Linear Growth

Study the double number line below. On the bottom of the number line is the number of gallons put into the car. The top of the number line is the price the customer pays. For each gallon of gas put into the car, $2.25 is added to the price the customer has to pay.

Instead of adding $2.25 over and over (repeated addition), we use multiplication. The price the customer has to pay is $2.25 times the number of gallons of gasoline put into the car. The linear growth model represented by the above double number line is:

y = 2.25x, where y is the price the customer owes and x is the number of gallons

Exponential Growth

Now, consider the double number line below. On the bottom of the number line is the number of years of an investment.  The top of the number line is the accumulated amount in the investment. The initial investment was $100. Each year, the amount in the account was multiplied by 1.05. The 1.05 tells us the investment grew 5% each year.

Instead of multiplying by 1.05 over and over (repeated multiplication), we use exponents. The base (or growth factor) of 1.05 is multiplied by the investment amount each year. The exponential growth model represented by the above double number line is:

y = 100 (1.05)^x, where y is the amount accumulated in the investment and x is the number of years

Comparing Linear and Exponential Growth

One way to determine if a model is linear or exponential, is to examine the graphs.

Linear Growth	Exponential Growth

Another way to determine if a model is linear or exponential is to examine the table of values.

In a linear model, each time x is increased by one, the same number is added to the y-value.	In an exponential model, each time x is increased by one, the same number is multiplied by the y-value.

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