Parabolas - Maths Excel Class€¦ · Parabolas – Sum and product of roots © Maths Excel Class...

Parabolas

©MathsExcelClass2016 1 4thYrMary

>

>

𝑥

𝑦

𝒚 = 𝒙𝟐

>

>

𝑥

𝑦

𝒚 = −𝒙𝟐

>

>

𝑥

𝑦

−1

𝒚 = 𝒙𝟐 + 𝟏

>

>

𝑥

𝑦

−

−1

𝒚 = (𝒙 + 𝟏)𝟐

Concaveup Concavedown

Parabolas


Sketchthefollowingparabolas:(a)𝑦 = 𝑥. (b)𝑦 = −𝑥.(c)𝑦 = 𝑥. + 2 (d)𝑦 = 𝑥 + 2 .(e)𝑦 = 𝑥. − 4 (f)𝑦 = −𝑥. + 5(g)𝑦 = 𝑥 − 7 . (h)𝑦 = − 𝑥 − 6 .

Parabolas


Sketchthefollowingparabolas:(i)𝑦 = −𝑥. − 1 (j)𝑦 = 𝑥 − 3 .(k)𝑦 = 𝑥 − 2 . + 1 (l)𝑦 = 𝑥 + 7 . − 5(m)𝑦 = 𝑥 − 3 . − 9 (n)𝑦 = − 𝑥 − 5 . + 6(o)𝑦 = 4 + 𝑥 − 1 . (p)𝑦 = 6 − 𝑥 + 6 .

Parabolas


Writetheequationofthefollowingparabolas.

𝑥

𝑦

𝑥

𝑦

−

5

𝑥

𝑦

−7

𝑥

𝑦

−

−8

𝑥

𝑦

−−6

𝑥

𝑦

−

−

2

1

𝑥

𝑦

−

−

−2

−3 𝑥

𝑦

−

−

5

9

Parabolas–Roots


Therootsarethe𝑥 −interceptsoftheparabola.Example1Findtherootsof𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟖andsketchtheparabola.Tofind𝑥 −intercepts,let𝑦 = 0.𝑥. − 6𝑥 + 8 = 0𝑥 − 2 𝑥 − 4 = 0Roots:𝒙 = 𝟐or𝒙 = 𝟒Findtherootsofthefollowingparabolasandsketchthem.1. 𝑦 = 𝑥. − 4𝑥 + 32. 𝑦 = 𝑥. + 𝑥 − 20

𝑥

𝑦

−

2

−

4

Parabolas–Roots


3. 𝑦 = 𝑥. − 7𝑥 − 8

4. 𝑦 = 𝑥. + 8𝑥 + 12

5. 𝑦 = −𝑥. + 6𝑥 − 5

6. 𝑦 = −𝑥. − 5𝑥 + 36

Parabolas–Roots


7. 𝑦 = 𝑥. − 6𝑥 + 9

8. 𝑦 = −𝑥. − 10𝑥 − 25

9. 𝑦 = 2𝑥. − 7𝑥 + 3

10. 𝑦 = 3𝑥. + 11𝑥 − 4

Parabolas–Roots


Writetheequationsofthefollowingparabolas.Expandallbrackets.

𝑥

𝑦

−

1

−

−2 𝑥

𝑦

−

4

𝑥

𝑦

− 2

−−8

𝑥

𝑦

− 5

−3

2

𝑥

𝑦

−−6 0

𝑥

𝑦

−

3

−

7

Parabolas–Quadraticformula


Thefollowingformulacanbeusedtofindtherootsof𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.Example1Sketchtheparabola𝑦 = 5𝑥. + 2𝑥 − 1.𝑦 = 5𝑥. + 2𝑥 − 1 𝑎 = 5, 𝑏 = 2, 𝑐 = −1Findtheroots: Vertex

𝒙 =−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

𝑥

𝑦

−

𝑥 = −𝑏2𝑎 +

√𝑏. − 4𝑎𝑐2𝑎

−

𝑥 = −𝑏2𝑎 −

√𝑏. − 4𝑎𝑐2𝑎

Vertex

𝑥 = −𝑏2𝑎

RootRoot

𝑥 =−𝑏 ± √𝑏. − 4𝑎𝑐

2𝑎

𝑥 =−2 ±E(2). − 4(5)(−1)

2(5)

𝑥 =−2 ± √24

10

𝑥 =−1 + √6

5 𝑜𝑟−1 − √6

5

= −𝑏2𝑎

= −2

2(5) = −15

𝑥

𝑦

−

−1 + √65

−−1 − √6

5

−15



1. Identify𝑎, 𝑏and𝑐forthefollowingquadraticequations.

a. 𝑦 = 2𝑥. + 7𝑥 + 9

b. 𝑦 = 4𝑥. − 8𝑥 + 11

c. 𝑦 = 𝑥. + 6𝑥 − 3

d. 𝑦 = 5𝑥. − 3𝑥 − 3

e. 𝑦 = 12𝑥. + 𝑥2. Usethequadraticformulatosolvethefollowing.Leaveyouranswerinexact

form.

a. 2𝑥. − 𝑥 − 3 = 0

b. 𝑥. − 3𝑥 + 1 = 0

c. 𝑥. + 2𝑥 − 6 = 0d. 2𝑥. − 5𝑥 + 1 = 0

e. 3𝑥. − 8𝑥 + 2 = 0



3. Findthevertexofeachparabolaandsketch.Youdonotneedtoshowtheroots.

a. 𝑦 = 𝑥. − 6𝑥 + 9

b. 𝑦 = 𝑥. + 10𝑥 − 5

c. 𝑦 = 3𝑥. − 24𝑥 + 7

d. 𝑦 = −2𝑥. − 8𝑥 + 1

e. 𝑦 = −3𝑥. + 5𝑥 − 10



4. Sketchthefollowingparabolas,showingtherootsandvertexofeach.

a. 𝑦 = 𝑥. − 5𝑥 − 4

b. 𝑦 = 2𝑥. + 6𝑥 + 1

c. 𝑦 = −2𝑥. − 3𝑥 + 4

d. 𝑦 = 8𝑥. + 2𝑥 − 3

Parabolas–Discriminant


Thediscriminanttellsus:1. Howmanyroots2. Iftherootsarerationalorirrational

𝑥 =−𝑏 ± √𝒃𝟐 − 𝟒𝒂𝒄

2𝑎

△= 𝒃𝟐 − 𝟒𝒂𝒄

𝑥

𝑦

. .

𝑥

𝑦

.

𝑥

𝑦

△> 𝟎2realroots

𝑥 = −𝑏+√△

2𝑎 and𝑥 = −𝑏−√△2𝑎

△= 𝟎1realroot

𝑥 =−𝑏 + √𝟎2𝑎

= −𝑏2𝑎

△< 𝟎Norealroots

𝑥 =−𝑏 +E(−𝒗𝒆)

2𝑎



Rationalroot–DiscriminantisasquarenumberExample𝑦 = 𝑥. + 10𝑥 + 9 Sketchtheparabola:Irrationalroot–DiscriminantisnotasquarenumberLeaveyouranswerasasurd.Example𝑦 = 𝑥. + 3𝑥 + 1 Sketchtheparabola:

△= 𝑏. − 4𝑎𝑐= 10. − 4(1)(9)= 64Thediscriminantisasquarenumber,thereforetherootsarerational.

𝑥 =−10 ± E10. − 4(1)(9)

2(1)

=−10 ± √64

2

=−10 ± 8

2 Roots:𝑥 = −1and𝑥 = −9

△= 𝑏. − 4𝑎𝑐= 3. − 4(1)(1)= 5Thediscriminantisnotasquarenumber,thereforetherootsareirrational.

𝑥 =−3 ± E3. − 4(1)(1)

2(1)

=−3 ± √5

2

Roots:𝑥 = −3+E52 and𝑥 = −3−E5

2



1. Howmanyrootsdoeachofthefollowingparabolashave?

𝑥

𝑦

𝑥

𝑦

𝑥

𝑦

𝑥

𝑦

𝑥

𝑦

𝑥

𝑦



2. Foreachofthefollowingparabolasstate:• Thenumberofroots• Iftherootsarereal/unreal• Iftherootsarerational/irrational

Sketcheachparabolashowing:• Roots• Vertex(𝑥 −coordinateonly)• 𝑦 −intercept

a. 𝑦 = 𝑥. − 𝑥 − 6

b. 𝑦 = 𝑥. − 8𝑥 + 16

c. 𝑦 = 2𝑥. + 3𝑥 − 2



d. 𝑦 = 𝑥. − 2𝑥 − 5

e. 𝑦 = 𝑥. + 6𝑥 + 10

f. 𝑦 = −𝑥. + 2𝑥 + 35

g. 𝑦 = 𝑥. + 𝑥 − 3



3. Forwhatvaluesof𝑘does𝑥. − 𝑘𝑥 + 9 = 0have1realroot?4. (2004)Forwhatvaluesof𝑘does𝑥. − 𝑘𝑥 + 4 = 0havenorealroots?5. Forwhatvaluesof𝑘does𝑥. + 𝑘𝑥 + 1haveatleast1realroot?6. (2009)Findthevaluesof𝑘forwhichthequadraticequation

𝑥. − 𝑘 + 4 𝑥 + 𝑘 + 7 = 0

hasequalroots.

Parabolas–Sumandproductofroots


Let𝛼and𝛽betherootsoftheparabola𝑦 = 𝑎𝑥. + 𝑏𝑥 + 𝑐.Example1Findthesumandproductoftherootsoftheparabola𝑦 = 3𝑥. + 5𝑥 + 4.Sumofroots Productofroots𝛼 + 𝛽 𝛼𝛽Example2Let𝛼and𝛽betherootsoftheparabola𝑦 = 𝑥. + 2𝑥 + 5.Findthevalueof𝛼. + 𝛽..Usingsumandproductofroots,wecanfindthat:Weknowthat:𝛼 + 𝛽 . = 𝜶𝟐 + 2𝛼𝛽 + 𝜷𝟐 ∴ 𝛼. + 𝛽.

𝜶 + 𝜷 = −𝒃𝒂

𝜶𝜷 =

𝒄𝒂

Sumofroots

Productofroots

= −𝑏𝑎

= −𝟓𝟑

=𝑐𝑎

=𝟒𝟑

𝛼 + 𝛽 = −2𝛼𝛽 = 5

= (𝛼 + 𝛽). − 2𝛼𝛽

= (−2). − 2(5)

= −6



Example3Let𝛼and𝛽betherootsoftheparabola𝑦 = 𝑥. + 4𝑥 + 3.Findthevalueof

UV+UW

Usingsumandproductofroots,wecanfindthat:UV+UW = 𝛽

𝛼𝛽 +VVW Makethedenominatorthesame.

= 𝛼+𝛽

𝛼𝛽 = −4

3Questions1. Findthesumandproductoftherootsforeachofthefollowingparabolas.

a. 𝑦 = 𝑥. + 6𝑥 + 3

b. 𝑦 = 2𝑥. + 7𝑥 − 4

c. 𝑦 = 5𝑥. − 4𝑥 + 2

d. 𝑦 = 11𝑥. − 2𝑥 − 4

𝛼 + 𝛽 = −4𝛼𝛽 = 3



2. Let𝛼and𝛽berootsofthequadraticequation𝑥. − 3𝑥 + 4.Findthevalueof:

a. 𝛼 + 𝛽

b. 𝛼𝛽

c. UV+UW

d. 𝛼. + 𝛽.

e. (𝛼 − 3)(𝛽 − 3)

f. 𝛼.𝛽 + 𝛼𝛽.

g. UVX+

UWX

h. WV+VW

i. 𝛼 − 𝛽 .



3. (2011)Thequadraticequation𝑥. − 6𝑥 + 2 = 0hasroots𝛼and𝛽.

a. Find𝛼 + 𝛽

b. Find𝛼𝛽

c. FindUV+UW

4. Aparabolahasroots𝛼and𝛽.Thesumoftherootsis7andtheproductofthe

rootsis10.Findtheequationoftheparabola.5. (2006)Let𝛼and𝛽bethesolutionsof𝑥. − 3𝑥 + 1 = 0.

a. Find𝛼𝛽

b. Hencefind𝛼 +UV

6. (2014)Therootsofthequadraticequation2𝑥. + 8𝑥 + 𝑘 = 0are𝛼and𝛽.

a. Findthevalueof𝛼 + 𝛽.

b. Giventhat𝛼.𝛽 + 𝛼𝛽. = 6,findthevalueof𝑘.

Parabolas–Maximumandminimum


Maximumandminimumpointsarestationarypoints.Theyoccuratthevertexoftheparabola.Example1Whattypeofstationarypointisontheparabola𝑦 = 𝑥. − 6𝑥 + 1?Finditscoordinates.Theparabola𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟏isconcaveup.∴Ithasaminimumpoint.Minimumpointoccursatthevertex.𝑥co−ordinate 𝑦co−ordinate

Minimumpoint𝑥

𝑦

. (𝑥, 𝑦)

Maximumpoint𝑥

𝑦

. (𝑥, 𝑦)

= −𝑏2𝑎

= −−62(1)

= 3

= (3). − 6(3) + 1= −8

∴Minimumpointoccursat(𝟑, −𝟖).



1. Statewhetherthefollowingparabolashavemaximumorminimumpoints.

a. 𝑦 = 𝑥. − 7𝑥 + 9

b. 𝑦 = −𝑥. + 8𝑥 + 4

c. 𝑦 = −2𝑥. − 5𝑥 + 4

d. 𝑦 = 3𝑥22 + 10𝑥 + 11

2. Findthecoordinatesofthefollowingmaximumorminimumpoints.

𝑥

𝑦

𝑦 = 𝑥. − 6𝑥 + 8

𝑥

𝑦

𝑦 = 𝑥. − 9

𝑥

𝑦𝑦 = −𝑥. + 10𝑥 − 16

𝑥

𝑦

𝑦 = −2𝑥. + 4𝑥 − 7



3. Sketchtheparabola𝑦 = 𝑥. − 8𝑥 + 15,showingtherootsandthecoordinatesofthestationarypoint.

4. Sketchtheparabola𝑦 = 𝑥. − 6𝑥 − 40,showingtherootsandthecoordinates

ofthestationarypoint.

5. Sketchtheparabola𝑦 = −𝑥. + 6𝑥 + 7,showingtherootsandthe

coordinatesofthestationarypoint.



6. Whattypeofstationarypointisontheparabola𝑦 = 𝑥. − 4𝑥 + 2?Finditscoordinates.

7. Whattypeofstationarypointisontheparabola𝑦 = 𝑥. + 8𝑥 + 1?Findits

coordinates.8. Apaperplaneislaunchedfromtheoriginandfollowsthepathofthe

parabola𝑦 = −𝑥. + 12𝑥.Findthemaximumheightreachedbytheplane.9. Asubmarinedescendsthroughtheoceanbeforereturningtothesurface.It

followsthepathoftheparabola𝑦 = 3𝑥. − 54𝑥.Findthemaximumdepthreachedbythesubmarine.

𝑥

𝑦

𝑥

𝑦

Parabolas - Maths Excel Class€¦ · Parabolas – Sum and product of roots © Maths Excel Class...

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