Exponential Functions

Stuck on a Desert Island! This investigation will simulate exponential decay. Each person will need roll a standard six- sided die. Each standing person represents a stranded person on the island. The people who sit down at each stage represent someone who was rescued off the island.

Follow these steps Collect data in the form (stage, number standing). All members of the class should stand up, except for the recorder. The recorder counts and records the number standing at each stage. Each standing person rolls a die, and anyone who gets a 1 sits down. Wait for the recorder to count and record the number of people standing. Repeat the last two steps until fewer than three students are standing.

Graph your data. What type of sequence does this resemble? Identify uo and the common ratio, r, for your sequence. Complete the table below. Use the values of uo and r to help you write an explicit formula for your data.

Example A Most automobiles depreciate as they get older. Suppose an automobile that originally costs $14,000 depreciates by one-fifth of its value every year. What is the value of this automobile after 2 1/2 years? When is this automobile worth half of its initial value?

To find when the automobile is worth half of its initial value, or 7,000 replace the y with 7000 and solve for x. 7,000= 14,000(1- 0.2)x 0.5= (1- 0.2)x 0.5 =(0.8)x You don’t yet know how to solve for x when x is an exponent, but you can experiment to find an exponent that produces a value close to 0.5. The value of (0.8)3.106 is very close to 0.5. This means that the value of the car is about $7,000, or half of its original value, after 3.106 years (about 3 years 39 days). This is the half-life of the value of the automobile, or the amount of time needed for the value to decrease to half of the original amount.

EXAMPLE B Rita wants to deposit $500 into a savings account so that its doubling time will be 8 years. What annual percentage rate is necessary for this to happen? (Assume the interest on the account is compounded annually.) Rita will need to find an account with an annual percentage rate of approximately 9.05%.

When an exponential graph models decay, the graph approaches the horizontal axis as x gets very large. When the context is growth, the graph approaches the horizontal axis as x gets increasingly negative. A horizontal line through any long-run value is called a horizontal asymptote.