COLLEGEALGEBRA
Lesson: Exponential Functions
Objectives: 1. To graph an exponential function;
2. To solve basic exponential equation.
An exponential function is a function of the form:
: → 0,∞,
,
where a is a positive real number with 1.
For an exponential function
, we have that
, ∀∈.
Proof: If
, then 1
, so that
.
Graph of exponential functions:
a. If a > 1 b. If 0 < a < 1:
Properties of the graphs:
1. If a > 1, the exponential functions are increasing; if a < 1 they decrease;
2. The exponential function
crosses the y-axis in the point (0,1) for any positive
value of a;
3. The exponential function
does not touch the x-axis; the x-axis is a horizontal
asymptote for the function;
4. The exponential function is one-to-one:
→.
-4-3-2-1 123456789
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
(0,1)
-4-3-2-1 123456789
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
(0,1)
