Transcript Section 3.1 Exponential Functions

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 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
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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.
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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) = 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?
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
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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