In the first chapter, we talked about linear functions as functions that add the same amount every time. For instance,
Exponential functions are conceptually very analogous: they multiply by the same amount every time. For instance,
Linear functions can go down, as well as up, by having negative slopes:
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
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  108,000  
1983  116,640  
1984  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
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 

1983 

1984 

1985 



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
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
So we have our house value function:
That is the pattern we needed in order to answer the question. So in the year 2001, the value of the house is
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.
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