Section 6.1Percent Growth

• 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 20th 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 because our salary is increasing by 5% each year– 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 one hour decay factor since our amount is decreasing by 24% each hour 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) = abt, 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?

• Sometimes it is easier to find a change factor that is for more than 1 unit of input– Let’s look at the following example

• According to City-Data.com, the population of the Phoenix area in 1990 was 2,238,498. By 2000 it was 3,251,876.– How would we calculate the percentage change

between 1990 and 200?–What is the percentage change between 1990 and 2000?• 45.3%

– This is know as the 10 year growth factor because it gives the percent change over a 10 year period

– How can we find the annual growth factor?• HINT: How many times to you have to multiply the base by

itself to get from 1990 to 2000?• 1/101.435 1.0368 3.68%

• According to City-Data.com, the population of the Phoenix area in 1990 was 2,238,498. By 2000 it was 3,251,876.– Now that we have an annual growth factor (1.0368) for

our population, how can we create a model that gives the population as a function of the number of years since 1990?• HINT: The annual growth factor will always be the base of

our exponential function

–What if we wanted to write our model in terms of its 10 year growth factor (1.453)

2,238,498(1.0368)tP

102,238,498(1.453)t

P

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 change factor is more than 1 it is a growth factor and our function is increasing

• When the change 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

• Let’s try a few from the chapter

6.1 - 5, 6, 8, 15, 16, 17, 19