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WARM-UP :There is only one pair of numbers that satisfied the equation xy = yx,
where x does not equal y. Can you figure out which two numbers?
UNIT e: Ln 1
EXPONENTIAL FUNCTIONS.
What are some real life examples of exponential
functions?????
Examples of exponential growth and decay:
-rich populations. (vine leaves vs. time)
(insiders vs. time)
all of fashions. (hats vs. time)
or medication in the human body.
(amount vs. time)
EXAMPLE:
The Million $ Mission
Bill Gates has a job offer for you. He's going to need you for 30
days. You'll have your choice of two payment options:
1. One cent on the first day, two cents on the second
day, and double your salary every day thereafter for
the thirty days; or
2. Exactly $1,000,000.
WHICH OPTION WILL YOU CHOOSE???
HOW MUCH MONEY WOULD YOU EARN AFTER 30
DAYS with option #1???
CHALLENGE: Find a function rule (aka
equation) that allows you to calculate the answer
Day 7 you made $1.27.
Day 14 = $163.83
Day 21 = $ 20 971.51
Day 30 = $ 10 737 418.23
Function Rule: f(x)=0.01((2)^x -1)
Explain each value in this equation
DEFINITION:
Exponential functions model rates of increase
or decrease that are proportional to the
value of the function.
The following two formulas are used most often in real life application.
Exponential GROWTH y  I (1  r )t
Exponential DECAY y  I (1  r )t
Guided practice:
CLASSWORK: Exponential Functions Application
Use the appropriate formula to solve each problem.
Ex. 1: In an experiment, bacteria are put into a petri dish and are allowed to grow. The
number of bacteria in the dish after n hours is found to be 2000  3n.
a. How many bacteria were put into the dish at the beginning of the experiment?
b. How fast is the population of bacteria growing?
c. How many bacteria are in the dish after 5 hours?
Ex 2: Growth of bacteria in food products causes a need to “time-date” some products (like
milk) so that shoppers will buy the product and consume it before the number of bacteria
grows too large and the product goes bad. Suppose that the initial count of bacteria is 500;
t(0) = 500. Each day, the amount of bacteria in food doubles. The product should not be
consumed after the bacteria count reaches 4,000,000.
a) Write the rule representing bacteria growth in food products.
b) After how many days should you dispose of the product ?
Ex 3: A total of $9,000 is invested at an annual interest rate of 2.5%, compounded annually.
Find the balance in the account after 5 years.
Ex. 4: The population of Rochester is 17,500 and is projected to grow at a rate of 4.5% per
decade.
a. Write an expression for the projected population of Rochester after n decades.
b. Predict the population, to the nearest hundred, of Rochester after 40 years.
Ex. 5: Insulin is an important hormone produced by the body. In 5% to 10% of all diagnosed
cases of diabetes, the disease is due to the body’s inability to produce insulin. Those people
have to take medicine containing insulin. Insulin breaks down very quickly once injected into
the bloodstream. When 10 units of insulin are delivered into the system, the amount
remaining after t minutes is decreasing by 15% per minute.
a. Write the function rule that shows the remaining traces of insulin in the
bloodstream after t minutes.
b. Calculate the amount of insulin remaining after 2 hours
c. When is i(t) = 0?? EXPLAIN!!
Ex. 6:. ‘CARGO’ company decides to buy a new delivery van for $25,000. Based on their
usage of the vans they own, the van’s resale value decreases at a rate of 20% per year. What
is the resale value of the van 10 years after its purchase?
a. Write an expression for the value of the van after n years.
Exponential Functions:
GENERAL RULE:
The exponential function f with base b is denoted by f ( x)  ab x where, b  0 , a  0 , b  1 and x is any
real number.
Evaluating Exponential Functions
Examples: Use a calculator to evaluate each function at the indicated value of x.
1. f ( x)  2 x , x  3.1 f(-3.1)0.117
2.
f ( x)  2 x , x  
f() 0.113
3.
f ( x)  0.62 x , x 
3
4
f(3/4) =0.65
Graphs of Exponential Functions
Example: In the same viewing window, graph the functions
f ( x)  2 x , g ( x)  4 x , h( x )  2  x , and j ( x)  4 x . Describe their similarities and differences.
Exponential growth , decay
Y-axis  axes of symmetry, reflection over x =0
Y-intercept: f(x) = 4x f(0) = 0
f(x) = 4x³
f(0) = 0
f(x) = 4^x f(0) = 1
F(x) = 4^x +12 f(0) = 13
Example: In the same viewing window, graph the functions
f ( x )  3x
f ( x )  3x  2
Identify the y-intercept for each function, identify the smallest y-value for each function.
DISCUSS ASYMPTOTES.
Solving Exponential Equations Using Equivalent Bases
Examples: Solve the following equations for x.
1. 42x = 48
2. 3x = 27
3. 6 x 
1
36
Advanced Algebra w/Trigonometry
Homework – Unit e, ln 1: Exponential Functions
Name ______________________________
Date: __________ Period: ___________
Use a calculator to evaluate each function at the indicated value of x.
1. f (x)  3.4 x
x = 6.8
__________________ 2. g(x)  5 x
3. f (x)  5 x
x = -
__________________ 4. h(x)  17 2 x x  3 __________________
5. g(x)  8.63x
x 2
__________________
x = -1.5 __________________
Solve the following exponential equations by using equivalent bases
6. 24x = 217
__________________
7. 5-6x = 554
__________________
Solve each application problem
8) According to the National Census Bureau, since 1980 there has been a consistent decrease
in birth rate 3% each 5 years. If there were 7.2 million children born in Chicago in 1980, what
will be the approximate number of children born in Chicago in 2010?
9) Pat bought a car for $9500.The salesperson projected that the value of the car would
decline by 20% per year for the next 5 years.
a. Write an expression for the projected value of Pat’s car after n years.
b. Calculate the value, to the nearest hundred dollars, of Pat’s car after 5 years.
10) A new car is purchased for 19600 dollars. The value of the car depreciates at 14.75% per year. What
will the value of the car be, to the nearest cent, after 14 years?
11) An element with mass 280 grams decays by 5% per minute. How much of the element is remaining
after 15 minutes, to the nearest 10th of a gram?
Advanced Algebra w/Trigonometry
Name ______________________________
Homework – Unit e, ln 1 #2 Exponential Functions Date: __________ Period: ___________
Sketch the graph of the exponential function by hand. Identify any asymptotes and intercepts
and determine whether the function is increasing or decreasing. Do not use the calculator!!
2. g ( x)  2 x  5
1. f(x) = 5x
3. h(x) = 4-x - 4
Solve the following exponential equations by using equivalent bases
4. 43 = 256x __________________
 1
5. (-3)2x = 81
__________________
1
49
__________________
x
6.    16 __________________
 2
1
8)  
 16 
7. 7 4 x 
2 x 1
 16 5 x 15 _______________
9) 64 x  9  322 x 14
Solve each application problem
8) 9500 dollars is placed in an account with an annual interest rate of 6%. How much will be in
the account after 14 years, to the nearest cent?
9) An element with mass 550 grams decays by 24.4% per minute. How much of the element is
remaining after 6 minutes, to the nearest 10th of a gram?
10) A town has a population of 7000 and grows at 2% every year. What will be the population
after 13 years?
11) A town has a population of 19000 and decreases in population at 0.5% every year. What will
be the population after 7 years?