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Graph of 2 to the x. Shifted one to the left
y = 2 2 x size 12{y=2 cdot 2 rSup { size 8{x} } } {} aka y = 2 x + 1 size 12{y=2 rSup { size 8{x+1} } } {}

Once again, this is predictable from the rules of exponents: 2 2 x = 2 1 2 x = 2 x + 1 size 12{2 cdot 2 rSup { size 8{x} } =2 rSup { size 8{1} } cdot 2 rSup { size 8{x} } =2 rSup { size 8{x+1} } } {}

Using exponential functions to model behavior

In the first chapter, we talked about linear functions as functions that add the same amount every time . For instance, y = 3x + 4 size 12{y=3x+4} {} models a function that starts at 4; every time you increase x size 12{x} {} by 1, you add 3 to y size 12{y} {} .

Exponential functions are conceptually very analogous: they multiply by the same amount every time . For instance, y = 4 × 3 x size 12{y=4 times 3 rSup { size 8{x} } } {} models a function that starts at 4; every time you increase x size 12{x} {} by 1, you multiply y size 12{y} {} by 3.

Linear functions can go down, as well as up, by having negative slopes : y = 3x + 4 size 12{y= - 3x+4} {} starts at 4 and subtracts 3 every time. Exponential functions can go down, as well as up, by having fractional bases : y = 4 × ( 1 3 ) x size 12{y=4 times \( { {1} over {3} } \) rSup { size 8{x} } } {} starts at 4 and divides by 3 every time.

Exponential functions often defy intuition, because they grow much faster than people expect.

Modeling exponential functions

Your father’s house was worth $100,000 when he bought it in 1981. Assuming that it increases in value by 8 % size 12{8%} {} every year, what was the house worth in the year 2001? (*Before you work through the math, you may want to make an intuitive guess as to what you think the house is worth. Then, after we crunch the numbers, you can check to see how close you got.)

Often, the best way to approach this kind of problem is to begin by making a chart, to get a sense of the growth pattern.

Year Increase in Value Value
1981 N/A 100,000
1982 8 % size 12{8%} {} of 100,000 = 8,000 108,000
1983 8 % size 12{8%} {} of 108,000 = 8,640 116,640
1984 8 % size 12{8%} {} of 116,640 = 9,331 125,971

Before you go farther, make sure you understand where the numbers on that chart come from . It’s OK to use a calculator. But if you blindly follow the numbers without understanding the calculations, the whole rest of this section will be lost on you.

In order to find the pattern, look at the “Value” column and ask: what is happening to these numbers every time? Of course, we are adding 8 % size 12{8%} {} each time, but what does that really mean? With a little thought—or by looking at the numbers—you should be able to convince yourself that the numbers are multiplying by 1.08 each time . That’s why this is an exponential function: the value of the house multiplies by 1.08 every year.

So let’s make that chart again, in light of this new insight. Note that I can now skip the middle column and go straight to the answer we want. More importantly, note that I am not going to use my calculator this time—I don’t want to multiply all those 1.08s, I just want to note each time that the answer is 1.08 times the previous answer .

Year House Value
1981 100,000
1982 100 , 000 × 1 . 08 size 12{"100","000" times 1 "." "08"} {}
1983 100 , 000 × 1 . 08 2 size 12{"100","000" times 1 "." "08" rSup { size 8{2} } } {}
1984 100 , 000 × 1 . 08 3 size 12{"100","000" times 1 "." "08" rSup { size 8{3} } } {}
1985 100 , 000 × 1 . 08 4 size 12{"100","000" times 1 "." "08" rSup { size 8{4} } } {}
y size 12{y} {} 100 , 000 × 1 . 08 something size 12{"100","000" times 1 "." "08" rSup { size 8{"something"} } } {}

If you are not clear where those numbers came from, think again about the conclusion we reached earlier: each year, the value multiplies by 1.08. So if the house is worth 100 , 000 × 1 . 08 2 size 12{"100","000" times 1 "." "08" rSup { size 8{2} } } {} in 1983, then its value in 1984 is 100 , 000 × 1 . 08 2 × 1 . 08 size 12{ left ("100","000" times 1 "." "08" rSup { size 8{2} } right ) times 1 "." "08"} {} , which is 100 , 000 × 1 . 08 3 size 12{"100","000" times 1 "." "08" rSup { size 8{3} } } {} .

Once we write it this way, the pattern is clear. I have expressed that pattern by adding the last row, the value of the house in any year y size 12{y} {} . And what is the mystery exponent? We see that the exponent is 1 in 1982, 2 in 1983, 3 in 1984, and so on. In the year y size 12{y} {} , the exponent is y 1981 size 12{y - "1981"} {} .

So we have our house value function:

v ( y ) = 100 , 000 × 1 . 08 y 1981 size 12{v \( y \) ="100","000" times 1 "." "08" rSup { size 8{y - "1981"} } } {}

That is the pattern we needed in order to answer the question. So in the year 2001, the value of the house is 100 , 000 × 1 . 08 20 size 12{"100","000" times 1 "." "08" rSup { size 8{"20"} } } {} . Bringing the calculator back, we find that the value of the house is now $466,095 and change.

Wow! The house is over four times its original value! That’s what I mean about exponential functions growing faster than you expect: they start out slow, but given time, they explode. This is also a practical life lesson about the importance of saving money early in life—a lesson that many people don’t realize until it’s too late.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is system testing?
AMJAD
preparation of nanomaterial
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how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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