Toexplorethepropertiesofquadraticfunctionsandtheirgraphs.

Toinvestigatethedifferentformsinwhichquadraticfunctionscanbeexpressed. Toexplorethetransformationsofquadraticfunctionsandtheirgraphs. http://www.youtube.com/watch?v=VSUKNxVXE4E&feature=player_embedded#

http://evmaths.jimdo.com/year11/functions/?logout=1

Ifa,bandcarerealnumbersanda0,thenthefunctionf(x)=ax2+bx+cisaquadraticfunction.

theexpressionax2+bx+cisapolynomialofdegree2

a,bandcarecoefficients

Thegraphofaquadraticfunctionisaparabola.

Whatisthegraphofaquadraticfunction?

UseyourGDCtoplotthegraphoff(x)=2x28x+1

UseyourGDCtoplotthegraphoff(x)=2x28x+1

1)selectmenuGraph

2)imputtheformula

usethisbuttonforvariable''x''

UseyourGDCtoplotthegraphoff(x)=2x28x+1

3)pressEXEtoentertheformula

4)PressF6toDRAW

5)Adjustthewindowtoviewthewholegraph.(F3)

vertex

Aparabolahasasingleturningpointthatiscalleditsvertexandalineofsymmetrythatpassesthroughthevertex

lineofsymmetry

Youcanuseyourcalculatortofindthevertexofaparabola

Forthisparabolathevertexisitsminimumvalue

Vertexisthepoint(2,7)

Youaregoingtoinvestigatetheeffectthatthecoefficientshaveonthegraphofaparabola.

Useyourcalculatortosketchthegraphofthefollowingfunctions:

y=x2 y=x2+3 y=x22

NowyouaregoingtoworkwithdynamicgraphsbyusingslidersinGeoGebra.

Verifyyourconclusionusingasliderkandy=x2+k.

Verticaltranslationofparabola.ggb

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

Sketch the graphs of and

y=x2

y=x22

y=x2+3

vertex:

lineofsymmetry:

vertex:

lineofsymmetry:

Conclusions:

parabola moves upwards

parabola moves downwards

vertex:

line of symmetry:

(0,k)

x=0

Translationk units along yaxis

k>0

k

UseGeoGebratoplotthefollowinggraphs:

Verifyyourconclusionusingasliderhandy=(xh)2.

y=(x2)2 y=(x+3)2y=x2

Consider the graphs of

vertex:

lineofsymmetry:

vertex:

lineofsymmetry:

y=(x+3)2

y=(x2)2

y=x2

and

What can you say about vertex and symmetry line?

Horizontaltranslationofparabola.ggb

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

Conclusions:

parabola moves to the right

parabola moves to the left

vertex:

line of symmetry:

vertex:

line of symmetry:

(h,0)

x=h

(h,0)

x=h

Translation alongxaxis

(h>0)

Wherewouldyouexpectthevertexofy=(x4)2+5tobe?

Describetheshapeandthepositionofthegraphofy=(xh)2+k.

Translationsinparabola.ggb

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

Conclusions:

vertex

(h,k)

(h,k)

Consider,and

y=x2 y=2x2

vertex:

lineofsymmetry:

Whatdoyouexpectif?

y=x2

y=x2

vertex:

lineofsymmetry:Verticalstretchofparabola.ggb

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

Conclusions:

y=ax2Thegraphof isaparabolawithvertex: (0,0) lineofsymmetry: x=0

a>0 a1 as"a"increasestheparabolagets"thinner"

0

y=(x1)2+3

vertex:

line of symmetry:

(1,3)

x=1

y=2(x3)2

vertex:

line of symmetry:

(3,0)

x=3

y=3x2+4

vertex:

line of symmetry:

(0,4)

x=0

y=3(x+1)22

vertex:

line of symmetry:

(-1,-2)

x=-1

http://members.shaw.ca/ron.blond/QFA.CSF.APPLET/index.html

TransformacionesFuncinCuadrtica.ggb

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

For

Parabolas of the form

Whatistheyintercept?

Findtherootsoff.

Concavity?

factorising (if possible) by formula

(y=0)

=0

yintercept=8

roots:4and2

lineofsymmetry?

vertex?

Line of symmetry is in the middle between the roots :

The vertex will be on the line of symmetry:

We can find the line of symmetry by doing :

y - intercept: (0,c)

a0

Cambioscuadratica.ggb

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

For find:

y- intercept:

line of symmetry:

vertex:

roots:

Now draw a sketch of the function.

y- intercept: line of symmetry:

vertex: roots:

Now draw a sketch of the function.

Express f(x) in the form

y=a(xx1)(xx2)Parabolas of the form :

y=(x3)(x+1)

Roots:

Lineofsymmetry:

Vertex:

In general:

x1andx2

axisofsymmetry

vertex

root

root

yintercept

(0,c)

http://www.univie.ac.at/future.media/moe/tests/fun1/erkennen.html

Solveworksheet:Quadraticfunctions

Attheendofthelesson:QUADRATICFUNCTIONSI2010.doc

Y11 SL EV

QUADRATIC FUNCTIONS

1) Complete

2) Find the minimum value of

3) i) Factorize

ii) Determine the integer values for which

is less than zero.

4) a) Factorise

b) Sketch the graph of

5) Complete

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

_1253110198.unknown

_1253110703.unknown

_1253111211.unknown

_1253111262.unknown

_1253111347.unknown

_1253111234.unknown

_1253110752.unknown

_1253110501.unknown

_1253110535.unknown

_1253110347.unknown

_1253110085.unknown

_1253110145.unknown

_1253109960.unknown

_1253110010.unknown

_1253109856.unknown

SMART Notebook

y=(x2)2

y=x2+1

y=x22

y=x2+3

y=(x3)2+5

y=2x2+1

Slidetoprint:

Attachments

Parabolacanonica.ggb

Cambioscuadratica.ggb

QUADRATICFUNCTIONSI2010.doc

TransformacionesFuncinCuadrtica.ggb

Verticaltranslationofparabola.ggb

Verticalstretchofparabola.ggb

Horizontaltranslationofparabola.ggb

Translationsinparabola.ggb

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

Y11 SL EV

QUADRATIC FUNCTIONS

1) Complete

2) Find the minimum value of

3) i) Factorize

ii) Determine the integer values for which

is less than zero.

4) a) Factorise

b) Sketch the graph of

5) Complete

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Vertex:

Line of symmetry:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

Roots:

Line of symmetry:

Vertex:

y-intercept:

EMBED Equation.DSMT4

_1253110198.unknown

_1253110703.unknown

_1253111211.unknown

_1253111262.unknown

_1253111347.unknown

_1253111234.unknown

_1253110752.unknown

_1253110501.unknown

_1253110535.unknown

_1253110347.unknown

_1253110085.unknown

_1253110145.unknown

_1253109960.unknown

_1253110010.unknown

_1253109856.unknown

SMART Notebook

geogebra_thumbnail.png

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

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