Section 3.1 Exponential Functions

Upon receiving a new job, you are offered a base salary of $50,000 plus a guaranteed raise of 5% for each year you work there –Find out how much you would make for your second and third years –Is there an easy way to figure how much you will be making for after your 20 th year on the job?

A population of bacteria decays at a rate of 24% per hour. If there are initially 4.3 million bacteria, how many are left after 1 hour? How about after two hours? –What if we wanted to know how many bacteria were left after 1 day (or 24 hours)?

Growth Factors In our first example our salary went up 5% each year –New Salary = Old Salary + 5% of Old Salary –New Salary = 100% of Old Salary + 5% of Old Salary –New Salary = 105% of Old Salary –New Salary = 1.05 of Old Salary We call 1.05 the annual growth factor –Notice this happens because it is greater than 1

Decay Factor In our second example our bacteria population decayed by 24% each hour.  New amount = old amount – 24% of old amount  New amount = 100% of old amount – 24% of old amount  New amount = 74% of old amount  New amount =.74 of old amount  We call this the decay factor since our amount is decreasing  Notice this happens because it is less than 1

Let’s get a general formula for our salary An exponential function Q = f(t) has the formula f(t) = ab t, b > 0, where a is the initial value of Q (at t = 0) and b, the base is the rate at which it grows or decays. –Note that it is called an exponential function because the input, t, is in the exponent –Note that b = 1 + r where r is the rate that our quantity is growing or decaying at Let’s get a general formula for our decaying bacteria

Example Say you invest $500 into an account paying 5% annually. –Create a general formula to figure out how much you will have after t years –Graph this function Is the graph increasing or decreasing? Is the graph concave up or concave down? What does this information tell us?

Example Carbon-14 is used to estimate the age of organic compounds. Over time, carbon-14 decays into a stable form. The decay rate is 11.4% every 1000 years. –Write a general formula for the quantity left after t years for an object that starts out with 200 micrograms of Carbon-14 –Graph this function Is the graph increasing or decreasing? Is the graph concave up or concave down? What does this information tell us? How does this graph compare to the last example?

A function is exponential if it has a constant percentage change –This is unlike a linear function which has a constant rate of change When the factor is more than 1 it is a growth factor and our function is increasing When the factor is less than 1 (but greater than 0) then it is a decay factor and the function is decreasing In both cases the functions are concave up so they both have increasing rates of change

In your groups work on 3, 13, and 20