# 5.3 Exponential curves  (Page 2/2)

 Page 2 / 2 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: $2\cdot {2}^{x}={2}^{1}\cdot {2}^{x}={2}^{x+1}$

## 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, $y=3x+4$ models a function that starts at 4; every time you increase $x$ by 1, you add 3 to $y$ .

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

Linear functions can go down, as well as up, by having negative slopes : $y=-3x+4$ starts at 4 and subtracts 3 every time. Exponential functions can go down, as well as up, by having fractional bases : $y=4×\left(\frac{1}{3}{\right)}^{x}$ 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 $8%\text{}$ 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 $8%\text{}$ of 100,000 = 8,000 108,000 1983 $8%\text{}$ of 108,000 = 8,640 116,640 1984 $8%\text{}$ 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 $8%\text{}$ 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 $\text{100},\text{000}×1\text{.}\text{08}$ 1983 $\text{100},\text{000}×1\text{.}{\text{08}}^{2}$ 1984 $\text{100},\text{000}×1\text{.}{\text{08}}^{3}$ 1985 $\text{100},\text{000}×1\text{.}{\text{08}}^{4}$ $y$ $\text{100},\text{000}×1\text{.}{\text{08}}^{\text{something}}$ 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 $\text{100},\text{000}×1\text{.}{\text{08}}^{2}$ in 1983, then its value in 1984 is $\left(\text{100},\text{000}×1\text{.}{\text{08}}^{2}\right)×1\text{.}\text{08}$ , which is $\text{100},\text{000}×1\text{.}{\text{08}}^{3}$ . 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 $y$ . 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 $y$ , the exponent is $y-\text{1981}$ . So we have our house value function: $v\left(y\right)=\text{100},\text{000}×1\text{.}{\text{08}}^{y-\text{1981}}$ That is the pattern we needed in order to answer the question. So in the year 2001, the value of the house is $\text{100},\text{000}×1\text{.}{\text{08}}^{\text{20}}$ . 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.

What is demand
the amount of a good that buyers are willing and able to purchase
Asit
what is population
The people living within a political or geographical boundary.
Ziyodilla
what happens to price and quantity when demand curves shift to the right
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?
price level will increase
Asit
quantity demand will decrease
Asit
what is inflation
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
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
what is elasticity
Elasticity is an economics concept that measures responsiveness of one variable to changes in another variable.
cynthia
right
Augustine
wooow!!
cynthia
Computer software represents
पर्यावरण राज्यों में से किस राज्य में शिष्य शिक्षक अनुपात 30 से अधिक वाले विद्यालयों का प्रतिशत न्यूनतम होता है
Hey what are you trying to mean?
Kenyana
what is Asset
MUBARAK
like a banana
Ahmed
demand is the process whereby consumers are willing and able to purchase a particular product at various price over a given period of time
The law of dinimish
What is the law of dinimish
Frank
What is the law of dinimish
Frank
What is the law of dinimish
Frank
opportunity cost is to forgo something for another.
yes
King
what is financial market
what is demand
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.
Ali
explain any three exceptions to the law of demand
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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