MatematikC.GrafiskLøsningAfLigninger History

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March 15, 2015, at 06:16 PM by 89.150.96.96 -
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Figuren herunder viser grafen for funktionen {$f(x)=x^5-15x^4+85x^3-225x^2+274x-120$}. Ligningens løsninger er x-koordinaterne for denne funktions skæringspunkter med x-aksen. Det ses, at der er fem løsninger: x=1, x=2, x=3, x=4 og x=5

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Figuren herunder viser grafen for funktionen {$f(x)=x^5-15x^4+85x^3-225x^2+274x-120$}. Ligningens løsninger er x-koordinaterne for denne funktions skæringspunkter med x-aksen. Det ses, at der er fem løsninger: x=1, x=2, x=3, x=4 og x=5 (og da der er tale om et femtegradspolynomium, som jo højst har fem rødder, kan det udelukkes, at der kunne være flere løsninger udenfor det vindue, som grafen er vist i).

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June 30, 2013, at 12:53 PM by 31.25.19.69 -
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October 16, 2012, at 11:56 PM by 89.239.216.135 -
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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder. Der er tale om lineære funktioner, hvis grafer er rette linier (i appletten, som startes ved at klikke på billedet, kan andre lineære funktioner tegnes).

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder. Der er tale om lineære funktioner, hvis grafer er rette linier.

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October 16, 2012, at 11:54 PM by 89.239.216.135 -
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I denne applet kan du justere a og b værdierne for begge de to lineære funktioner og se hvordan grafernes skæringspunkt ændres

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October 16, 2012, at 11:51 PM by 89.239.216.135 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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October 16, 2012, at 11:50 PM by 89.239.216.135 -
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October 16, 2012, at 11:45 PM by 89.239.216.135 -
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October 16, 2012, at 11:43 PM by 89.239.216.135 -
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(:toggle div=test button=1 init=hide lshow="Vis interaktiv figur" lhide="Skjul interaktiv figur":)

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July 18, 2012, at 08:21 PM by 89.239.216.135 -
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Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi - ligningen {$x-2 = -x+5$} er altså sand - og x=3 er derfor en løsning til ligningen (det kontrolleres let ved at indsætte x=3 i ligningen).

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Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-1$} og {$-x+5$} altså samme værdi - ligningen {$x-1 = -x+5$} er altså sand - og x=3 er derfor en løsning til ligningen (det kontrolleres let ved at indsætte x=3 i ligningen).

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January 29, 2012, at 02:01 AM by 89.239.234.18 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://jars.geogebra.org/webstart/4.0" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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January 29, 2012, at 02:00 AM by 89.239.234.18 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="600" height="400" MAYSCRIPT filename="pub/applets/ggbfiler/geo.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../Geogebra/geogebra.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://jars.geogebra.org/webstart/4.0" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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January 29, 2012, at 01:56 AM by 89.239.234.18 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="600" height="400" MAYSCRIPT filename="pub/applets/ggbfiler/geo.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../Geogebra/geogebra.png" clickalign="center":)

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January 29, 2012, at 01:54 AM by 89.239.234.18 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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January 29, 2012, at 01:52 AM by 89.239.234.18 -
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August 07, 2011, at 03:12 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="../pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 03:02 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="../pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="../pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="../pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:42 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:39 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:39 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../uploads/MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:38 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../uploads/MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:37 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:37 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads/MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:32 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http:/www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads/MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:30 AM by 87.58.31.236 -
Changed lines 15-16 from:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http:/www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:26 AM by 87.58.31.236 -
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August 07, 2011, at 02:22 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads/MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:15 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads//MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads/MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:13 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="http://www.buhlweb.dk/matwiki/uploads//MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:08 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="http://www.buhlweb.dk/matwiki/pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:05 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:03 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http:/www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 02:02 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http:/www.buhlweb.dk/matwiki/pub/applets/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 01:56 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.geogebra.org/webstart/3.2/unsigned/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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August 07, 2011, at 01:55 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

to:

(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="http://www.geogebra.org/webstart/3.2/unsigned/" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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September 02, 2010, at 07:34 PM by 87.58.31.236 -
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(:noleft:) (:noheader:) (:notitle:)

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August 31, 2010, at 12:17 AM by 87.58.31.236 -
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(:toggle div=glopgaver button=0 init=hide lshow=Opgaver lhide="Skjul opgaver":)

(:include MatematikC/OpgaverTilGrafiskLøsningAfLigninger:)

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August 30, 2010, at 11:55 PM by 87.58.31.236 -
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(:if ! group MatematikC:) test (:ifend:)

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August 30, 2010, at 11:49 PM by 87.58.31.236 -
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(:if group!=MatematikC:)

to:

(:if ! group MatematikC:)

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August 30, 2010, at 11:47 PM by 87.58.31.236 -
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(:if !group=MatematikC:)

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(:if group!=MatematikC:)

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August 30, 2010, at 11:47 PM by 87.58.31.236 -
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(:if group=MatematikC:)

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(:if !group=MatematikC:)

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August 30, 2010, at 11:47 PM by 87.58.31.236 -
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(:endif:)

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(:ifend:)

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August 30, 2010, at 11:46 PM by 87.58.31.236 -
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(:if group=MatematikC:) test (:endif:)

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August 29, 2010, at 10:34 PM by 87.58.31.236 -
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(:noaction:)

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August 29, 2010, at 10:32 PM by 87.58.31.236 -
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(:noaction :)

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(:noaction:)

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August 29, 2010, at 10:31 PM by 87.58.31.236 -
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(:noaction :)

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August 29, 2010, at 10:29 PM by 87.58.31.236 -
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(:redirect MatematikC/OpgaverTilGrafiskLøsningAfLigninger:)

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(:redirect MatematikC/OpgaverTilGrafiskLøsningAfLigninger:)

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Opgaver

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(:noprint [[Opgaver->MatematikC/OpgaverTilGrafiskLøsningAfLigninger]]:)

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Opgaver

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Opgaver

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(:noprint [[Opgaver->MatematikC/OpgaverTilGrafiskLøsningAfLigninger]]:)

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Opgaver

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{$$x^2-x = 2 \quad \text{eller} \quad x^2x-2=0$$}

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{$$x^2-x = 2 \quad \text{eller} \quad x^2-x-2=0$$}

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(:include MatematikC/Opgaver til grafisk løsning af ligninger:)

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(:include MatematikC/OpgaverTilGrafiskLøsningAfLigninger:)

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(:toggle div=glopgaver button=0 init=hide lshow=Opgaver lhide="Skjul opgaver":)

(:include MatematikC/Opgaver til grafisk løsning af ligninger:)

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Figuren herunder viser grafen for funktionen {$f(x)=x^5-15x^4+85x^3-225x^2+274x-120$}. Ligningens løsninger er x-koordinaterne for denne funktions skæringspunkter med x-aksen. Det ses, at der er fem løsninger: x=1, x=2, x=3, x=4 og x=5

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Figuren herunder viser grafen for funktionen {$f(x)=x^5-15x^4+85x^3-225x^2+274x-120$}. Ligningens løsninger er x-koordinaterne for denne funktions skæringspunkter med x-aksen. Det ses, at der er fem løsninger: x=1, x=2, x=3, x=4 og x=5


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Figuren herunder viser grafen for funktionen {$f(x)=x^5-15x^4+85x^3-225x^2+274x-120$}. Ligningens løsninger er x-koordinaterne for denne funktions skæringspunkter med x-aksen. Det ses, at der er fem løsninger: x=1, x=2, x=3, x=4 og x=5

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August 28, 2010, at 10:36 PM by 87.58.31.236 -
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Da udtrykkene på de to sider af lighedstegnet herved ændres, skal der tegnes andre grafer, men som det fremgår af figurerne herunder, vil de nye grafers skæringspunkter stadig have x-koordinaterne x=-1 og x=2

to:

Da udtrykkene på de to sider af lighedstegnet herved ændres, skal der tegnes andre grafer, men som det fremgår af figurerne herunder, vil de nye grafers skæringspunkter stadig have x-koordinaterne x=-1 og x=2. Det er selvfølgelig ikke så overraskende, for de omskrevne ligninger har jo de samme løsninger som den oprindelige.

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Lad os som et sidste eksempel se på en ligning, som ville være umulig at løse ved beregning: {$$x^5-15x^4+85x^3-225x^2+274x-120=0$$}

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Her er der to skæringspunkter med x-koordinaterne x=-1 og x=2, som altså er løsninger til ligningen (tjek selv ved at indsætte dem). Der er to løsninger fordi ligningen jo er en andengradsligning

to:

Her er der to skæringspunkter med x-koordinaterne x=-1 og x=2, som altså er løsninger til ligningen (tjek selv ved at indsætte dem). Der er to løsninger fordi ligningen jo er en andengradsligning.

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(:table border=0 width=80% cellpadding=10 align=center bgcolor=#cccc99 cellspacing=0 :)

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Da udtrykkene på de to sider af lighedstegnet herved ændres, skal der tegnes andre grafer, men som det fremgår af figurerne herunder, vil de nye grafers skæringspunkter stadig have x-koordinaterne x=-1 og x=2


(:table border=0 width=80% cellpadding=10 align=center cellspacing=0 :)

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Men hvad nu hvis vi omskriver ligningen til en af formerne: {$$x^2-x = 2 \quad \text{eller} \quad x^2x-2=0$$}

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(:table border=1 width=60% cellpadding=10 align=center bgcolor=#cccc99 cellspacing=0 :) (:cellnr:)

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(:table border=1 width=80% cellpadding=10 align=center bgcolor=#cccc99 cellspacing=0 :) (:cellnr align=center:)

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(:cell:)

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(:cell align=center:)

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(:table border=1 width=60% cellpadding=10 align=center bgcolor=#cccc99 cellspacing=0 :) (:cellnr:)

(:cell:)

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Her er der to skæringspunkter med x-koordinaterne x=-1 og x=2, som altså er løsninger til ligningen (tjek selv ved at indsætte dem). Der er to løsninger fordi ligningen jo er en andengradsligning

to:

Her er der to skæringspunkter med x-koordinaterne x=-1 og x=2, som altså er løsninger til ligningen (tjek selv ved at indsætte dem). Der er to løsninger fordi ligningen jo er en andengradsligning

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Her er der to skæringspunkter med x-koordinaterne x=-1 og x=2, som altså er løsninger til ligningen (tjek selv ved at indsætte dem). Der er to løsninger fordi ligningen jo er en andengradsligning

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\\

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\\

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[$$x^2 = x+2$$}

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{$$x^2 = x+2$$}

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Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi - ligningen {$x-2 = -x+5$} er altså sand - og x=3 er derfor en løsning til ligningen (det kontrolleres let ved at indsætte x=3 i ligningen).

to:

Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi - ligningen {$x-2 = -x+5$} er altså sand - og x=3 er derfor en løsning til ligningen (det kontrolleres let ved at indsætte x=3 i ligningen).

Lad os se på et andet eksempel: [$$x^2 = x+2$$} På figuren herunder ses graferne for funktionerne {$y = x^2$} og {$y = x+2$}.

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder. Der er tale om lineære funktioner, hvis grafer er rette linier.

to:

Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder. Der er tale om lineære funktioner, hvis grafer er rette linier (i appletten, som startes ved at klikke på billedet, kan andre lineære funktioner tegnes).

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Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi, og ligningen {$x-2$ = -x+5$} er dermed sand. Derfor er x=3 en løsning til ligningen (det kontrolleres selvfølgelig let ved at indsætte x=3 i ligningen).

to:

Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi - ligningen {$x-2 = -x+5$} er altså sand - og x=3 er derfor en løsning til ligningen (det kontrolleres let ved at indsætte x=3 i ligningen).

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Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3).

to:

Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3). For x-værdien x=3 har udtrykkene {$x-2$} og {$-x+5$} altså samme værdi, og ligningen {$x-2$ = -x+5$} er dermed sand. Derfor er x=3 en løsning til ligningen (det kontrolleres selvfølgelig let ved at indsætte x=3 i ligningen).

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)


Det ses, at der er en enkelt x-værdi, hvor y-værdierne er ens for de to grafer, nemlig x-værdien for det punkt, hvor de skærer hinanden (x=3).

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder.

to:

Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder. Der er tale om lineære funktioner, hvis grafer er rette linier.

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De to udtryk, som står på henholdsvis venstre- og højresiden af lighedstegnet repræsenterer størrelser, hvis værdi afhænger af variablen x. Disse størrelser kunne vi kalde y, og vi får så udtrykkene:

to:

De to udtryk, som står på henholdsvis venstre- og højresiden af lighedstegnet - altså {$x-2$} og {$-x+5$} - repræsenterer størrelser, hvis værdi afhænger af variablen x. Disse størrelser kunne vi kalde y, og vi får så udtrykkene:

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen)

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen), som vist på figuren herunder.




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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="460" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="440" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="460" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/grafisklosning.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/grafisklosning.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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De to udtryk, som står på henholdsvis venstre- og højresiden af lighedstegnet repræsenterer størrelser, hvis værdi afhænger af variablen x. Disse størrelser kunne vi kalde y og får så udtrykkene:

to:

De to udtryk, som står på henholdsvis venstre- og højresiden af lighedstegnet repræsenterer størrelser, hvis værdi afhænger af variablen x. Disse størrelser kunne vi kalde y, og vi får så udtrykkene:

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Disse to ligninger kan afbildes i et koordinatsystem med x-værdierne ud ad den vandrette akse (x-aksen) og y-værdierne op ad den lodrette akse (y-aksen)

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Ikke alle ligninger kan løses ved beregning, og

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Ikke alle ligninger kan løses ved beregning, og derfor kan man have brug for andre metoder. Her vil vi se på, hvordan man kan finde løsninger til ligninger ved hjælp af grafer.

Lad os for eksemplets skyld se på en simpel lineær ligning: {$$x-1 = -x+5$$} De to udtryk, som står på henholdsvis venstre- og højresiden af lighedstegnet repræsenterer størrelser, hvis værdi afhænger af variablen x. Disse størrelser kunne vi kalde y og får så udtrykkene: {$$y=x-1 \quad \text{og} \quad y=-x+5$$}

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Grafisk løsning af ligninger

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Grafisk løsning af ligninger

Ikke alle ligninger kan løses ved beregning, og

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Grafisk løsning af ligninger

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Grafisk løsning af ligninger

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="true" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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Grafisk løsning af ligninger

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Grafisk løsning af ligninger

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="true" clickimg="../MatematikC/parabel.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="true" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="true" clickimg="../MatematikC/parabel.png" clickalign="center":)

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August 26, 2010, at 03:05 AM by 87.58.31.236 -
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(:applet name="ggbApplet" code="geogebra.GeoGebraApplet" codebase="./" archive="geogebra.jar" width="584" height="412" MAYSCRIPT filename="pub/applets/ggbfiler/parabel.ggb" java_arguments="-Xmx512m" framePossible="true" showResetIcon="false" showAnimationButton="true" enableRightClick="false" errorDialogsActive="true" enableLabelDrags="false" showMenuBar="false" showToolBar="false" showToolBarHelp="false" showAlgebraInput="false" clickimg="../MatematikC/parabel.png" clickalign="center":)

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August 26, 2010, at 03:04 AM by 87.58.31.236 -
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