Funktioner 2C

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Funktion och graf

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

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

Celler de Sant Cugat lateral

Det kan vara intressant att som bakgrund titta på denna sida om kägelsnitt.

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: Malin C GGB-övning

GeoGebra med styrlinje och fokus

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Andragradsfunktionens graf

Parábola con foco y directriz

vertex är kurvans vändpunkt

nollställen

positivt före x²-termen betyder minimipunkt

negativt före x²-termen betyder maximipunkt

symmetrilinje genom vertex

Digitala rutan

Sidan 159.

Gör den i GeoGebra.

Kvadratiska modeller

Square root

Så här ser andragradsfunktionen ut på allmän form:

y(x) = ax² + 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

ParabolicWaterTrajectory

Exempel 1 handlar om att man har en måttsatt bild och ska anpassa den allmänna funktionen y(x) = ax² + 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)², 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 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)

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) = ax² + bx + c

Gör en matematisk modell av kastbanan.

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^oC från början. När kaffet svalnar sjunker temperaturen med 10^oC per minut. Förändringsfaktorn är alltså 0.9

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Definitioner

y = Ca^x

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^x = 1 + 3x grafiskt.

Lös även olikheten 2^x < 1 + 3x

Linjära och exponentiella modeller

må lektion 2 v 16

Logaritmer och funktionen y = 10^x

Logaritmfunktioner, ritade för olika baser. Röd graf svarar mot basen e, grön graf mot basen 10, och lila graf mot basen 1.7. Varje ruta på axlarna är en enhet. Samtliga grafer avbildar punkten (1,  0) då alla tal upphöjda till 0 är lika med 1 och dessutom punkten (b, 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^x

Logaritmernas uppfinnare anses skotten John Napier (1600-talet) vara.

Texten ovan från Wikipedia

Vad är logaritmer?

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^x <==> x = log₁₀a

Andra beteckningssätt för log₁₀ a är log a och lg a.

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 ap = p lg a

Logaritmiska modeller

Torsdag v 16

Ekvationen 2^x = 3

Mån v 17

Tillämpningar på exponentiell förändring

Lektion 2, måndag v 17

Fler funktioner

Tisdag v 17

y = 1 / x är diskontinuerlig

y = lg x

y = x^0.5 (roten ur x)

Logaritmer på andra baser

2-logaritmen och 2^x

1/2-logaritmen

Repetition

Som planeringen ser ut har vi tre lektioner för repetition. Det är bra med tanke på att något kan gå bort tidigare.

• Onsdag v 17
• 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

torsdag den 3 maj, v 18