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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 is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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
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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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