1) quadratic functions
TRANSCRIPT
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Toexplorethepropertiesofquadraticfunctionsandtheirgraphs.
Toinvestigatethedifferentformsinwhichquadraticfunctionscanbeexpressed. Toexplorethetransformationsofquadraticfunctionsandtheirgraphs. http://www.youtube.com/watch?v=VSUKNxVXE4E&feature=player_embedded#
http://evmaths.jimdo.com/year11/functions/?logout=1
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Ifa,bandcarerealnumbersanda0,thenthefunctionf(x)=ax2+bx+cisaquadraticfunction.
theexpressionax2+bx+cisapolynomialofdegree2
a,bandcarecoefficients
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Thegraphofaquadraticfunctionisaparabola.
Whatisthegraphofaquadraticfunction?
UseyourGDCtoplotthegraphoff(x)=2x28x+1
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UseyourGDCtoplotthegraphoff(x)=2x28x+1
1)selectmenuGraph
2)imputtheformula
usethisbuttonforvariable''x''
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UseyourGDCtoplotthegraphoff(x)=2x28x+1
3)pressEXEtoentertheformula
4)PressF6toDRAW
5)Adjustthewindowtoviewthewholegraph.(F3)
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vertex
Aparabolahasasingleturningpointthatiscalleditsvertexandalineofsymmetrythatpassesthroughthevertex
lineofsymmetry
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Youcanuseyourcalculatortofindthevertexofaparabola
Forthisparabolathevertexisitsminimumvalue
Vertexisthepoint(2,7)
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Youaregoingtoinvestigatetheeffectthatthecoefficientshaveonthegraphofaparabola.
Useyourcalculatortosketchthegraphofthefollowingfunctions:
y=x2 y=x2+3 y=x22
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NowyouaregoingtoworkwithdynamicgraphsbyusingslidersinGeoGebra.
Verifyyourconclusionusingasliderkandy=x2+k.
Verticaltranslationofparabola.ggb
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
SMART Notebook
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Sketch the graphs of and
y=x2
y=x22
y=x2+3
vertex:
lineofsymmetry:
vertex:
lineofsymmetry:
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Conclusions:
parabola moves upwards
parabola moves downwards
vertex:
line of symmetry:
(0,k)
x=0
Translationk units along yaxis
k>0
k
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UseGeoGebratoplotthefollowinggraphs:
Verifyyourconclusionusingasliderhandy=(xh)2.
y=(x2)2 y=(x+3)2y=x2
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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
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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)
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Wherewouldyouexpectthevertexofy=(x4)2+5tobe?
Describetheshapeandthepositionofthegraphofy=(xh)2+k.
Translationsinparabola.ggb
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
SMART Notebook
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Conclusions:
vertex
(h,k)
(h,k)
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Consider,and
y=x2 y=2x2
vertex:
lineofsymmetry:
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Whatdoyouexpectif?
y=x2
y=x2
vertex:
lineofsymmetry:Verticalstretchofparabola.ggb
geogebra_thumbnail.png
geogebra_javascript.js
function ggbOnInit() {}
geogebra.xml
SMART Notebook
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Conclusions:
y=ax2Thegraphof isaparabolawithvertex: (0,0) lineofsymmetry: x=0
a>0 a1 as"a"increasestheparabolagets"thinner"
0
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y=(x1)2+3
vertex:
line of symmetry:
(1,3)
x=1
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y=2(x3)2
vertex:
line of symmetry:
(3,0)
x=3
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y=3x2+4
vertex:
line of symmetry:
(0,4)
x=0
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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
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For
Parabolas of the form
Whatistheyintercept?
Findtherootsoff.
Concavity?
factorising (if possible) by formula
(y=0)
=0
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yintercept=8
roots:4and2
lineofsymmetry?
vertex?
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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
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For find:
y- intercept:
line of symmetry:
vertex:
roots:
Now draw a sketch of the function.
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y- intercept: line of symmetry:
vertex: roots:
Now draw a sketch of the function.
Express f(x) in the form
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y=a(xx1)(xx2)Parabolas of the form :
y=(x3)(x+1)
Roots:
Lineofsymmetry:
Vertex:
In general:
x1andx2
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axisofsymmetry
vertex
root
root
yintercept
(0,c)
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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
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y=(x2)2
y=x2+1
y=x22
y=x2+3
y=(x3)2+5
y=2x2+1
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Slidetoprint:
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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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