1. Lesson 6.2 The Natural Base e
Learning Goal: (F.IF.C.7e, F.LE.B.5)
I can graph exponential functions, showing intercepts and end behavior.
I can interpret the parameters in an exponential function in terms of context.
Essential Question: What is the natural base e?
Homework Discussion

2. Group Consensus
Use a graphing calculator to graph the following. Describe the end behavior of the function as x approaches

3. Natural Base e
called the Euler number - named after mathematician Leonhard Euler
an irrational number
Simplifying Natural Base Expressions
Simplify each expression.
a b c.
Simplify the expression.
Your Turn

4. Transformation of Natural Base Exponential Functions
Graphing Natural Base Exponential Functions
Growth or Decay?
Transformation of Natural Base Exponential Functions
What value will affect whether the function is growth or decay?
Example: Determine if the function is growth or decay. Then graph the function.
a. y = 3ex b. f (x) = e−0.5x
Your Turn
c. y = d. y = 4e−x e. f(x) = 2e2x

5. Continuously Compounded Interest
A: amount in account after t years
P: the initial amount deposited
r: the annual interest rate
t: the number of years
Example: Modeling with Mathematics
You and your friend each have 
accounts that earn annual interest 
compounded continuously. The 
balance A (in dollars) of your 
account after t years can be modeled 
by A = 4500e0.04t. The graph shows 
the balance of your friend’s account 
over time. Which account has a 
greater principal? Which has a 
greater balance after 10 years?
You deposit $4250 in an account that earns 5% annual interest compounded continuously. Compare the balance after 10 years with the accounts in the above example.
Your Turn

6. Practice to Strengthen Understanding
Exit Question:
What is the natural base e?
Practice to Strengthen Understanding
Hmwk #3 BI p #3-8, 13, 15-25odd,35, 36