Funktioner 2C: Skillnad mellan sidversioner

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(214 mellanliggande sidversioner av 2 användare visas inte)
Rad 1: Rad 1:
kan du rita en sån här?
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= 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).
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= 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 ===
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=== 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]] ==


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== 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.
 
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=== Avtagande ===
 
==== GGB ====
 
RegressionExp[avsvalning]
 
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==== 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
 
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=== 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
 
 
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=== 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)
 
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=== 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'''
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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

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Hämtad från ”https://wikiskola.se/index.php?title=Funktioner_2C&oldid=37985”