1. Aim: What is the exponential function?
Do Now: Given y = 2x, fill in the table x
1/8 ¼ ½ y
HW: Worksheet

2. y = 2x

3. Exponential Functions
Graph the curves y = 2x, y = 3x, y = 4x on
the same coordinate system
What about y=(1/2)x ? x 2x 3x 4x -3
1/8
1/27
1/64 -2
1/4
1/9
1/16 -1
1/2
1/3 1 2 3 4 9 16 27 64
y = 4x
(1/2)x 8 4 2 1
1/2
1/4
1/8
y = 3x x y
y = 2x
y=(1/2)x 1

4. a = constant b = base x = variable
All base exponential functions are similar because they all go through the point (0,1), regardless of the size of their base number
Exponential Functions are written in the form:
f(x) = abx
a = constant b = base x = variable

5. Properties of the basic f(x) = abx function
The domain consist of all real numbers x
The range consists of all positive all real numbers y
The function is increasing when b >1 and decreasing when 0 < b < 1
It is one to one function
The x –axis is the horizontal asymptote to curve, toward the left when b>0 and toward the right for 0 < b <1

6. if b > 1, the graph represents
Exponential Function: Any equation in the form f(x) = Cbx.
if 0 < b < 1 , the graph represents exponential decay – the y-value is going down as x increases
if b > 1, the graph represents
exponential growth – the y-value is going up as x increases
Examples: f(x) = (1/2)x f(x) = 2x
Exponential Decay
Exponential Growth
We will take a look at how these graphs “shift” according to changes in their equation...

7. f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3
Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x :
f(x) = (1/2)x f(x) = (1/2)x f(x) = (1/2)x – 3
Vertical Shift: The graphs of f(x) = Cbx + k are shifted vertically by k units.

8. f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3
Take a look at how the following graphs compare to the original graph of f(x) = (2)x :
f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3
(0,1)
(0,-1)
(-2,-4)
Notice that f(0) = 1
This graph
is a reflection of
f(x) = (2)x . The graph is
reflected over the x-axis.
Shift the graph of
f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down

9. e = ….
This number is called the natural base.
It is called this because the value seems to occur naturally in many situations. e is an irrational number that is similar to the property of .
e is a very important in calculus since the derivative of e is itself.
The function f(x) = ex is the natural exponential function.

10. f(x) = ex

11. The population of the United States can be modeled by the function p(x) = 80.12e0.131x where x is the number of decades since 1900 and p(x) is the population in millions
graph p(x) over the interval 0  x  15
If the population of United States continues to grow at this rate, predict the population in the years 2010 and 2020.

12. Sketch the graph of y = 3x over -3  x  3

13. Sketch the graph of over -2  x  2