warm-up given the function: find the inverse function and complete the table below. given the...
DESCRIPTION
Warm-up What are the characteristics of an exponential function, based on the following list of examples. Examples of exponential functions. NOT an Example of exponential function. NOT an Example of exponential function.TRANSCRIPT
Warm-upGiven the function:
Find the inverse function and complete the table below.
x-interceptsy-interceptsvertical asymptoteshorizontal asymptotes
f 1f
Sketch the graph of both on the same set of axes
312)(
xxxf
4.3 Exponential Functions
Warm-upWhat are the characteristics of an exponential function, based on the following list of examples.
Examples of exponential functions.
NOT an Example of exponential function.
x2x10
xe x13 x
1
32
x 2x x1
x1
xxx
21
An exponential function is of the form where:
BASE is: a with requirement: a > 0 and a ≠ 1
EXPONENT is: x
I. Exponential Function
f (x) ax
ApplicationExample: The function:
describes the number of O-Rings expected to fail when the temperature is x (degrees F). The space-shuttle Challenger exploded in 1986 when the temperature was 31 o F .
What is the expected number of O-rings to fail at 31 o F ?
at 60 o F ?
1)967.0(49.13)( xxf
Do you recall these rules?
II. Laws of Exponents
yxaa
yxa
xab
xa 0a
Example: Plot points and sketch:
III. Graphing an Exponential Functionxxf 2)(
Example: Plot points and sketch: x
xf
21)(
III. Properties of the graph of Exponential Function
f (x) ax
Domain/Rangex-interceptsy-intercept
Horizontal Asymptotes
Increasing or Decreasing ?
Key Points (0,1), (1, 2) (0,1),(1, ½)
x2 :Examplex
Example
21 :
If base is greater than 1 If base is less than 1
IV. Transformations of the graph of Exponential Function
What do transformations do to the graph of an exponential function?
xy 2 xy 2 p. 283, # 29-36
IV. Transformations of the graph of Exponential Function
What do transformations do to the graph of
cay x
xay
cxay xay
xay xcay
cxay
V. Graphing on the Calculator
Graph
Does your graph demonstrate the correct shape for exponential growth?
If not, what happened?
xy 23
Watch your parentheses!!!!!
Enter y = 3^(2x) on the calculator
VI. A “Natural” Base. The Number e
First approximation for was found in studying continuous compound interest.
is the number given by the value of
n
n
n
as 11
e
e
The calculator gives us an approximation to the number.
Where does the graph of lie?
...718281828.21 e
xey
A Theorem for Exponential Functions
What does the base of an exponential function represent?
Rumors: Scenario 1: # people you tell is 2 per day.
Scenario 2: # people you tell is 3 per day.
What can you say about the ratio of consecutive days ?
Theorem: An exponential function satisfies the property:
xaxf )(
axf
xf
)(
)1(
A Theorem for Exponential Functions
How can we solve for x using the graphing calculator ?
Algebraically ?
800012 1 x
If an exponential equation can be expressed in the form: (same base on each side of equal sign)
Then we are HAPPY, ‘cause it’s pretty easy to solve
)(u(x) xvaa
If a problem is not written in this form, we can TRY to get it that way:
1422 1) xx
where u and v are expressions in x
If au av, then u v
xx 461 2)x
x3
812 3)
2
422 x
52 33 x
VII. Solving Exponential Equations
If a problem can be not written in this form, we will use logarithms to solve for the unknown
Exponential Equations with base eTreat as a number.
Solve for x: If we have a problem of the form:
3
2 1 2)2
eee xx
vuee v then ,u
e
124 2
1) eee xx
4ee x