1. Exponential Functions, Growth and Decay Understand how to write and evaluate exponential expressions to model growth and decay situations.
Do Now: - What is the Domain and Range. Is there a x and y intercept? If so where?
Success Criteria:
I can graph an exponential function
Identify exponential growth and decay
Today’s Agenda
Do Now
Lesson
Review
Quiz Tomorrow

2. Exponential Functions and Their Graphs Section 3-1

3. Definition of Exponential Function
The exponential function f with base a is defined by
f(x) = ax
where a > 0, a  1, and x is any real number.
For instance,
f(x) = 3x and g(x) = 0.5x
are exponential functions.
Definition of Exponential Function

4. Example: Exponential Function
The value of f(x) = 3x when x = 2 is
f(2) = 32 = 9
The value of f(x) = 3x when x = –2 is
f(–2) = 3–2 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
0.0625
Example: Exponential Function

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

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

7. Exponential Function 3 Key Parts 1. Pivot Point (Common Point)
2. Horizontal Asymptote
3. Growth or Decay

8. Manual Graphing Lets graph the following together: f(x) = 2x
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9. 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
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Example: Graph f(x) = 2x

10. Definition of the Exponential Function
The exponential function f with base b is defined by
f (x) = bx or y = bx
Where b is a positive constant other than and x is any real number.
Here are some examples of exponential functions.
f (x) = 2x g(x) = 10x h(x) = 3x
Base is 2.
Base is 10.
Base is 3.
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11. Calculator Comparison
Graph the following on your calculator at the same time and note the trend
y1 = 2x
y2= 5x
y3 = 10x

12. When base is a fraction
Graph the following on your calculator at the same time and note the trend
y1 = (1/2)x
y2= (3/4)x
y3 = (7/8)x

13. Lesson YOU TRY
In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year.
1. Write a function for world population. Does the function represent growth or decay?
P(t) = 6.08(1.0121)t
2. Use a calculator to predict the population in 2020.
≈ 7.73 billion
The value of a $3000 computer decreases about 30% each year.
3. Write a function for the computer’s value. Does the function represent growth or decay?
V(t)≈ 3000(0.7)t
4. Use a calculator to predict the value in 4 years.
≈ $720.30
Gabby purchased a car for $ A year later the car was valued at $4500.
5. Write a function that represents the value of the car.
C(t)≈ 5000(1-0.1)t
6. At this rate of depreciation, how many years until her car is worth $1000?
≈ ?? years

14. HW#13
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15. Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward c units if c > 0.
Shifts the graph of f (x) = bx downward c units if c < 0.
g(x) = bx+ c
Vertical translation
Reflects the graph of f (x) = bx about the x-axis.
Reflects the graph of f (x) = bx about the y-axis.
g(x) = -bx
g(x) = b-x
Reflecting
Multiplying y-coordintates of f (x) = bx by c,
Stretches the graph of f (x) = bx if c > 1.
Shrinks the graph of f (x) = bx if 0 < c < 1.
g(x) = cbx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left c units if c > 0.
Shifts the graph of f (x) = bx to the right c units if c < 0.
g(x) = bx+c
Horizontal translation
Description
Equation
Transformation

16. 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

17. 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

18. Discuss these transformations
y = 2(x+1)
Left 1 unit
y = 2x + 2
Up 2 units
y = 2-x – 2
Ry, then down 2 units

19. Special Symbols
Math uses special symbols at times to represent special numbers used in calculations.
The symbol  (pi) represents 3.14…..
The symbol “i” represents

20. (The Euler #) e is an irrational #, where e  2.718281828…
is used in applications involving growth and decay.
The number e

21. 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
Natural Exponential Function 6 4 2 x –2 2
Graph of Natural Exponential Function f(x) = ex

22. Homework
WS 6-1