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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 demand
TECK Reply
the amount of a good that buyers are willing and able to purchase
Asit
what is population
Amadou Reply
The people living within a political or geographical boundary.
Ziyodilla
what happens to price and quantity when demand curves shift to the right
Asha Reply
price level goes up. quantity demand increases
Asit
example- inferior goods
Asit
demand law
Athony
Its states that higher the price the of the commodity, and lower the quantity demanded
Kosiso
I am confused but quantity demand will increase.
Asit
No. That's the law of supply
Kosiso
what happens to price and quantity when supply curve shifts left?
Asha Reply
price level will increase
Asit
quantity demand will decrease
Asit
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Pop Reply
inflation is a general and ongoing rise in the level of prices in an entire economy.
cynthia
is the pasistance increase in the price of a country economy
Liyu
kk
Duppy
yes
Aadi
how does inflation affects the economy of a country? what is deflation?
Augustine
deflation can simply be define as the persistence decrease in price of a countrys economy
Liyu
the revenge of malthus relates "revenge" with "commodity prices". collect data for 3 commodoties and check their price evolution
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Elasticity is an economics concept that measures responsiveness of one variable to changes in another variable.
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Ahmed
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The law of dinimish
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Frank
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What is the law of dinimish
Frank
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jackie Reply
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King
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Levinel Reply
Demand is an economic principle referring to a consumer's desire to purchase goods and services and willingness to pay a price for a specific good or service.
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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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