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State the domain and range of each function
21)y x
: ; : 0Domain all reals Range y
2)y x
: 0; : 0Domain x Range y
3.1 Graphs of Exponential Functions
Exponential Function
Notice that now the variable is the exponent.
Where base (b), b > 0, b 1, and x is any real number.
( ) xf x b
Graphs of Exponential Equations
To see the basic shape of an exponential function such as
make a table of values and plot points.
( ) 2xf x
x y-2
-1
0123
1
41
21248
Graph f(x) = 2x
Notice the end behavior of the graph. As x → +∞, f(x) → +∞, which means that the graph moves up to the right.
As x → -∞, f(x) → 0, which means that the graph has the line y=0 as an asymptote.
An asymptote is a line that a graph approaches (gets close to) as you move away from the origin.
1Graph (2 ) and 3(2 ).
3
Compare the graphs with the graph of 2
x x
x
y y
y
1Graph (2 ) and 5(2 ).
5
Compare the graphs with the graph of 2
x x
x
y y
y
Describe the effect of on the graph of (2 )
when is positive and when a is negative.
xa y a
a
1(2 )
3xy
3(2 )xy 2xy
1(2 )
5xy
5(2 )xy 2xy
Exponential Growth Function:
The graph passes through the point (0,a)
[the y-intercept].
Have these characteristics…
xy abWhere a > 0 and b > 1
The x-axis is an asymptote of the graph.
The domain is all real numbers.
The range is y > 0 if a > 0
Identify the y-intercept and asymptoteState the domain and range
11) (2 )
3xy
1int : 0,
3y
: 0Asymptote y
:Domain all reals
: 0Range y
2) 3(2 )xy
int : 0,3y
: 0Asymptote y :Domain all reals
: 0Range y
Identify the y-intercept and asymptoteState the domain and range
3) 2xy
int : 0,1y
: 0 or -Asymptote y x axis:Domain all reals
: 0Range y
What do you think would happen to the shift of the graph?
x hy ab k h determines the horizontal shift
k determines the vertical shift
4) 4xy 15) 3xy Domain: All Real Numbers
Range: All Positive Real Numbers
Domain: All Real Numbers
Range: All Positive Real Numbers
Asymptote: y = 0 Asymptote: y = 0
26) 2xy 7) 5 3xy Domain: All Real Numbers
Range: All Positive Real Numbers
Domain: All Real Numbers
Range: y > -3
Asymptote: y = -3Asymptote: y = 0
18) 5 2xy 39) 2 1xy Domain: All Real Numbers
Range: y > 2
Domain: All Real Numbers
Range: y > 1
Asymptote: y = 2 Asymptote: y = 1
Exponential Decay Function
Where a > 0, base b is 0 < b < 1 (b is a fraction), and x is any real number.
( ) xf x ab
Decide whether f(x) is an exponential growth or exponential decay function.
2( ) 5
3
x
f x
3( ) 8
2
x
f x
Because 0 < b < 1 f is an exponential decay function
Because b > 1 f is an exponential growth function
1)
2)
Graphs of Exponential Equations
To see the basic shape of an exponential function such as
make a table of values and plot points.
1( )
2
x
f x
x y
-2-1
01
2
8421
½¼
Graph f(x) = (½)x
-3
State the domain, range, make a table of values and graph the following
2)3/1()( xxf
1)4/3(3)( xxf
where e is an irrational constant whose value is
2.718281828459045...e
The exponential function with base e is called “The Natural Exponential Function”
( ) xy f x e
The Number e
The Natural Exponential Function
Homework:Page 185 #7-13 odd, 15-18, 29-37 odd