Funktioner 2C: Skillnad mellan sidversioner

Från Wikiskola
Hoppa till navigering Hoppa till sök
 
(214 mellanliggande sidversioner av 2 användare visas inte)
Rad 1: Rad 1:
kan du rita en sån här?
<ggb_applet width="681" height="450"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
= Funktion och graf =
= Funktion och graf =
[[Fil:Celler de Sant Cugat lateral.JPG|thumb|Celler de Sant Cugat lateral]]


s 146
s 146
Rad 15: Rad 13:


Värdemängd = y-värdena
Värdemängd = y-värdena
=== Hur ritar man en parabel om man vet funktionen? ===
Man gör en värde tabell. Tag ett lämpligt x-värde och skriv i tabellens x-kolumn. Räkna ut vad y blir genom att sätta in x-värdet i funktion. Skriv y-värdet i dess kolumn. Nu har du det första talparet. Upprepa med ett antal lämpliga x-värden tills du fått minst tre gärna fem talpar. Det är viktigt att du väljer talparen så att du hittar vertex (min- eller maxpunkten).
<ggb_applet width="852" height="423"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />


= Andragradsfunktioner =
= Andragradsfunktioner =
[[File:Celler de Sant Cugat lateral.JPG|thumb|Celler de Sant Cugat lateral]]
Det kan vara intressant att som bakgrund titta på denna sida om [http://sv.wikipedia.org/wiki/K%C3%A4gelsnitt kägelsnitt].
{{clear}}
== Parabelns ekvation ==
'''Definitioner'''
Brännpunkt kallas också fokus
Styrlinje är en linje som används för att konstruera parabeln
=== GeoGebra som visar samma avstånd ===
<ggb_applet width="918" height="406"  version="4.0" ggbBase64="UEsDBBQACAAIAIqJikAAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiu5QIAUEsHCEXM3l0aAAAAGAAAAFBLAwQUAAgACACKiYpAAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbNVY247bRhJ9dr6iwWdL6jtJQ+NAk4WxBiZxgPEuFvvWJHuk9lAkl6Q0kpHfyZfsj211N0lR0lieSxJgg4z7Vl3VVadu1PzH3TpHW103piyuAjLFAdJFWmamWF4Fm/ZuEgU/vv9hvtTlUie1QndlvVbtVcAtpcmugkSFklCeTaIklhMeSzGJYhzBUogM8zAVcRYgtGvMu6L8Ra11U6lU36YrvVY3ZapaJ3jVttW72ezh4WHai5qW9XK2XCbTXQMM4JlFcxV0k3fA7ujSA3PkFGMy+9fPN579xBRNq4pUB8iqsDHvf3gzfzBFVj6gB5O1q6sgpjJAK22WK9BJcBGgmSWqwCCVTluz1Q1cHS2dzu26ChyZKuz5Gz9D+aBOgDKzNZmurwI8pYwQTAQhURhjMFAUoLI2umg7YtIJnfXs5lujHzxfO3MieYDasswTZVmi335DFFOM3tqB+IHCIKU/wn4PMz9QP3A/CE/D/XXuSbmn4Z6GswBtTWOSXF8FdypvwISmuKsBvmHdtPtcu/d0Gwf1yVvQqTFfgZhh8BNvc9jH+K39k/DH7cHsWEkyktrWm2cK7UXKOH66SPoakawXSQk/F0nFN7SUF4zr3/AUNYkYWRZEuf/d35lERp8h0a9fJ1Dyv0TF+ayPlHkXHKhZWdoOyVavGxsuLEYitl5PkIDQkCE4uUAkhiGkCIIBEYG4gCWJkLRjiFgIBxwxFCFLRxhysSEi+IeHjplEApjZ3RBCEhEQxJFgiLiQ4ggCCbmwhBClDCiEQAIuWfGEWhZMIi5hxSLE4Y02IkMChAwuwhrEU8QIYvYyCRGVSFp+hNtIl5F9OrCkSGIkiWUIQQ0B7YMZ6CPErDayM5cpqk17ZKJ0nfXTtqwGLIAa0tEh6/n0dJQU38xzlegc6sStRRKhrcptRDhBd2XRoh5E7veWtapWJm1uddvCrQZ9UVt1o1q9+wDUTS/b0aZl0fxal+1PZb5ZFw1CaZnj4c1lTkZzOrwaFmx0wMcHYnQgR/PwUbklnKBNo0F+WTc9ucqyj5bikBrAkp+KfH9da3VfleZYjfnMlZy53qS5yYwq/gnOaqVYu6C+Arls1VcgFon+IWWd3e4b8GC0+7euS8gxsZyyWESYS0FkFIUB2ncnAqoxDSEJhYJyyuxRkyobewJPGWc45hxTyQiH0No/fkTjDie9HRBSOz0ov6xtYHeK28XH5rrMD1tO/Z9U1W5q1ztAiNZWqUWxzLVzERfYUJjT+6Tc3XrfYJ7X532l7RX3gGTpzI5qqxlU5GU3Jn50NPZlAxV2NNhR4N7ZTDack5g6CjcmfnRU4L3+aZ2mpFeT4F6MaVxCw8FR2DjXt2V+U5j2pl+0Jr0/aGrpf9msEz040DFL8gexnM9OHGx+r+tC550/A5KbctP48By5eqZTs4alP+gMoixY/4AH+N1ML2vdvzt3XZk3lzvFY1c923asPtTl+mOx/QyecPKA+ax/5bxJa1NZh0MJ1IB7ffCpzDQKSkg2vmcDEFRPbakA87TWNBCam3ZV1q7xgowCo427XK+hy0Ktcy64Z9LBzKnr36w9UZl8gaR2AsPIUHB+8EfMj3wNqbxaKdvkdVrnaq/rIzs4hj+X2al1wPhOBQjxyoNbae3dou2iAVXAzgXT6DkHn20hm95Dxwi+IEaX7OTvJsu0q6Q+pM1SF1vQEpIZtOLYPxjtcdfxf+0nO0isEzfbk47oK/FnjhG8uTY7tAD6CTS2AVr0ZAva31ywfov3EzFiof9T+Oc33q31rspNatrOlT1mZ+hZrQdw4H4NG1/0d0A8BN0Yw3G2IIK9AEPyOIZ4+nQUXdA0aNcj0QMBLjY84yVAv8i4LnUPVvtQ3m+aV1vWZew/1bLkqZZlnWUHF3T62gLkhYrx7kkeGxstLddrVWSocF3sr85oh65KYZtVvE02bb+z8Ey6q9+x/OJFVu9rJMH0j85KL7M7n1L4T2ApcRiFPCQOBTqlEQ3he0hyKuBz1X6TXAAlfhwUehmUv5nug/8YlwWYhPSefQqR2jbtf38vssVw/G28is1a16MycnLXfburfNM9lp+ozC+HzBg4clnPW720+89S87JmTcdx0OzVGeAlfkhPfQmqTEhCKuKYCXClOLS/noA3TR5xMutNdCqlDAWJI8ok5YRdyKX0ebnUrA+51Fk2tzHwsWihG9OuHTlvsu61rmxv+6n4XKuisT+eeZpR8/YNiD/V0NEsy0LlN7b2PYr0bgHN5BnSyWWkj0ppcg5z/zl+sRF6LdD4DGjSZWvcZetzgP8UJL8NwIDsie0P3YfDIDmz//Vzcv71i+JM+jbUDokfXo/A42l7Qr9TPdmLqufFBHZ9ZtLseckr+wuT16e7u0a31oCMe4sR+rTcdloe2JH/T0g0xRGNMAVEqIxh/n+Uy2bjzzT3U0j3q/77/wFQSwcIg6ng8SEHAAByGAAAUEsBAhQAFAAIAAgAiomKQEXM3l0aAAAAGAAAABYAAAAAAAAAAAAAAAAAAAAAAGdlb2dlYnJhX2phdmFzY3JpcHQuanNQSwECFAAUAAgACACKiYpAg6ng8SEHAAByGAAADAAAAAAAAAAAAAAAAABeAAAAZ2VvZ2VicmEueG1sUEsFBgAAAAACAAIAfgAAALkHAAAAAA==" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
=== Parabelns egenskaper i GeoGebra 1 ===
'''Datorövning:''' [http://www.malinc.se/math/functions/parabolasv.php Malin C GGB-övning] {{clear}}


=== GeoGebra med styrlinje och fokus ===
== [[Parabelns ekvation]] ==


<ggb_applet width="792" height="319"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />


== Andragradsfunktionens graf ==
== [[Fyra sätt att beskriva andragradaren]] ==
[[File:Parábola con foco y directriz.svg|thumb|Parábola con foco y directriz]]


vertex är kurvans vändpunkt
== [[Andragradsfunktionens graf]] ==


nollställen
== [[Testa dina kunskaper om andragradsfunktioner]] ==
 
positivt före x<sup>2</sup>-termen betyder minimipunkt
 
negativt före  x<sup>2</sup>-termen betyder maximipunkt
 
symmetrilinje genom vertex
{{clear}}


== Digitala rutan ==
== Digitala rutan ==
Rad 66: Rad 31:
Gör den i GeoGebra.
Gör den i GeoGebra.


== Kvadratiska modeller ==
== [[Kvadratiska modeller]] ==
[[File:Square root.svg|thumb|Square root]]
 
Så här ser andragradsfunktionen ut på allmän form:
 
y(x) = ax<sup>2</sup> + bx + c
 
c anger var grafen skär y-axeln. a gör bland annat parabeln smalare eller bredare. bx-termen ger en diagonal förflyttning av hela kurvan (något förenklat uttryckt).
 
=== Exempel 1 ===
[[File:ParabolicWaterTrajectory.jpg|thumb|ParabolicWaterTrajectory]]
 
Exempel 1 handlar om att man har  en måttsatt bild och ska anpassa den allmänna funktionen y(x) = ax<sup>2</sup> + bx + c till dessa mått.
 
Här är det smart att placera origo symmetriskt i bilden och att kika på ställena där grafen skär x-axeln och där den skär y-axeln.
 
'''Uppgift:''' Anpassa den allmänna funktionen till vattenstrålen i bilden. Strålen når 2 m långt och är 1.5 m hög.
{clear}}
 
=== Exempel 2 ===
 
Exempel 2 i boken handlar om att titta på nollställena för en funktion för att hitta vertex mitt emellan nollställena och sätta in x-värdet och räkna ut y-värdet (högsta punkten i detta fall).
 
=== Parabelns egenskaper i GeoGebra 2 ===
 
I Malins övning skriv kurvan på annan form (x-k)<sup>2</sup>, osv. Nyttigt men vi hinner inte göra den på lektionstid. Gör den gärna hemma!
 
Digitala rutan samt detta avsnitt sid 160-164 ersätts av en [http://www.malinc.se/math/functions/vertexformsv.php Övning i Geogebra på Vertex och faktorform av Malin C].
 
'''Överkurs:''' [http://www.malinc.se/math/functions/otherconicssv.php Andra kägelsnitt] Av Malin C. Pröva själv att konsttruera med hjälp av mittpunktsnormaler.
{{clear}}
 
=== Överbliven provupgift (svår) ===
[[File:Parabolic trajectory.svg|thumb|Parabolic trajectory]]
 
Bilden visar en kastparabel.
 
Tänk dig att kastbanans högsta punkt är 35 m.
 
Längden på kastet är 110 m.
 
Utgå från formen för andragradsfunktionen
y(x) = ax<sup>2</sup> + bx + c
 
Gör en matematisk modell av kastbanan.
{{clear}}
 
= Exponentialfunktioner och logaritmer =
 
== Exponentialfunktioner ==
 
må lektion 1
 
=== Växande ===
 
Årlig tillväxt med 15 % innebär en tillväxtfaktor om 1.1. Antag att man har 2000 från början. Tillväxten blir då exponentiell. Det tar bara fem år till en fördubbling.
 
<ggb_applet width="640" height="383"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 
=== Avtagande ===
 
==== GGB ====
 
RegressionExp[avsvalning]
 
<ggb_applet width="1007" height="487"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "false" allowRescaling = "true" />
 
==== Vatten i termos ====
 
Kaffet i en kopp är 100<sup>o</sup>C från början. När kaffet svalnar sjunker temperaturen med 10<sup>o</sup>C per minut. '''Förändringsfaktorn''' är alltså 0.9
 
<ggb_applet width="557" height="383"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 
=== Definitioner ===
 
y = Ca<sup>x</sup>
 
växande a > 1
 
avtagande a < 1
 
C är skärningspunkt med y-axeln
 
a ej lika med 1, a > 0
 
 
<ggb_applet width="640" height="379"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 
=== Övning - GeoGebra ===
 
Rita själv funktionerna i bildebn överst på sid 167
 
=== Exempel 1 ===
 
Bestäm exponentialfunktionen där grafen går genom punkterna (0,2) och (5,6)
 
<ggb_applet width="700" height="407"  version="4.0" ggbBase64="UEsDBBQACAAIALK7j0AAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiuBQBQSwcI1je9uRkAAAAXAAAAUEsDBBQACAAIALK7j0AAAAAAAAAAAAAAAAAMAAAAZ2VvZ2VicmEueG1szVhbb9s2FH5uf8WBnrahsUld7cJukWYoWiDtCqQbhj0MoCTaZiOJgkj5UvTH75CUZDnJ2qTdigZNeDs8l+/cqC6e78sCtrxRQlZLj06IB7zKZC6q9dJr9eps5j1/9nix5nLN04bBSjYl00svNJQiX3pZsgpXs4ScZXHKz0KW0bM0yfFeOichp1GQsdwD2CvxtJJvWclVzTJ+lW14yS5lxrQVvNG6fjqd7na7SS9qIpv1dL1OJ3uFDFDNSi29bvIU2Z1c2gWW3CeETv98c+nYn4lKaVZl3ANjQiuePX602IkqlzvYiVxvll5CZh5suFhv0KYojD2YGqIaAal5psWWK7w6WlqbdVl7loxV5vyRm0ExmONBLrYi583SIxM/CMM48JN4FsQ0mvkeyEbwSne0tJM57bkttoLvHFszsxJDD7SURcoMR/j0CXziE3hiBuoGH4c4dkfE7ZHADb4bQjdEjiZ010NHGjqa0NGEgQdboURa8KW3YoVCBEW1atB7w1rpQ8GtPt3G0Xr6BG1S4iMSBwTDxEGO+4Q8Mb8x/obmYHpqJB1J1U37QKG9yMif31+k/y0ig14kjePbIv3oX6yMPwOu0+E+ZtJohCyKsv/s7y2Jgf8AiW79bQLj8LuYuJj2mbLokgPUxtB2ntS8VCZdgjlEcxP1FCJMjTjBII+AznFIfMBkABpBGOGSziA2YwJBggchBDADQ0cDsLkRzfBPmFhmMUTIzOwmmJJAUVAIUQDUplQImEhg0xJT1A+QIoogwktGPPUNiyCGMMZVMIMQdTQZmVAkDPAirlG8DwGFwFymCfgxxIYfDU2mxzOjOrL0ISYQU8MQkxoT2iUz0s8gMNb0VU1UdatPIMrKvJ9qWQ++QGosR8ei58rTSU18tChYygtsE1fGkwBbVpiMsIJWstLQOzF0e+uG1RuRqSuuNd5S8IFt2SXTfP8SqVUv29JmslLvGqkvZNGWlQLIZEEGnWVBR3N/0BoXweggHB9Eo4N4NE/ulCvxBFrFUb5sVE/O8vy1oTiWBkTyt6o4vGg4u66lODVjMbUdZ8HbrBC5YNUfGKxGisEF+gZkq1XfgIJZ3Csim/zqoDCCYf8Xb+TSm2PHHf9gih3cSRDOJvPxDx6pjJnci8jpJUytQ3cUzk8v9aL5dvAQ2/PB+HUj8vH8tXohi3yAwlp/wWrdNvblgBWnMTadV+uC2wixeY1tObtO5f7KhUbgeL0/1LgiTn66tqgDVgY/ipAAR2NsitZQR2L0GoiIJSGWgPShJvIb55T4lsKMlgYj1+nVWUl7EynphQhlixnxTlLGhr3p8G0l9GW/0CK7Pppp6N+2ZcqH4DllSf8LlpbgV+FeOO7pZqPuRrwtrnlT8aILb3RsK1vlsnUU+TnPRIlLd9BhxIzzfked3G7O1w3vTSnsG80haE/JOHJvbVtWLxtZvq627zEybiiwmPZaLlTWiNrEH6TYEq75McZyoRh2lHx8z+QjopGZzoGAaIMWZmqrN7KxzzAsMDhayrJkVQ6VbULvTLx6x6LIsLwcztE/Tl3Z6n733OnXXTf5XPASX2+gbdTawB9ceG5ZGl+BTD9gsbzh4mMs4PGdIWxjnRX1hpmnY4dewQ68OcHT8nsj85sooxMtFFg5asPARE7NuYs5pzFOamRo0/Sk9qHjFOydWDigKmb8OEQWPnKNrSZ1ndBovHvD4xiGDqYvAPbixwLsa+CKOrji/wWuqi15I7IBEGYBw5ttd59O/LCri98CIxmBSO8JYi+2MF86UIrKXMbvqZLtjV5YvFmqsIVr/NzDylEdP/eccl0LxMeygW5v+l/QgUkD+4G5Ens+tB1MafERSxg7MehYyDU+L67xE0rZ9qO7pmInr0Se82rQ+HaeYGskt31P7vL9/X11cdNXZBL9MJ46o52jkq9wU9K7aRZ9VzfdkZ9f9tG+blCaYdNhvPIAN3H8af8zLOECfgH2N87dw/fUp6u2ss3FO17+vANHNeI+HnxgwXo4jGMopuN2aV+o3f+1PPsHUEsHCAvfnDLuBQAACBIAAFBLAQIUABQACAAIALK7j0DWN725GQAAABcAAAAWAAAAAAAAAAAAAAAAAAAAAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgAsruPQAvfnDLuBQAACBIAAAwAAAAAAAAAAAAAAAAAXQAAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAgACAH4AAACFBgAAAAA=" showResetIcon = "false" enableRightClick = "true" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 
=== Exempel 2 ===
 
Lös ekvationen 2<sup>x</sup> = 1 + 3x grafiskt.
 
Lös även olikheten 2<sup>x</sup> < 1 + 3x
 
== Linjära och exponentiella modeller ==
 
må lektion 2 v 16
 
= Logaritmer och funktionen y = 10<sup>x</sup> =
[[Fil:Logarithms.png|miniatyr|300px|Logaritmfunktioner, ritade för olika baser. <span style="color:red">Röd</span> graf svarar mot basen ''<span style="color:red">e</span>'', <span style="color:green">grön</span> graf mot basen <span style="color:green">10</span>, och <span style="color:purple">lila</span> graf mot basen <span style="color:purple">1.7</span>.
 
Varje ruta på axlarna är en enhet. Samtliga grafer avbildar punkten (1, &nbsp;0) då alla tal upphöjda till 0 är lika med 1 och dessutom punkten (''b'',&nbsp;1) för basen ''b'', då ett tal upphöjt till 1 är lika med talet självt. Graferna har högergränsvärdet -∞ då x -> 0 från höger.]]
 
'''Logaritmen''' för ett tal ''a'' är den exponent ''x'' till vilket ett givet tal, basen ''b'', måste upphöjas för att anta värdet ''a'':
:a = b<sup>x</sup>
 
Logaritmernas uppfinnare anses skotten John Napier (1600-talet) vara.
{{clear}}
''Texten ovan från Wikipedia''
 
 
== Vad är logaritmer? ==
 
[[Fil:Graph of common logarithm.png|300px|miniatyr|Graf över tiologaritmen]]
 
Tisdag
 
Ett praktiskt val av bas när man använder den decimala notationen är  (10-logaritmen): den exponent ''x'' till vilken man ska upphöja 10 för att få talet ''a'':
 
:a = 10<sup>x</sup> <==> x = log<sub>10</sub>a
Andra beteckningssätt för log<sub>10</sub> ''a'' är log ''a'' och lg ''a''.
{{clear}}
 
== Räkneregler för logaritmer ==
 
Onsdag v 16
 
'''Sats:''' Multiplikation
lg(a b) = lg a + lg b
 
'''Sats:''' Division
lg (a/b) = lg a - lg b
 
'''Sats:''' Potensräkning
  lg a<big>p</big> = p lg a
 
== Logaritmiska modeller ==
 
Torsdag v 16
 
== Aktivitet richterskalan ==
 
== Ekvationen 2<sup>x</sup> = 3 ==
 
Mån v 17
 
== Tillämpningar på exponentiell förändring ==
 
Lektion 2, måndag v 17
 
== Aktivitet: När kan du dricka ditt kaffe? ==
 
== Fler funktioner ==
 
Tisdag v 17
 
y = 1 / x är diskontinuerlig
 
y = lg x
 
y = x<sup>0.5</sup> (roten ur x)
 
=== Logaritmer på andra baser ===
[[File:Function-log-animation.gif|thumb|2-logaritmen och 2^x]]
[[File:Function-log-animation2.gif|thumb|1/2-logaritmen]]{{clear}}


== Repetition ==
== Kortdiagnos 4 ==


Som planeringen ser ut har vi tre lektioner för repetition. Det är bra med tanke på att något kan gå bort tidigare.
{{print|[[Media:Kortdiagnos_4.pdf|Kortdiagnos4]]}}


* '''Onsdag v 17'''
== Utmaning ==
* To v 17 går bort pga NP Sv
* '''Må v 18''' Valborg = skoldag
* Tisdag v 18 = Ledig = 1:a maj
* '''Onsdag v 18''' lektion som vanligt


'''Prov'''
Klarar du denna övning?


torsdag den 3 maj, v 18
<html>
<script type='text/javascript' src='http://demonstrations.wolfram.com/javascript/embed.js' ></script><script type='text/javascript'>var demoObj = new DEMOEMBED(); demoObj.run('FunctionIdentificationGame', '', '439', '682');</script><div id='DEMO_FunctionIdentificationGame'><a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/FunctionIdentificationGame/' target='_blank'>Function Identification Game</a> from the <a class='demonstrationHyperlink' href='http://demonstrations.wolfram.com/' target='_blank'>Wolfram Demonstrations Project</a> by Izidor Hafner</div>
</html>

Nuvarande version från 14 november 2016 kl. 12.37

Funktion och graf

Celler de Sant Cugat lateral

s 146

Teori funktionen f(x)

Vad står f(x) för? Funktionen f med variabeln x.

Lösa ekvationer med grafer

Definitionsmängd = x-värdena

Värdemängd = y-värdena

Andragradsfunktioner

Parabelns ekvation

Fyra sätt att beskriva andragradaren

Andragradsfunktionens graf

Testa dina kunskaper om andragradsfunktioner

Digitala rutan

Sidan 159.

Gör den i GeoGebra.

Kvadratiska modeller

Kortdiagnos 4

Du kan printa denna! Kortdiagnos4


Utmaning

Klarar du denna övning?