1. Exponential Functions and Their Graphs

2. Exponential Function Families
We’ve already learned about
This is the parent function
We’ll expand this to
Note a=1 and h=0 in the parent function

3. Horizontal shift to the right
Component
Property
+ k
Vertical shift up
- k
Vertical shift down
- h
Horizontal shift to the right
+ h
Horizontal shift to the left
a > 1
Stretch
a < 1
Compression -a
Reflection

4. Graph of Exponential Function (a > 1)
The graph of f(x) = ax, a > 1 y 4
Range: (0, )
(0, 1) x 4
Horizontal Asymptote y = 0
Domain: (–, )
Graph of Exponential Function (a > 1)

5. Graph of Exponential Function (0 < a < 1)
The graph of f(x) = ax, 0 < a < 1 y 4
Range: (0, )
Horizontal Asymptote y = 0
(0, 1) x 4
Domain: (–, )
Graph of Exponential Function (0 < a < 1)

6. Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x))y x
f(x)
(x, f(x)) -2
(-2, ¼) -1
(-1, ½) 1
(0, 1) 2
(1, 2) 4
(2, 4) 4 2 x –2 2
Example: Graph f(x) = 2x

7. Example: Translation of Graph
Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y
f(x) = 2x
The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2
Domain: (–, ) x
y = –1
Range: (–1, )
Example: Translation of Graph

8. Example: Reflection of Graph
Example: Sketch the graph of g(x) = 2-x. State the domain and range. y
f(x) = 2x
The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4
Domain: (–, ) x –2 2
Range: (0, )
Example: Reflection of Graph

9. Graph of Natural Exponential Function f(x) = ex
The graph of f(x) = ex y x
f(x) -2
0.14 -1
0.38 1
2.72 2
7.39 6 4 2 x –2 2
Graph of Natural Exponential Function f(x) = ex

10. The irrational number e, where e  2.718281828…
is used in applications involving growth and decay.
The number e

11. Pert

12. Continuously Compounded Interest
Time
Interest Rate (Annual)
Final Amount
Principal (beginning amount)
Euler’s Number ( …)

13. Pert Example
Suppose you won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar.