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Chapter 4 – Exponential and Logarithmic Functions 4.1 - Exponential Functions.
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Section 4.1 Exponential Functions Chapter 4 – Exponential and Logarithmic Functions 4.1 - Exponential Functions
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Transcript of Chapter 4 – Exponential and Logarithmic Functions 4.1 - Exponential Functions.
4.1 - Exponential Functions
Section 4.1 Exponential Functions
Chapter 4 – Exponential and Logarithmic Functions
4.1 - Exponential Functions
Exponential Functions
What is an exponential function?
An exponential function has its variable in the exponent.
It may be used to model rapidly increasing or decreasing situations such as population growth, growth of epidemics, radioactive decay, cooling or heating of objects, etc.
4.1 - Exponential Functions
Review – Laws of Exponents
Simplify the following:
2 3
11
7
13
23
64
42 3
a a
a
a
a
a
a
a b
2 7
19
17
24
27
32
53 4
3 3
4
4
2
2
6
2
x
x y
0
6
0
0
3
3
3
10
1
2
0.5
a
a
4.1 - Exponential Functions
EvaluatingAnswer the question below.
In college, we study large volumes of information that, unfortunately, we do not often retain for very long. The function
describes the percentage of information that a person can be expected to remember x weeks after learning it.
0.5( ) 80 20xf x e
4.1 - Exponential Functions
Evaluating0.5( ) 80 20xf x e
1.) Let x = 0 and give the value of f (0).
2.) Let x = 1 and determine the value of f (1)
accurate to the nearest ten-thousandth.
3.) Let x = 52 and determine the value of f (52) accurate to the nearest thousandth.
f (0)= 100
f (1)= 68.5225
f (52)= 20
4.1 - Exponential Functions
Exponential Functions
The exponential function with base a is defined for all real numbers x by
where a > 0 and a 1.
xf x a
4.1 - Exponential Functions
Exponential Functions
What does an exponential function look like?
What is its domain and range?
Does it increase or decrease or both?
Does it have vertical or horizontal asymptotes?
( ) 2xf x
4.1 - Exponential Functions
Exponential Functions
What does an exponential function look like?
What is its domain and range?
Does it increase or decrease or both?
Does it have vertical or horizontal asymptotes?
( ) 2 xf x
4.1 - Exponential Functions
Graphs of Exponential Functions
4.1 - Exponential Functions
Transformations of FunctionsWe will now see how to graph certain functions by
looking at the basic graphs of exponential functions and applying the shifting and reflecting transformations.
4.1 - Exponential Functions
ExampleUse the graph of to sketch the graph of
each function. 2xf x
31. ( ) 2
2. ( ) 2 3
3. ( ) 2
x
x
x
j x
k x
l x
4.1 - Exponential Functions
Example – pg. 308 # 19 & 20Find the exponential function f (x) = ax whose graph
is given.
4.1 - Exponential Functions
ApplicationExponential functions occur in calculating
compound interest.
4.1 - Exponential Functions
Compounding – Key Words
Compounding
n
Annual 1
Semiannual 2
Quarterly 4
Monthly 12
Daily 365
4.1 - Exponential Functions
Example – pg. 309 # 52
If $2500.00 is invested at an interest rate of 2.5%, compounded daily, find the value of the investment after the given amount of years.
a) 2 years
b) 3 years
c) 6 years
4.1 - Exponential Functions
Example – pg. 309 #56
Find the present value of $100,000 if interest is paid at a rate of 8% per year, compounded monthly, for 5 years.
