Upload
gordon-perry
View
212
Download
0
Embed Size (px)
Citation preview
Aim: What is the exponential function?
Do Now: Given y = 2x, fill in the table
x -3 -2 -1 0 1 2 3
1/8 ¼ ½ 1 2 4 8
y
HW: Worksheet
Math Composer 1. 1. 5http: / / www. mathcomposer. com
-4 -3 -2 -1 1 2 3 4
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
y = 2x
Graph the curves y = 2x, y = 3x, y = 4x onthe same coordinate system
x
y
y = 2x
y = 3xy = 4x
1
x 2x 3x 4x
-3 1/8 1/27 1/64-2 1/4 1/9 1/1
6-1 1/2 1/3 1/40 1 1 11 2 3 42 3 9 163 4 27 64
Exponential FunctionsExponential Functions
(1/2)x
8
4
2
1
1/2
1/4
1/8
y=(1/2)x
What about y=(1/2)x ?
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
Properties of the Properties of the basicbasic ff((xx) = ab) = abxx function 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
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...
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 + 1 f(x) = (1/2)x – 3
Vertical Shift: The graphs of f(x) = Cbx + k are shifted vertically by k units.
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
Notice that f(0) = 1
(0,1)
This graphis a reflection of f(x) = (2)x . The graph isreflected over the x-axis.
(0,-1)
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
(-2,-4)
e = 2.71828….
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.
Math Composer 1. 1. 5http: / / www. mathcomposer. com
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
f(x) = ex
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
a. graph p(x) over the interval 0 x 15
b. If the population of United States continues to grow at this rate, predict the population in the years 2010 and 2020.
Sketch the graph of y = 3x over -3 x 3Math Composer 1. 1. 5http: / / www. mathcomposer. com
-3 -2 -1 1 2 3
2
4
6
8
10
12
14
16
18
20
22
24
26
x
y
Sketch the graph of over -2 x 2 x
y
4
1
Math Composer 1. 1. 5http: / / www. mathcomposer. com
-3 -2 -1 1 2 3
2
4
6
8
10
12
14
16
18
x
y