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I den här lektionen går vi igenom följande begrepp:

Exponentialfunktion
Exponentiell ökning och minskning

Förkunskaper

Funktion
Linjär funktion

Utforska

Investigating Functions Through Tables

Functions might look similar at first glance. For example, both could have increasing values — yet, one might grow much faster than the other. Investigating a function's values helps tell how they might differ. Consider following tables of values which belong to two functions derived from different situations.

How do the

y -

values of Function I and Function II change according to their corresponding

x -

values? What do the graph of these functions look like? Is Function I a linear function? What conclusions can be made about Function II?

Teori

Exponentialfunktion

Funktioner som innehåller uttryck på formen $a^{x},$ alltså där variabeln $x$ finns i exponenten, kallas exponentialfunktioner. Generellt skrivs en exponentialfunktion på följande sätt.

$y = C \cdot a^{x}$

Koefficienten $C$ anger det $y -$ värde där funktionens graf skär $y -$ axeln, vilket också kan tolkas som funktionens startvärde. Basen $a$ i potensen kan tolkas som en förändringsfaktor. För båda dessa konstanter finns det villkor som anger vilka värden de får anta.

C a - startv \ddot{a} rde - f \ddot{o} r \ddot{a} ndringsfaktor

Koefficienten $C$ får inte vara noll eftersom det skulle ge en vågrät linje linje längs med $y = 0$ , vilket då inte längre skulle vara en exponentialfunktion. Multipliceras $a^{x}$ med $0$ blir ju produkten $0$ oavsett potensens värde.

Graph of y=C*2^x where the value of 'C' can be changed from -2.5 to 2.5

Konstanten

a

får inte vara negativ eftersom funktionen då ger odefinierade resultat för vissa

x -

värden. T.ex. skulle det inte gå att upphöja ett negativt

a

till

x = \frac{1}{2},

eftersom det är samma sak som att dra kvadratroten ur

a,

vilket inte går för ett negativt tal. Det ger villkoret

a \geq 0 .

Vidare ger

a = 0

och

a = 1

inte exponentialfunktioner utan vågräta linjer. När

a = 0

är funktionen alltid lika med

0,

vilket ger en vågrät linje vid

y = 0,

och när

a = 1

får man en vågrät linje längs med startvärdet

C

eftersom

1^{x} = 1,

oavsett exponentens värde. Det ger villkoren

a \neq = 0 och a \neq = 1 .

Dessa villkor kan sammanfattas som

a > 0

och

a \neq = 1 .

Graph of y=2*a^x where the value of 'a' can be changed from 0.1 to 2

Grafen för en exponentiell funktion är alltid ökande eller minskande, beroende på värdena av $C$ och $a .$

Graph of y=a*b^x for a-values less than and greater than 0, and for b-values less than and greater than 1.

Övning

Identifying Functions

Consider the definitions of an exponential function and a linear function. Now, identify the functions given by the following table of values.

Identifying functions

Teori

Exponentiell ökning och minskning

För en exponentialfunktion gäller:

Om förändringsfaktorn $a > 1$ är det en exponentiell ökning. Funktionen kallas då växande eftersom $y -$ värdet ökar när $x$ ökar.

Graph of exponential growth functions

Om förändringsfaktorn $0 < a < 1$ är det en exponentiell minskning. Funktionen kallas då avtagande eftersin $y -$ värdet minskar när $x$ ökar.

Exponential Decay Applet

Exempel

Calculating the salary

The monthly salary of a certain worker in a company, in kr, is described by the exponential function $y = 32000 \cdot 1, 0 3^{x},$ where $x$ is the number of years they have worked in the company.

a What is the monthly salary of a new employee?

b By what percentage the monthly salary increases per year?

c What is the monthly salary of someone who has been working at the company for

4

years? Round the answer to the closest whole number.

Ledtråd

a What is the initial value of the given exponential function?

b What is the growth factor of the given exponential function?

c Evaluate the given exponential function when

x = 4 .

Lösning

a The monthly salary of a new employee is described by the function when

x = 0,

that is, when they have been working

0

years at the company. This corresponds to the function's initial value.

y = 32000 \cdot 1, 0 3^{x}

This means that the monthly salary of a new employee is

32000

kr.

b Find by what percentage the monthly salary increases per year by looking at the function's growth rate.

y = 32000 \cdot 1, 03^{x}

The growth rate is

1, 03 .

This rate is

0, 03

greater than

1,

which means that the monthly salary increases by

3 % .

c The given function describes the monthly salary of a worker who has been working for

x

years at the company. This means that evaluating the function when

x = 4

yields the monthly salary of a worker that has been working at the company for

4

years.

y = 32000 \cdot 1, 0 3^{x}

$x = 4$

y = 32000 \cdot 1, 0 3^{4}

Beräkna potens

y = 32000 \cdot 1, 125508 \dots

Multiplicera faktorer

y = 36016, 28192 \dots

Avrunda till närmaste heltal

y = 36016

The monthly salary of a worker that has been working

4

years at the company is

36016

kr.

Exempel

Calculating the price of a bike

The price $y$ of a certain model of a mountain bike can be described by the exponential function $y = 21000 \cdot 0, 7 5^{x},$ where $x$ is the number of years after it was released to the market.

a What is the price of the mountain bike on the year it is released into the market?

b By what percentage does its price decrease every year?

c What would be the price of the bike

3

years after its release into the market? Round the answer to the nearest whole number.

Ledtråd

a What is the initial value of the given exponential function?

b What is the growth factor of the given exponential function?

c Evaluate the given exponential function when

x = 3 .

Lösning

a The release price of the mountain bike corresponds to the exponential function's initual value.

y = 21000 \cdot 0, 7 5^{x}

This means that the price of the bike is

21000

kr when just released into the market.

b Find by what percentage the price of the bike decreases by looking at the function's growth rate.

y = 21000 \cdot 0, 75^{x}

The growth rate is

0, 72,

which is

0, 25

less than

1,

so the price of the bike decreases by

25 %

every year.

c The given function describes the price of a mountain bike

x

years after its release into the market. This means that evaluating the function when

x = 3

results in the bike's price

3

years after its release.

y = 21000 \cdot 0, 7 5^{x}

$x = 3$

y = 21000 \cdot 0, 7 5^{3}

Beräkna potens

y = 21000 \cdot 0, 421875

Multiplicera faktorer

y = 8859, 375

Avrunda till närmaste heltal

y = 8859

The price of the mountain bike

3

years after it was released into the market is

8859

kr.

Övning

Identifying Exponential Growth or Decay

The following applet shows a function in the form of an expression or a table of values. Select the option that best describes it.

Identifying functions

Övning

Identifying Rate of Growth or Decay

Exponential functions $y = a b^{t}$ can model exponential growth and decay. Identify the rate of decay or growth $r$ for the given function. Write the corresponding rate in decimal form.

Exponential Growth or Decay Equation

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