Section 2.1 - Finding the Equation of a Parabola Lambrick

Section 2.1 - Finding the Equation of a Parabola ♦ 41

Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.

Bylookingatthesketchofaparabola,theequationfortheparabolacanbedetermined.

Considertheparabola:

Thevertexoftheparabolais ,2 5-^ h,thereforey a x 2 52= + +^ h .

Threeotherpointsonthegraphare ,0 3^ h, ,4 3-^ h,and ,2 3-^ h.Todeterminea,anyoneofthesepointscan besubstitutedintoy a x 2 52= + +^ h .

Using ,0 3^ h: y = a x 2 52+ +^ h 3 = a 0 2 52+ +^ h 3 = a4 5+

a4 = 2- a =

21-

Thereforetheequationoftheparabolaisy x21 2 52=- + +^ h .

Note: The points ,4 3-^ h or ,2 3-^ h will give the same result.

Findtheequationofaquadraticfunctionwhosegraphhasvertex ,4 8^ handx-intercept6.

►Solution: Ifthevertexis ,4 8^ hthen f x a x 4 82= - +^ ^h h .

Thex-interceptisthepoint ,6 0^ hwhichisonthegraph.

Solvingfora: f a6 6 4 82- +=^ ^h h =0 a4 8+ =0 a4 = 8- a = 2-

Therefore f x x2 4 82=- - +^ ^h h .

Example 1

x

y

Radical Equations1.6 Finding the Equation of a Parabola2.1

Lambrick Park Secondary

42 ♦ Chapter 2 - Quadratic Transformations


Findtheequationofaquadraticfunctionwithpoints , , , , ,1 0 023 3 0-^ ` ^h j h.

►Solution: Let f x a x h k2= - +^ ^h h .

,1 0-^ hand ,3 0^ haresymmetricbecausetheyhavethesamey-value.Thereforetheaxisof symmetrymustbeaverticallineequidistantfromthetwopoints.Itsequationisx

21 3 1= - + =

Therefore f x a x k1 2= - +^ ^h h .

Substituting ,3 0^ hintotheequation: f a k3 3 1 2- +=^ ^h h =0 a k4 + =0 k = a4-

Therefore f x a x a1 42= - -^ ^h h .

Using ,023` j: f a a0 0 1 42= - -^ ^h h =

23

a a4- =23

a3- =23

a =21-

Thereforek 2=- ,and f x x21 1 22=- - +^ ^h h .

Findanequationofaquadraticfunctionwithpoints , , , , ,3 4 3 2 1 2- -^ ^ ^h h h.

►Solution: Let f x a x h k2= - +^ ^h h .

,3 2-^ hand ,1 2^ haresymmetricbecausetheyhavethesamey-value.Thereforetheaxisof symmetrymustbeaverticallineequidistantfromthetwopoints.Itsequationisx

23 1 1= - + =-

Therefore f x a x k1 2= + +^ ^h h .

Substituting ,1 2^ hintotheequation: f a k1 1 1 2= + +^ ^h h =2 a k4 + =2 k = a2 4-

Therefore f x a x a1 2 42= + + -^ ^h h .

Using ,3 4-^ h: f a a3 3 1 2 42= + + -^ ^h h = 4- a a16 2 4+ - = 4- a12 = 6- a =

21-

Thereforek 4= ,and f x x21 1 42=- + +^ ^h h .

Example 2

Example 3




2.1 Exercise Set

1. Determinetheequationfortheparabola.

a) b)

c) d)

e) f)

g) h)

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y




2. Findtheequationofaquadraticfunctionwhosegraphsatisfiesthegivenconditions.

a) vertex: ,2 9^ hx-intercept:5 b) vertex: ,2 12-^ hx-intercept:−4

c) vertex: ,1 4-^ hx-intercept:−2 d) vertex: ,4 12-^ hx-intercept:4

e) vertex ,3 5- -^ hy-intercept:1 f) vertex: ,2 4^ hy-intercept:−3

g) vertex: ,1 4^ hpoint: ,2 3^ h h) vertex: ,2 4- -^ hpoint ,3 1- -^ h

i) vertex: ,0 2^ hpoint: ,2123` j j) vertex: ,3 0-^ hpoint: ,

2321-` j

k) vertex: ,2 5^ hpoint: ,3 2 3- -^ h l) vertex: ,3 6- -^ hpoint: ,3 3 2^ h




3. Findtheequationofaquadraticfunctionwhosegraphcontainsthegivenpoints.

a) , , , , ,2 0 4 0 1 2- - -^ ^ ^h h h b) , , , , ,3 0 1 8 5 0- -^ ^ ^h h h

c) , , , , ,1 0 0 3 3 0-^ ^ ^h h h d) , , , , ,1 0 0 5 5 0-^ ^ ^h h h

e) , , , , ,1 0 025 5 0-^ ` ^h j h f) , , , , ,5 0 1 0 0

25- -^ ^ `h h j

g) , , , , ,2 3 0 3 4 5- - -^ ^ ^h h h h) , , , , ,3 1 1 1 2 5- - - -^ ^ ^h h h

i) , , , , ,2 1 6 1 2 7- - -^ ^ ^h h h j) , , , , ,4 8 3 23 8 8- - -^ ^ ^h h h




4. Ifaparabolahasx-interceptsaandb,whatisthe 5. Findallvaluesofksothatthevertexofthegraph x-coordinateofthevertex? y x kx 162= + + liesonthex-axis.

6. Determinethequadraticfunctionf,withzeros 7. Determinethequadraticfunctionfwithpoints −2and6,andrangey 4# . ,3 5-^ hand ,1 5^ h,andrangey 2$- .

8. Ifthegraphofthequadraticfunction 9. Considerthequadraticequationy ax ax b2= + + . f x ax bx c2= + +^ h passesthroughtheorigin, Findthevertexifa b4= . whatisthevalueofc?

10. Howfarfromtheoriginisthevertexofthe 11. Findthedistancebetweentheverticesofthe parabolay x2 3 42=- - +^ h ? parabolas 6y x 22

1 2= + +^ h and y x2 4 22=- - -^ h .


Section 2.2 - General Form to Standard Form ♦ 47


Thevertexofaquadraticfunctionisnotobviousfromitsgeneralform,y f x ax bx c2= = + +^ h . However,inthestandardformofaparabola,y f x a x h k2= = - +^ ^h h ,thevertex, ,h k^ h,iseasilyapparent. Changingthegeneralformofaparabolatothestandardformofaparabola,simplifiestheprocessoffindingthe vertex,andtographingtheparabola.

Perfect Square Trinomials

Atrinomialthatisthesquareofabinomialiscalledaperfectsquaretrinomial.

Forexample: x x x x x6 9 3 3 32 2+ + = + + = +^ ^ ^h h h x x x x x4 4 2 2 22 2- + = - - = -^ ^ ^h h h

Inaperfectsquaretrinomial: 1.Thelasttermmustbepositiveandaperfectsquare. 2.Thefirsttermmustbeaperfectsquare. 3.Themiddletermisthesquarerootofthefirsttermmultipliedbythesquare rootofthelastterm,thendoubled.Itcanbepositiveornegative.

Tofindthemissingconstanttermthatmakesatrinomialaperfectsquare: 1.Ifthecoefficientofthex2 termisnot1,factorthatvaluefrombothterms. 2.Taketheconstantofthelinearterm,dividethevalueby2,thensquareit. 3.Thesquaredvalueistheconstantterm.

Findtheconstantthatwillmakeeachexpressionaperfectsquaretrinomial.

a) x x102 - +

b) x x22 + +

c) x x2 82 - +

d) x x2 122- - +

e) x x1021 2 - +

►Solution: a) x x x x x10 25 5 5 52 2- + = - - = -^ ^ ^h h h

b) x x x x x2 1 1 1 12 2+ + = + + = +^ ^ ^h h h

c) x x x x x x x x x2 8 2 4 2 4 4 2 2 2 2 22 2 2 2- + = - + = - + = - - = -^ ^ ^ ^ ^h h h h h

d) x x x x x x x x x2 12 2 6 2 6 9 2 3 3 2 32 2 2 2- - + =- + + =- + + =- + + =- +^ ^ ^ ^ ^h h h h h

e) 10 20x x x x x x x21

21

21 20 100

21 102 2 2 2- + = - + = - + = -^ ^ ^h h h

Constantsare:a)25b)1c)8d) 18- e)50

Example 1

Radical Equations1.6 General Form to Standard Form2.2




Changing from General Form to Standard Form

1. Groupthevariabletermsandconstanttermsseparately. 2. Factortheleadingcoefficient‘a’fromthevariableterms. 3. Determineaconstantvaluethatmakesthevariabletermsaperfectsquaretrinomial.Addandsubtractthe thevaluetothevariableterms.(Itisnecessarytoaddandsubtractthesamevalueinordertokeepthe balanceoftheequation.) 4. Regroupthevariabletermsintoaperfectsquaretrinomial.Whenthevaluethatisremovedfromthevariable termsisregrouped,itmustbemultipliedbythecoefficientfromstep2. 5. Factorthetrinomialasaperfectsquareandaconstantterm.

Findthevertexandx-interceptsof f x x x2 4 52=- + -^ h .

►Solution: f x^ h= x x2 4 52- + + -^ h = x x2 2 52- - + -^ h = x x 52 2 1 12 -- - + -^ h = x x2 2 1 2 52- - + + -^ h ←When regrouping, 2 1 2#- - =^ h = x x2 1 1 3- - - -^ ^h h = x2 1 32- - -^ h Thevertexis ,1 3-^ h

Atthex-intercepts, 0 2 3 0y f x x 1 2"= = - - - =^ ^h h x 1

232- =-^ h

Asquarevaluecannotequalanegativevalue,thereforethereisnox-intercept.

Findthevertexandx-interceptsof f x x x2 5 32= + -^ h .

►Solution: f x x x2 5 32= + + -^ ^h h

x x225 32= + + -` j

x x225

1625

1625 32= + + - -` j

x x225

1625

825 32= + + - -` j ←When regrouping, 2

1625

825# - =-` j

x245

8492

= + -` j

Thevertexis ,45

849- -` j

Atthex-intercepts, 0 2 0y f x x45

8492

"= = + - =^ `h j

x45

16492

+ =` j

x45

47!+ =

,x4547

21 3!=- = -

Thex-interceptsare ,21 3- .

Example 2

Example 3




2.2 Exercise Set

1. Findanumberthatmakestheequationvalid.

a) x x x1 22 2+ = + +^ h b) x x x1 22 2- = - +^ h

c) x x x2 42 2+ = + +^ h d) x x x2 42 2- = - +^ h

e) x x x3 62 2+ = + +^ h f) x x x3 62 2- = - +^ h

g) x x x4 82 2+ = + +^ h h) x x x4 82 2- = - +^ h

2. Findanumberthatmakestheexpressionaperfectsquareoftheform x h 2+^ h .

a) x x22 + + b) x x22 - +

c) x x42 + + d) x x42 - +

e) x x52 + + f) x x72 - +

g) x bx2 + + h) x bx2 - +

3. Findanumbertomakeeachsideequalinvalue.

a) x x x62 2+ + = +^ h b) x x x102 2- + = -^ h

c) x x x52 2+ + = +^ h d) x x x72 2- + = -^ h

e) x x x322 2+ + = +^ h f) x x x

432 2- + = -^ h

g) x bx x2 2+ + = +^ h h) xab x x2 2- + = -^ h




4. Findthevertex.

a) f x x x4 32= + +^ h b) g x x x6 102= + +^ h

c) h x x x8 152= - +^ h d) j x x x10 182= - +^ h

e) k x x x3 82= + -^ h f) l x x x3 52=- - +^ h

g) m x x x3 18 252= - +^ h h) n x x x3 5 32=- + -^ h

i) p x x x21 3 42= - +^ h j) q x x x

32 4 12=- + -^ h

k) .r x x x0 6 2 32= + -^ h l) s x x x43

32

212=- + +^ h




5. Findthexandy-intercepts,ifpossible,foreachquadraticfunction.

a) f x x 3 42= - -^ ^h h b) g x x 3 42= - +^ ^h h

c) h x x4 1 362= + -^ ^h h d) j x x2 3 112=- + +^ ^h h

e) k x x31

21 2

=- -^ `h j f) l x x x52= +^ h

g) m x x x2 4 62=- - -^ h h) x x xn21 3

252= - +^ h

i) p x x x3 52=- -^ h j) x x xq21 4

3192=- - -^ h

k) .r x x0 4 22=- +^ h l) s x x x6 2 22= - -^ h




6. Grapheachquadraticfunction.Findthevertex,axisofsymmetry,x-intercept(s),andy-interceptofthegraph.

a) f x x x42=- -^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

b) g x x x4 32=- + -^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

c) h x x x4 52= - +^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

d) i x x x2 4 32=- - +^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

e) j x x x3 2= -^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

x

y

x

y

x

y

x

y

x

y




6. f) k x x x2 8 62=- - -^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

g) l x x x3 6 32=- + +^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

h) m x x x31 2 12=- - +^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

i) x xn 42= -^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

j) x x xp 2 32= - + +^ h

vertex

axisofsymmetry

x-intercept(s)

y-intercept

x

y

x

y

x

y

x

y

x

y




7. Findthevertexof f x x kx 42= + +^ h . 8. Findthevertexofg x x kx k2 2 2= + +^ h .

9. Findthevertexofh x x ax b2 2 2= + +^ h . 10. Findthevertexof i x px x p32= - +^ h .

11. Findthevertexof j x kx x8= -^ ^h h. 12. Findasuchthat f x ax x4 42= + -^ h hasa maximumvalueatx 6= .

13. Findbsothatthefunction f x x bx2 32= + -^ h 14. Findcsothatthefunction .f x x x c0 1 72= + +^ h hasaminimumvalueof−5. hasaminimumvalueof−120.5.


Section 2.3 - Vertex of a Parabola ♦ 55


Thevertexofaparabolaingeneralformcanbefoundusingthevertexformula.Thevertexformulaisderived bycompletingthesquareofthegeneralformofaparabola, f x ax bx c2= + +^ h .

f x^ h=ax bx c2 + +

= ax bx c2 + + +^ h

=a xab x c2 + + +` j

=a xab x

ab

ab c

4 42

2

2

2

2

+ + - +c m

=a xab x

ab a

ab c

4 42

2

2

2

2

+ + + - +c cm m

=a xab x

ab c

ab

2 2 4

2

+ + + -` `j j

=a xab c

ab

2 4

2 2

+ + -` j

withhab2

=- andk cab4

2

= -

The Vertex Formula

Thegraphofthefunction ,f x ax bx c a 02 != + +^ h isaparabolawithvertex ,ab c

ab

2 4

2

- -c m

Note: It is easier to find the y-coordinate by calculating the value of the x-coordinate, then substituting the value b a2- ,for x in the function.

Findthevertexfor f x x x21 4 32=- + -^ h .

►Solution: Usingthevertexformula:a21=- ,b 4= ,andc 3=-

xab2 2

4 421=- =-

-=^ h

y cab4

2

= - or y = x x21 4 32- + -

344

21

2

=- --^ h =

21 4 4 4 32- + -^ ^h h

5= = 5

Thevertexis ,4 5^ h.

Example 1

Radical Equations1.6 Vertex of a Parabola2.3




Graphthefunction f x x x21 3

252= - +^ h .

►Solution: Usingthevertexformula:a21= ,b 3=- ,andc

25=

xab2 2

33

21=- =--

=^^hh

y cab4

2

= - or y = x x21 3

252 - +

25

4

3

21

2

= --

^^hh =

21 3 3 3

252 - +^ ^h h

2=- = 2-

Thevertexis ,3 2-^ h

y-intercept: ,f 021 0 3 0

25

25 0

252

"= - + =^ ^ ^ `h h h j

x-intercept: f x x x21 3

25 02= - + =^ h

x x6 5 02 - + =

x x1 5 0- - =^ ^h h

, , , ,x 1 5 1 0 5 0"= ^ ^h h

Graphthefunction f x x x2 4 32=- + +^ h .

►Solution: Usingthevertexformula:a 2=- ,b 4= ,andc 3=

xab2 2 2

4 1=- =--

=^ h

y cab4

2

= - or y = x x2 4 32- + +

34 242= --^ h = 1 4 1 32 2 + +- ^ ^h h

5= =5

Thevertexis ,1 5^ h

y-intercept: ,f 0 2 0 4 0 3 3 0 32"=- + + =^ ^ ^ ^h h h h

x-intercept: f x x x2 4 3 02=- + + =^ h

Byquadraticformula: ,x2

2 102

2 10 0"! != c m

x

y

Example 2

x

y

Example 3




2.3 Exercise Set

1. Fillintheblankstomakeatruestatement.

a) Aquadraticfunctioningeneralformisdefinedbytheequation.

b) Thex-coordinateofthevertexofy ax bx c2= + + is.

c) They-coordinateofthevertexofy ax bx c2= + + is.

d) Theaxisofsymmetryofy ax bx c2= + + is.

e) Theisthelowestpointonaparabolathatopensup,orthehighestpointonaparabola thatopensdown.

f) Theaxisofsymmetryofaparabolaisalinethroughthevertex.

g) Aparabolaisthegraphofafunctionintwovariables.

h) Ifthevertexofthegraphof f x ax bx c2= + +^ h isabovethex-axis,anda 02 ,thenthegraphof

f x 0=^ h hasintercept(s).

i) Ifthevertexofthegraphof f x ax bx c2= + +^ h isabovethex-axis,anda 01 ,thenthegraphof

f x 0=^ h hasintercept(s).

j) Ifthegraphof f x ax bx c2= + +^ h hascab4

02

- = ,thenthegraphhasintercept(s).

2. Writeatwo-variablequadraticfunction, f x ax bx c2= + +^ h ,withthegivenvaluesofcoefficientsa,b,c.

a) , ,a b c3 1 4= =- = b) , ,a b c21 3 2=- =- =

c) , ,a b c2 3 0=- = = d) , ,a b c32 0 3=- = =




3. Findthevertexofeachparabola.

a) f x x 42= -^ h b) g x x 32=- +^ h

c) h x x x6 92= - +^ h d) i x x x6 32= + -^ h

e) j x x x3 24 302= + +^ h f) k x x x21 2 72= - -^ h

g) l x x x2 12 172=- + -^ h h) m x x x2 16 332= + +^ h

i) n x x x2 5 32= - -^ h j) p x x x31 5 12=- + -^ h

4. Findthemaximumorminimumvalueofeachparabola.

a) f x x x3 7 82= - +^ h b) f x x x21 4 52=- + -^ h

c) f x x x5 10 32=- - +^ h d) f x x x4 2= -^ h

e) f x x x2 82= - -^ h f) f x x1 3 2= -^ h




5. Sketchthegraphofeachfunctionf.Labelthevertexplusfourotherpoints.

a) f x x x2 32= - -^ h b) f x x x2 3 22= + -^ h

c) f x x x3 4 42=- - +^ h d) f x x x4 12 52=- + -^ h

e) .f x x x 3 752= + -^ h f) f x x x31 2 12=- - +^ h

x

y

x

y

x

y

x

y

x

y

x

y




5. g) f x x x3 5 2 2= + -^ h h) f x x x3 4 12= - +^ h

i) f x x x4 82=- +^ h j) f x x x2 4 52= + +^ h

k) f x x x3 3 52= + -^ h l) f x x x2 2 42= + -^ h

x

y

x

y

x

y

x

y

x

y

x

y




6. Findthevertexofeachquadraticfunction.

a) f x bx cx a2=- + -^ h b) g x cx ax b2= - +^ h

7. Correcttheerrorinthefollowingstatements.

a) Theaxisofsymmetryofaparabolais2ifthevertexis ,2 4^ h.

b) Thevertexof f x x x 122= - -^ h and x x xg 12 2= + -^ h areequivalentsincetheaxisofsymmetryand x-interceptsareequal.

c) Ifthevertexisnotonthex-axis,theparabolacanhave0,1,or2x-intercepts.

d) Theaxisofsymmetryisthehorizontallinethroughthevertex.

e) Thegraphofaparabolacanhave0or1y-intercepts.

f) Thevertexofy x2 3 42= + +^ h is ,3 4^ h.

8. Abaseballisprojectedupwardsatananglefromgroundlevel.Itsheights x^ hisgivenbythefunction s x x x2

4001=- +^ h ,wherexisthedistancetheballtravelledhorizontallyinfeet.

a) Whatisthemaximumheightreachedbytheball? b) Howfardidtheballtravelhorizontally?




Thereareseveralapplicationsofmaximum/minimumproblemsthatcanbemodeledusingquadraticfunctions.

Arancherhas800moffencingtoenclosearectangularcattlepenalongariverbank. Thereisnofencingneededalongtheriverbank.Findthedimensionsthatwouldenclosethe largestarea.

►Solution: Ifx = thewidthofthepen,thelengthofthepenis x800 2- .

Area=length# width =x x800 2-^ h = x x800 2 2-

Method1: Area= x x800 2 2-

= x x2 4002- -^ h = x x2 400 200 2002 2 2- - + -^ h = x x2 400 200 2 2002 2 2- - + - - -^ ^ ^h h h = x2 200 80 0002- - +^ h

Method2: Area= x x800 2 2-

Usingthevertexformula:a 2=- ,b 800= ,c 0=

Vertex: , , ,ab c

ab

2 4 2 2800 0

4 2800 200 80 000

2 2- - =-- -

-=c ^ ^c ^m h h m h

Method3: Usingagraphingcalculator,graphy x x800 2 2= -

Setwindow: Calculatemaximum:

Themaximumareaofthepen=200m# 400m=80000m².

Example 1

xx

Radical Equations1.6 Applications of Quadratic Functions2.4


Section 2.4 - Applications of Quadratic Functions ♦ 63


Asmallmanufacturerofwindowshasadailyproductioncostof .C x x300 4 0 2 2= - + ,where Cisthetotalcostindollarsandxisthenumberofunitsmade.Howmanywindowsshouldbe producedeverydaytominimizecosts?

►Solution: Method1: C = . x x0 2 4 3002 - +

= . x x0 2 20 3002 - + +^ h = . x x0 2 20 100 100 3002 - + - +^ h = . x x0 2 20 100 20 3002 - + - +^ h =0. x2 10 2802- +^ h

Vertexisaminimumat ,10 280^ h

Method2: .C x x300 4 0 2 2= - +

Usingthevertexformula: .a 0 2= ,b 4=- ,c 300=

Vertex: ,.,

.,

ab c

ab

2 4 2 0 2

4300

4 0 2

410 280

2 2

- - =- -

--

=c ^^

^^c ^m h

hhh m h

Method3: Usingagraphingcalculator,graph .y x x300 4 0 2 2= - +


Themanufacturershouldproduce10windowsperdayataproductioncostof$280.00.

Example 2




A400roomhotelisthree-quartersfullwhentheroomrateisanaverageof$80.00pernight. Asurveyshowsthateach$5.00increaseincostwillresultin10fewercustomers.Findthe nightlyrateandnumberofroomsoccupiedthatwillmaximizeincome.

►Solution:43 of400=300roomsoccupied.

Income=numberofroomsoccupied# costpernight.

Let I x^ h=incomefromhotel.

Letx = thenumberof$5increasesinprice.

Theaveragecostofahotelroomis x80 5+^ h.

Thenumberofroomsrentedis x300 10-^ h.

Therefore I x x x xx300 10 24 000 700 5080 5 2= - + -+ =^ ^ ^h h h .

Method1: Income= x x24 000 700 50 2+ -

= x x50 14 24 0002- - + +^ h = x x50 14 49 49 24 0002- - + - +^ h = x x50 14 49 50 49 24 0002 $- - + + +^ h = x50 7 26 4502- - +^ h

Vertexisamaximumat ,7 26 450^ h

Method2: Income= x x24 000 700 50 2+ -

Usingthevertexformula:a 50=- ,b 700= ,c 24 000=

Vertex: , , ,ab c

ab

2 4 2 50700 24 000

4 50700 7 26 450

2 2- - =-- -

-=c ^ ^c ^m h h m h

Method3: Usingagraphingcalculator,graph x xy 24 000 700 50 2= + -


Thenightlyrateshouldaverage$80 5 7+ =^ h $115 Thenumberofroomsoccupiedwouldbe300 10 2307- =^ h Themaximumincomeis230 115# = $26450

Example 3




Murraystandsonthetopofabuildingandfiresagunupwards.Thebullettravelsaccordingto theequationh t t16 384 502=- + + ,wherehistheheightofthebulletoffthegroundin metresattsecondsafteritwasfired.

a) HowfarisMurrayabovethegroundwhenheshootsthegun? b) Howhighdoesthebullettravelverticallyrelativetotheground? c) Howlongdoesittakeforthebullettoreachitsgreatestheight? d) Afterhowmanysecondsdoesthebullethittheground?

►Solution: a) when t 0= ,h 50= m.

b) h = t t16 384 502- + +

= t t16 24 502- - + +^ h = t t16 24 144 144 502- - + - +^ h = t t16 24 144 2304 502- - + + +^ h = t16 12 23542- - +^ h

Vertexisamaximumat ,12 2345^ h,thereforethebullettravels2354mhigh.

c) t t12 0 12"- = = seconds.

d) Thebulletreachesthegroundwhenh 0= .

t16 12 23542- - +^ h =0

t 12 2-^ h =162354

t 12- =162354!

t =12162354!

t =24.13secondsor−0.13seconds.

Rejectt=−0.13seconds,thereforeittakesthebullet24.13secondstohittheground.

Example 4




2.4 Exercise Set

1. TheAcmeautomobilecompanyhasfoundthatthe 2. Acattleranchwith6000metresoffencingwants revenuefromsalesofcarsisafunctionoftheunit toenclosearectangularfeedlotthatbordersona pricepthatitcharges.Iftherevenue,R,is river.Ifthecattlewillnotgointheriver,whatis R p p20002

1 2=- + ,whatunitprice,p,shouldbe thelargestareathatcanbeenclosed? chargedtomaximizerevenue?Whatisthe maximumrevenue?

3. Findtherectangleofmaximumareathatcanbe 4. Whatistheminimumproductoftwonumbers constructedwithaperimeterof36cm. thatdifferby8?Whatarethenumbers?

5. Acannonshellisfiredfromacliff100mabovethe 6. Theschoolplaycharges$10foradmission,andon water.Theheight,h,ofthecannonabovethewateris average80peopleattendeachshow.Foreach$1 givenby .h x x0 005 1002=- + + ,wherexisthe increase,attendancedropsby5people.Whatprice horizontaldistanceofthecannonfromthebaseofthe shouldtheschoolchargetomaximizerevenue? cliffinmetres. a) Howfarfromthebaseofthecliffistheheightof thecannonshellatamaximumheight? b) Findthemaximumheightofthecannonshell. c) Howfarfromthebaseofthecliffwillthecannon shelllandinthewater?




7. Acompanycansellxstereosatapriceof 8. A20cmwidesheetof $ x500 -^ h.Howmanystereosmustbesoldto metalisbentintoa maximizeincome?Findthemaximumincome. rectangulartrough. Howhighmustthe troughbetomaximize thetrough’svolume?

9. Thesumoftwointegersis10,andthesumoftheir 10. Aparabolicarchhasaspanof50manda squaresisaminimum.Findtheintegers. maximumheightof10m.Whatistheheightof thearchatapoint10mfromthecentre?

11. Arancherhas1200moffencingtoenclosetwo 12. SemiahmooFishandGameClubchargesits adjacentrectangularcorrals.Whatdimensionswill members$200peryear.Foreachincreaseover produceamaximumenclosedareaifthecommon 60members,themembershipcostisdecreased sidesareofequallength? by$2.Whatnumberofmemberswouldproduce themaximumrevenuefortheclub?




13. Whenpricedat$10,onetypeofsoftwarehasannual 14. ShipAis50nauticalmileswestofshipB.ShipA salesof300units.Foreachdollarthesoftwareis isheadingeastat10knotsandshipBisheading increasedinprice,thestoreexpectstolosethesale southat5knots.Findtheminimumdistance of10unitsofsoftware.Findthepricethatwill betweentheships,andatwhattimeitoccurred. maximizethetotalrevenue.

15. Anequestrianclubwantstoconstructacorralnext 16. Thecablesupportingasuspensionbridgeis tothebarn,usingthesideofthebarn.Theyhave parabolicinshape.Thesupportcablesarevertical 300ftoffencing.Findthedimensionsthatgivethe cables25ftapart.Iftheshortestsupportcableis maximumarea. 10ftlong,howlongisthelongestsupportcable?

17. ANormanwindowisarectanglewithasemi-circle 18. Apieceofstring36cminlengthiscutintotwo ontop.Iftheperimeterofthewindowis24ft, pieces.Onepieceformsasquareandtheother whatdimensionswillmaximizetheareaofthe pieceacircle.Howshouldthestringbecutso window? thesumoftheareasareaminimum?

60ft

x

y

longestsupportcable

200ft

10ft

60ft


Section 2.5 - Chapter Review ♦ 69


Section 2.1

1. Determinetheequationoftheparabolaintheform f x a x h k2= - +^ ^h h .

a) b)

2. Findtheequationofaquadraticfunctionwhosegraphsatisfiesthegivenconditions.

a) vertex: ,3 4-^ h,x-intercept:2 b) vertex: ,2 1-^ h,x-intercept:−3

c) vertex: ,1 4- -^ h,point: ,2 1-^ h d) , , , , ,2 0 4 0 0 3-^ ^ ^h h h

Section 2.2

3. Grapheachquadraticfunction.Findthevertex,axisofsymmetry,x-intercept(s),andy-interceptofthegraph.

a) f x x x31 2=- +^ h vertex

axisofsymmetry

x-intercept(s)

y-intercept

b) g x x x21 42= - -^ h vertex

axisofsymmetry

x-intercept(s)

y-intercept

Radical Equations1.6 Chapter Review2.5

x

y

x

y

x

y

x

y




4. Findthevertex,x-intercept(s)(ifpossible),andy-interceptsforeachquadraticfunction.

a) f x x x2 6 42=- - -^ h b) g x x x21

25

8252= - +^ h

Section 2.3

5. Findthevertexofeachquadraticfunction.

a) f x x x2 12 72= - +^ h b) g x x x3 6 22= + +^ h

c) h x x x2 12 202=- + -^ h d) i x x x2 221 2=- + -^ h

6. If ,8 0^ hisony bx x2= + ,whatistheleastvalue 7. If ,6 0-^ hisony bx x2= - ,whatisthegreatest ofthefunction? valueofthefunction?

Section 2.4

8. Findtherectangleofmaximumareathatcanbe 9. Abustouringcompanycharges$10perpassenger, constructedwithaperimeterof44cm. andcarriesanaverageof300passengersperday. Thecompanyestimatesitwilllose15passengers foreachincreaseof$1inthefare.Whatisthe mostprofitablefaretocharge?


Documents

Section 2.1 - Finding the Equation of a Parabola Lambrick