Exponential Function

Document Sample

```					              THE EXPONENTIAL FUNCTION
A function where the x variable is contained within an exponent is called an
exponential function. This is a new family of functions with unique
characteristics and peculiarities. This kind of function is required to describe
situations involving compound interest, population growth, depreciation and

The simple form of this function is shown below:

f :             where c must be
positive but not equal
x  cx         to 1.

Also can be written as:

f :
x  expc x
Observe the list of functions that are exponential and the list that are NOT
exponential functions.

EXAMPLES OF
EXAMPLES OF FUNCTIONS
FUNCTIONS THAT ARE
THAT ARE EXPONENTIAL
NOT EXPONENTIAL
f (x )  2x
x
f (x )  x 2
1
f (x )                                f (x )  1x
2
f (x )  exp18 x                       f (x )  exp2 x
1x
f (x ) 
2
The following function is the simplest of all exponential functions. We can
construct a table of values to shape the graph.

f (x )  2x
x          f(x) = 2x        (x, f(x) )
0        f(x) = 20 = 1        (0, 1)
1       f(x) = 21 = 2        (1, 2)
2        f(x) = 22 = 4       (2, 4)
3        f(x) = 23 = 8       (3, 8)
1
-1      f (x )  21  2    (-1, 0.5)
1
-2      f (x )  22  4   (-2, 0.25)

Notice that for every x-value the function
goes to the right the y-value doubles.
What is the maximum of this graph?

Notice that for every x-value the function
goes to the right the y-value is halved.
What is the minimum of this graph?
The following function is the simplest of all exponential functions. We can
construct a table of values to shape the graph.

f (x )  3x
x          f(x) = 3x        (x, f(x) )
0        f(x) = 30 = 1        (0, 1)
1       f(x) = 31 = 3        (1, 3)
2        f(x) = 32 = 9       (2, 9)
3       f(x) = 33 = 27       (3, 27)
1
-1      f (x )  31  3    (-1, 0.3)
1
-2      f (x )  32  9    (-2, 0.1)

Notice that the function with the base of
3 increases more quickly.
What do both graphs have in common?
The following function is the simplest of all exponential functions. We can
construct a table of values to shape the graph.
x
1
f (x )   
2
x                                 (x, f(x) )
x
1
f (x )   
2
0

0                   1
f (x )     1
2
(0, 1 )
1

1                1
f (x )    
2 2
1
(1, 0.5 )
2

2                1
f (x )    
2
1
4
(2, 0.25 )
1

-1               1
f (x )     2
2
(-1, 2 )
2

-2               1
f (x )     4
2
(-2, 4 )
3

-3                 1
f (x )     8
2
(-3, 8 )

What happens as a result of making a
reciprocal base?
What is the relationship between the base
and whether the function is increasing or
decreasing?
f(x) = cx
For the preceding function, there are several characteristics that are
constant:
- passes through the point (0,1)
- asymptote is y= 0
- graph is located in quadrants I and II only
- Dom = R
- Ran = ] 0,∞
However, depending on the value of parameter ‘c’, the function may be
increasing or decreasing.
If c > 1, then     If 0 < c < 1, then
the function is    the function is
increasing.        decreasing.
(0,1) and (2,4)    (0,1) and (-2,4)
0<2                0 > -2
1<4                1<4
f(x) = a•cb(x – h) + k
There are 4 other parameters that we can use with the exponential function.
These parameters will be familiar from MTH-5106 (a, b, h and k). Everything
that we learned about those functions from 5106 applies to this level.
a – vertical scale change
b – horizontal scale change
h – horizontal translation
k – vertical translation
If given the ordered pairs from the simple exponential function f(x) = cx, then
it is possible to determine the ordered pairs for the function f(x) = a•cb(x – h)
+ k, using the following formulas:    x              
  h , ay  k 
b             
f(x) = a•cb(x – h) + k
f(x) = cx
f(x) = 1.5•22(x + 3) + 4
f(x) = 2x          a = 1.5; b = 2; h = -3; k = 4
 x  h , ay  k 
                
b               
0                       
x   f(x)         ( 3), 1.5(1)  4   ( 3, 5.5)
2                       
0    1
1                        
1    2              ( 3), 1.5(2)  4   ( 2.5, 7)
2                        
2    4          2                       
  ( 3), 1.5( 4)  4   ( 2, 10)
3    8          2                       
-1   0.5         3                       
    ( 3), 1.5(8)  4   ( 1.5, 16)
-2   0.25        2                       
 1                         
      ( 3), 1.5(0.5)  4   ( 3.5, 4.75)
 2                          
2                            
      ( 3), 1.5(0.25)  4   ( 4, 4.375)
 2                            
f(x) = a•cb(x – h) + k
f(x) = 1.5•22(x + 3) + 4
a = 1.5; b = 2; h = -3; k = 4
 x  h , ay  k 
                 
b                
0                     
  ( 3), 1.5(1)  4   ( 3, 5.5)
2                     
1                      
    ( 3), 1.5(2)  4   ( 2.5, 7)
2                      
2                      
  ( 3), 1.5( 4)  4   ( 2, 10)
2                      
3                      
    ( 3), 1.5(8)  4   ( 1.5, 16)
2                      
 1                        
     ( 3), 1.5(0.5)  4   ( 3.5, 4.75)
 2                         
2                           
     ( 3), 1.5(0.25)  4   ( 4, 4.375)
 2                           
f(x) = c      x-h

Parameter h causes the function to translate horizontally.
In the function below each point is translated 2 to the right because h = 2.

f(x) = 2x                                 f(x) = 2       x-2
f(x) = c      x-h

Parameter h causes the function to translate horizontally.
In the function below each point is translated 3 to the left because h = -3.

f(x) = 2x                                 f(x) = 2       x+ 3

Notice that the asymptote is unaffected by parameter h; it is y = 0 for both
functions.
f(x) = c x + k
Parameter k causes the function to translate vertically.
In the function below each point is translated 2 to the right because k = -2.

f(x) = 2x                                    f(x) = 2 x - 2
f(x) = c x + k
Parameter k causes the function to translate vertically.
In the function below each point is translated 3 up because k = 3.

f(x) = 2x                                   f(x) = 2 x + 3
f(x) = c x + k
Parameter k causes the function to translate vertically.
In the function below each point is translated 2 down because k = -2.

f(x) = 2x                                   f(x) = 2 x - 2

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 31 posted: 9/16/2012 language: Unknown pages: 14
How are you planning on using Docstoc?