1. Exponential Functions and Their Graphs Section 3-1

2. 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

3. 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

4. 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)

5. 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)

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

7. Manual Graphing Lets graph the following together: f(x) = 2x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

8. Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) -2 ¼ (-2, ¼) 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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph f(x) = 2x

9. 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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

10. Calculator Comparison
Graph the following on your calculator at the same time and note the trend
y1 = 2x
y2= 5x
y3 = 10x

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

19. Homework
WS 6-1