Name: ______________________________
Period: ___________
Exponential Functions—Basics and
Applications
According to http://www.purplemath.com/modules/expofcns.htm :
“Exponential functions look somewhat similar to functions you have seen
before, in that they involve exponents, but there is a big difference, in that the
variable is now the power, rather than the base.
Previously, you have dealt with such functions as f(x) = x2, where the variable
x was the base and the number 2 was the power. In the case of exponentials,
however, you will be dealing with functions such as g(x) = 2x, where the base
is the fixed number and the power is the variable.”
Recall the graph of 𝑦 = 𝑥 2 :
What do you remember/notice
about the graph of quadratics?
_____________________________
_____________________________
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_____________________________
Compare this to the graph of 𝑦 = 2𝑥
What do you remember/notice about
the graph of quadratics (solid line)
compared with the exponential
(dashed line)?
______________________________
______________________________
______________________________
Name: ______________________________
Period: ___________
Evaluate the following exponentials at the given values. Be careful of order of
operations!
𝑦 = 3𝑡 ; 𝑤ℎ𝑒𝑛 𝑡 = 0 𝐴𝑁𝐷 𝑤ℎ𝑒𝑛 𝑡 = 4
𝑦 = 2.5𝑥 ; 𝑤ℎ𝑒𝑛 𝑥 = 3
𝑦 = 100(2.5)𝑥 ; 𝑤ℎ𝑒𝑛 𝑥 = −1 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 = 3
Name: ______________________________
Period: ___________
Exponentials have many applications to real world situations. Can you think
of times when you have heard about exponential growth and/or exponential
decay? ________________________________________________________________________________
_________________________________________________________________________________________
Let’s try a few examples of application problems using exponentials:
Ten grams of Carbon 14 is stored in a container. The amount C (in grams) of
Carbon 14 present after t years can be modeled by C = 10(0.99987) t. How
much is present after 1000 years?
The population of Winnemucca, Nevada, can be modeled by P=6191(1.04) t
T is the number of years since 1990.
What was the population in 1990?
When was the population about 7833?
Hint: use the graphing calculator.
Try this window:
X min =0
X max =10
Y min = 7500
Y max = 8000
Name: ______________________________
Period: ___________
Comparing linear growth to exponential growth:
Suppose you receive $150 for your birthday. You are considering two options for
investing your money.
Option 1: You will put your money in a back account paying interest according to
the equation 𝑦 = 𝐵(1.05)𝑡 where y is the amount saved, B is the initial balance,
and t is the time in months. You will not add to the account, but you will not
make any withdrawals.
Option 2: You will hide the $150 in your mattress. Each month you will add
$7.50. You will not make any withdrawals.
Write the two equations that represent these options here:
When would the balance be the same for both options?