Funktioner 2C: Skillnad mellan sidversioner
Hakan (diskussion | bidrag) |
Hakan (diskussion | bidrag) |
||
Rad 23: | Rad 23: | ||
[[File:ParabolicWaterTrajectory.jpg|thumb|ParabolicWaterTrajectory]] | [[File:ParabolicWaterTrajectory.jpg|thumb|ParabolicWaterTrajectory]] | ||
Brännpunkt | '''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 === | |||
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== Andragradsfunktionens graf == | == Andragradsfunktionens graf == |
Versionen från 11 april 2012 kl. 05.56
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
s 146
f(x)
Lösa ekvationer med grafer
Definitionsmängd = x-värdena
Värdemängd = y-värdena
Andragradsfunktioner
Det kan vara intressant att som bakgrund titta på denna sida om kägelsnitt.
Parabelns ekvation
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/ParabolicWaterTrajectory.jpg/300px-ParabolicWaterTrajectory.jpg)
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: Malin C GGB-övning
GeoGebra med styrlinje och fokus
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Andragradsfunktionens graf
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Par%C3%A1bola_con_foco_y_directriz.svg/langsv-300px-Par%C3%A1bola_con_foco_y_directriz.svg.png)
vertex är kurvans vändpunkt
nollställen
positivt före x2-termen betyder minimipunkt
negativt före x2-termen betyder maximipunkt
symmetrilinje genom vertex
Kvadratiska modeller
![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Square_root.svg/langsv-300px-Square_root.svg.png)
y(x) = ax2 + bx + c
Digitala rutan samt detta avsnitt sid 160-164 ersätts av en Övning i Geogebra på Vertex och faktorform av Malin C.
Överkurs: Andra kägelsnitt Av Malin C. Pröva själv att konsttruera med hjälp av mittpunktsnormaler.
Överbliven provupgift (svår)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Parabolic_trajectory.svg/langsv-300px-Parabolic_trajectory.svg.png)
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) = ax2 + bx + c
Gör en matematisk modell av kastbanan.
Exponentialfunktioner och logaritmer
Exponentialfunktioner
y = Cax
växande a > 1
avtagande a < 1
skärningspunkt med y-axeln
a ej lika med 1, a > 0
Linjära och exponentiella modeller
Logaritmer och funktionen y = 10x
![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Logarithms.png/300px-Logarithms.png)
Logaritmen för ett tal a är den exponent x till vilket ett givet tal, basen b, måste upphöjas för att anta värdet a:
- a = bx
Logaritmernas uppfinnare anses skotten John Napier (1600-talet) vara.
Texten ovan från Wikipedia
Vad är logaritmer?
![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Graph_of_common_logarithm.png/300px-Graph_of_common_logarithm.png)
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 = 10x <==> x = log10a
Andra beteckningssätt för log10 a är log a och lg a.
Räkneregler för logaritmer
Sats: Multiplikation
lg(a b) = lg a + lg b
Sats: Division
lg (a/b) = lg a - lg b
Sats: Potensräkning
lg ap = p lg a
Logaritmiska modeller
Aktivitet richterskalan
Ekvationen 2x = 3
Tillämpningar på exponentiell förändring
Aktivitet: När kan du dricka ditt kaffe?
Fler funktioner
y = 1 / x är diskontinuerlig
y = lg x
y = x0.5 (roten ur x)
Logaritmer på andra baser
![](https://upload.wikimedia.org/wikipedia/commons/4/45/Function-log-animation.gif)
![](https://upload.wikimedia.org/wikipedia/commons/3/3e/Function-log-animation2.gif)