the rearranged
If we usedthediscriminant
how many pointsof contactthereare betweenthe lineandthe curue.Thereare3 options:
1
o
Two points of contact
2 distinct roots
b2- 4ac> o
commonusefor this techniqueis to show tangencybetweenlines and
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Erample 12: Showthat the liney = 3x - 13 is a tangentto the cUF/€)z= xz - 7x +L2, and find the
coordinatesof the point of contact.
Example13: Findtwo values
of rn suchthaty- mx-7 is a tangent
to y= x'+2x-3
PastPaperExample:Showthatthe rootsof (k- 2)x' - 3k x + 2k = -2x arealways real.
QuadraticFunctions;TopicCheck st
Topic
Completing
thesquare
Ouadratic
Ineouations
Rootsusing
E-4ac
Tanqencvusinql/ - 4ac
Ouestions
5. p 325,O L,2
c Exercise
8D.o 146.O 2; Exercise
AIB Exercise
8D.o 147.O 4. 6: Exercise
5, o 326.O 3. 4
c Exercise
8F,p 149,Q 3
8I, p152,O 1, 2, 5
c Exercise
8H,p151,Q L,2; Exercise
A/B Exercise8I, p 152,O 6, 8; Exercise8K, p 156,O 10, 12
c Exercise
81.o 155.O l,2
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