Quadratic Functions Maximum and Minimum

8/9/2019 Quadratic Functions Maximum and Minimum

1/4

CXCDirect Institute Turning Point, Maximum and minimum and roots of a quadratic equation

cxcDirect Institute - 876 469-2775Email: admin@cxcdirect!r" #e$site: ###cxcdirect!r" %at& clu$ : cxcDirect %at& Clu$ '

CXC MATHEMATICS

Quadratic Equation

Maximum & Minimum

(ll ri"&ts reser)ed

CXCDirect Institute

Email: admin@cxcDirect!r"*e$site: ###cxcDirect!r"

%at& clu$ : cxcDirect %at& Clu$

+ele,&!ne: 876 469-2775 876 462-6'.9
mailto:[email protected]:[email protected]://www.cxcdirect.org/http://www.cxcdirect.org/http://cxclink.50.forumer.com/http://cxclink.50.forumer.com/http://cxclink.50.forumer.com/mailto:[email protected]:[email protected]://www.cxcdirect.org/http://cxclink.50.forumer.com/


2/4

CXCDirect Institute Turning Point, Maximum and minimum and root of a quadratic equation

The Quadratic Equation

!ra"hs of the quadratic function, and Maximum &

Minimum #a$ue

+&e "ra,& !/ t&e 0uadratic /uncti!n is a sm!!t& cur)e

called a "ara%o$a I/ t&e c!e//icient !/ t&e x2 term is

ne"ati)e t&e ,ara$!la #ill &a)e a maximum )alue

1! t&e /uncti!n 2x2+x+5 #ill &a)e a maximum )alue$ecause t&e c!e//icient !/ t&e x

2 term is '3

ne"ati)e

1imilarl t&e .x2+x 5 #ill &a)e a minimum )alue

$ecause t&e x2 term is ( )3 !siti)e

!enera$ *orms of the quadratic Equation

' y=ax2+bx+c

2 y=a(x+h)2+k /uncti!n #it& a minimum

. y=ka(x+h)2 /uncti!n #it& a maximum

*inding the maximum or minimum #a$ue

+! /ind t&e maximum !r minimum )alue !/ t&e /uncti!n it

is /irst use/ul t! ex,ress t&e /uncti!n in eit&er /!rm 2 !r

/!rm . as s&!#n a$!)e

1! i/ #e &a)e a /uncti!n in t&e /!rm y=ax2+bx+c #e

must /irst c!n)ert it t! t&e ne# /!rm y=a xh 2k

#&ere hand +are t#! ne# c!nstants

*inding h and +

i)en t&e "eneral /!rm ax2+bx+c #e can c!n)ert

t! t&e ne# /!rm a (x+h )2+k usin" a met&!d n!#n ascom"$eting the square:

+&is met&!d results in t&e /!ll!#in" t#! e0uati!ns #&ic&

are )er use/ul and s&!uld $e mem!ried

h=b

2a and k=

4acb2

4a

Case x = h

;!# i/ #e examine y=a xh 2k < e0n '#e n!te t&at x t&e inde,endent )aria$le t&at can tae !n

an )alue

;!# at a certain )alue #&en: x = h t&e e0uati!n/!r $ec!mes:

y=a (h+ h )2+k

y=a(=)2+k

y=k

+&is is an im,!rtant c!nclusi!n as it dem!nstrates t&at:

#&en x = - h: y = k

#&ere is:

+&e minimum)alue !/ t&e /uncti!n i/ a is "ositi#e3 >?

+&emaximum)alue !/ t&e /uncti!n i/ a is negati#e3

cxcDirect Institute - 876 469-2775

Email:admin@cxcdirect!r" #e$site: ###cxcdirect!r" %at& clu$ : cxcDirect %at& Clu$ 2
https://docs.google.com/a/cxcdirect.org/file/d/0B3IUSwj5k_aiZDkzNmNlYzgtMDc2NC00Mzc4LTg1MDUtZjYzM2EzMGEwMDZl/editmailto:[email protected]:[email protected]://www.cxcdirect.org/http://www.cxcdirect.org/http://cxclink.50.forumer.com/mailto:[email protected]://www.cxcdirect.org/http://cxclink.50.forumer.com/https://docs.google.com/a/cxcdirect.org/file/d/0B3IUSwj5k_aiZDkzNmNlYzgtMDc2NC00Mzc4LTg1MDUtZjYzM2EzMGEwMDZl/editmailto:[email protected]


3/4

CXCDirect Institute Turning Point, Maximum and minimum and roots of a quadratic equation

Exam"$e

i)en t&e /uncti!n 2 (x')28

Determine:

' +&e minimum )alue !/ t&e /uncti!n

2 +&e )alue !/ x at #&ic& t&e minimum !ccurs

Ans-er*e n!te t&at t&e /uncti!n is ex,ressed in t&e /!rm

y=a xh 2k #&ere: a=2 h=' andk=8

'3 %inimum )alue !/ t&e /uncti!n is y=k s! min= - 8

23 %inimum )alue !/ !ccurs #&en x=h s! x '

Exam"$e '

i)en t&e /uncti!n '.. (x+2)2

Determine:

. +&e maximum )alue !/ t&e /uncti!n

4 +&e )alue !/ x at #&ic& t&e maximum !ccurs

Ans-er*e n!te t&at t&e /uncti!n is ex,ressed in t&e /!rm

y=ka(x+h)2 #&ere: a=.- h=2 and k='.

%aximum )alue !/ t&e /uncti!n s! ymax

'.

%aximum )alue !ccurs #&en x=h s! x -2

Equation of the axis of s.m metr./ x 0 - h

+&e minimum !r maximum )alue !/ t&e 0uadratic

/uncti!n !ccurs at a turnin" ,!int

;!te t&at a line ,assin" t&r!u"& t&is turnin" ,!int is a line

!/ smmetr and s! t&e )alue !/xat t&is maximum !rminimum ,!int is als! called t&e e0uati!n !/ t&e axis !/

smmetr

s!: +&e e0uati!n !/ t&e axis !/ smmetr isx = - h

and

t&e c!!rdinates !/ t&e turnin" ,!int are: ymin,h 3

cxcDirect Institute - 876 469-2775Email: admin@cxcdirect!r" #e$site: ###cxcdirect!r" %at& clu$ : cxcDirect %at& Clu$ .

axis of symmetry

Turning Point

x
mailto:[email protected]:[email protected]://www.cxcdirect.org/http://www.cxcdirect.org/http://cxclink.50.forumer.com/mailto:[email protected]:[email protected]://www.cxcdirect.org/http://cxclink.50.forumer.com/


4/4

CXCDirect Institute Turning Point, Maximum and minimum and root of a quadratic equation

Exam"$e:

i)en 2x24x6

'3 C!n)ert t&is /uncti!n t! t&e /!rm y=a (x+h)2+k

23 Aind:

i +&e e0uati!n !/ t&e axis !/ smmetr

ii +&e minimum )alue !/ .

iii+&e interce,t #&ere t&e /uncti!n cuts t&e -axis3

(ns#er:

;!# a=2 b=4 and c=6

s! h=b2a

422

-'

and k=4acb2

4a

42(6) (4)2

42 8

s! ( a = 2, h = -1, k = - 8)

+&ere/!re:

'3 t&e ne# /!rm is: y=2(x')28

23

i3 +&e e0uati!n !/ t&e axis !/ smmetr is: x = - h = 1ii3 +&e minimum )alue !/ is: y = k = - 8

iii3 B-interce,t c - 6

1oots of the quadratic function 2equation3

The roots of the quadratic equation is defined as the two

oints where the function cuts the x- axis!

+! /ind t&e r!!ts #e set t! er! and s!l)e $ trans,!siti!n

so if " y=2(x')28

then 2(x')28==

so 2(x')2=8

2(x')2=8

(x')=82

x='4

x='2

so x = 1 # 2 = $

or x = 1 2 = - 1

See !ra"h %e$o- sho-ing the roots

*atc& !n

n$:

+&e r!!ts !/ t&e e0uati!n ma als! $e /!und usin" t&e

0uadratic /!rmula 2x24x6

now" a=2 , b=4 and c=6

so" x = bb

24ac2a

= 4(4)

242(6)22

= 4'6+48

4

=48

4

so" x = 4+84

= $

!r x 48

4 - '

Ans: ( x = 3, or - 1 )

cxcDirect Institute - 876 469-2775

Email:admin@cxcdirect!r" #e$site: ###cxcdirect!r" %at& clu$ : cxcDirect %at& Clu$ 4
mailto:[email protected]:[email protected]://www.cxcdirect.org/http://www.cxcdirect.org/http://cxclink.50.forumer.com/http://www.youtube.com/watch?v=rqSC9pek_xsmailto:[email protected]://www.cxcdirect.org/http://cxclink.50.forumer.com/mailto:[email protected]

Documents

Quadratic Functions Maximum and Minimum