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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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