Exponential Growth and Decay

Which is more? Which option would you choose? Suppose you have an offer to work for one hour a day for 6 days. You are told that you will only be paid for day 6. You can choose one of two payment options. One option: your pay begins at $5 an hour, and is in-creased by $5 each day. (For example, you get $5 per hour the first day, $10/hr the second day, $15/hr the third day, etc.) The other option: your pay begins at $2 an hour and is doubled each day. (for example, you get $2/hr the first day, $4/hr the second day, $8/hr the third day, etc.) Which option would you choose? FHS Functions

How can we compare the options? Let’s make a table of values for each option. The $5 increase each day: The $2 doubling each day: Days Pay 1 $5 2 $10 3 $15 4 $20 5 $25 6 $30 Days Pay 1 $2 2 $4 3 $8 4 $16 5 $32 6 $64 FHS Functions

Daily pay (each square is $5) Graph options If we graph the points from the tables on the previous slide and draw lines through the points, we get these two graphs. The green line represents the $5 increase option. The red line represents the $2 doubling option. The $5 increase option is a straight line (a linear function). The $2 doubling option is what we call an exponential function. Daily pay (each square is $5) 80 70 60 50 40 30 20 10 2 4 6 8 Days worked FHS Functions

Why is this called exponential? Let’s look at the table of values for the $2 doubling option. If we were to make an equation to represent this function, it would look like this: To see how this works, put in a value for x (one of the days) and see if you get the pay for that day. Days Pay 1 $2 2 $4 3 $8 4 $16 5 $32 6 $64 FHS Functions

How can you compare functions? An exponential growth function is one that increases by a fixed rate. A linear function is one that increases by a fixed amount. Tell whether the following are growing by a fixed rate or a fixed amount. 0, 3, 6, 9, . . . 1, 3, 9, 27, 81, . . . Fixed amount Fixed rate FHS Functions

Where in life do you see exponential functions? Compounding interest that you may receive in your bank account is an example of exponential growth. The growth of bacteria often is an example of exponential growth. Often we see something decrease in an exponential way (for example, radioactive decay) A function that is decreasing exponentially is called exponential decay. Let’s look at this. FHS Functions

Radioactive Decay All radioactive elements decay over time (some of the radio-active atoms will decay into non-radioactive atoms as time passes) A “half-life” of a radioactive element is the time it takes for half of the elements to decay. After a another half-life, another half will decay. A graph might look like this: Amount Time FHS Functions

Formula Using exponential functions we can create a formula to use to help calculate exponential growth and exponential decay. We will want to express the rate as a decimal. For example, the rate for doubling is 100% or 1.00. We use A to represent the beginning amount, r to represent the rate of growth or decay and t to represent the time period of the rate. The resulting equations would be: Exponential growth (interest) ___________ Exponential decay (depreciation) __________ FHS Functions