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Sections 4.1 & 4.2 Exponential Growth and Exponential Decay
What You Will Learn:
1. How to graph exponential growth functions.
2. How to graph exponential decay functions.
Exponential Growth
This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly
allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1
and increases by $1 each week.
• Although the second option, growing at a constant rate of $1/week, pays more in the short run, the
first option eventually grows much larger:
Exponential Growth!
The equation for option 1 is: y = 2n where n is the number of weeks.
The equation for option 2 is y = 1 + n where n is the number of weeks
Exponential Growth Equation:
An exponential function involves the expression bx where the base “b” is a positive number other
than 1.
The variable is going to be in the “position” of the exponent.
86
Let’s Graph an Example
An asymptote is a line that a graph approaches as you move away from the origin.
xy ab is an exponential growth function when a is greater than 0 and b is greater than 1.
It is important to know the ordered pairs to get your starting points for your graph. They will always be
11, , (0,1), and (1, )b
b
.
x y
-1
0
1
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
𝑦 = 2𝑥
87
In the exponential model, 0, 1a b . If we transform1
( ) ( ) ( )
x
x x
bf x ab f x ab f x a
. So the
exponential growth function reflects over the y-axis (multiplying all x-values by -1).
QUESTION? Is it still exponential growth?
Let’s Graph an Another Example
12
2
x
xy
x y
-1
0
1
88
Graphing by Transformation:
The generic form of an exponential function is: bx hy ax k Where h is movement along the x-axis
and k is movement along the y-axis. The value of a will give a reflection and/or a vertical stretch or
compression. The value of b will give a horizontal reflection when b is negative.(New Transformation!)
Graph 22 3 1xy
a. State the domain and range. D: _________ R:________ b. Give the equation of the horizontal asymptote. HA:_____________
Graph 13 2 4xy
a. State the domain and range. D: _________ R:________ b. Give the equation of the horizontal asymptote. HA:_____________
x y
-1
0
1
x y
x y
x y
-1
0
1
x y
x y
An Example of Graphing by Translation
x-10 -5 5 10
y
-10
-5
5
10
x-10 -5 5 10
y
-10
-5
5
10
89
Application:
1. State whether the following model exponential growth or exponential decay.
2. Describe the transformation of the parent function.
3. State the domain and range.
4. Identify the y-intercept and the asymptote.
3(2)xy
1. Growth Decay
2. _____________________________
3. D:_________ R:_________
4. y-int:________ Asymptote:_______
41
26( )xy
1. Growth Decay
2. _____________________________
3. D:_________ R:_________
4. y-int:________ Asymptote:_______
1
42( )xy
1. Growth Decay
2. _____________________________
3. D:_________ R:_________
4. y-int:________ Asymptote:_______
24 xy
1. Growth Decay
2. _____________________________
3. D:_________ R:_________
4. y-int:________ Asymptote:______
5 92xy
1. Growth Decay
2. _____________________________
3. D:_________ R:_________
4. y-int:________ Asymptote:_______
An Exponential Decay Word Problem:
When a real=life quantity increases or decreases by a fixed percent each year (or other time period) the
amount of y of the quantity after t years can be modeled by the formula: (1 )ty a r
is called the growth factor is called the decay factor
(1 )r(1 )r
90
Example:
You buy a new car for $24,000. The value y of the car decreases by 16% each year.
Initial amount is: ___________ Annual % increase or decrease__________
Growth or decay? Growth/decay factor: _________
a. Write an exponential model for the value of the car.______________ b. Use the model to estimate the value after 2 years._______________ c. When will the car have a value of $12,000? ________________
In January 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of
hosts increased by about 100% per year.
Initial amount is: ___________ Annual % increase or decrease__________
Growth or decay? Growth/decay factor: _________
a. Write an exponential model for the number of hosts.______________ b. Use the model to estimate the number of hosts in 1996._______________ c. When will the number of hosts reach 30 million?________________
Another example:
A motor scooter purchased for $1000 depreciates at an annual rate of 15%.
a. Write an exponential function and graph the function. ____________ b. Use the function to predict when the value will fall below $100. ____________
91
Example: Your parents want to deposit $500 in the bank as an investment for you the day you are born!
They will give you the money when you turn 18. The banks they can use all pay a yearly interest rate of
6.5%, but compound the interest differently. Which bank should they choose to invest their money in?
Bank A: Interest Compounded Annually: Bank D: Interest Compounded Monthly:
Bank B: Interest Compounded Semi-Annually: Bank E: Interest Compounded Daily:
Bank C: Interest Compounded Quarterly:
92
Section 4.3 The Number e
The number e is known as Euler’s number. Leonard Euler (1700’s) discovered its importance. An irrational
number, symbolized by the letter e, appears as the base in many applied exponential functions. It models a
variety of situations in which a quantity grows or decays continuously: money, drugs in the body,
probabilities, population studies, atmospheric pressure, optics, and even spreading rumors!
The number e is defined as the value that approaches as n gets larger and larger.
n 1 5 10 100 1000 100,000 1,000,000 1,000,000,000
________ ________ ________ ________ ________ ________ ________ ________
1 , 1 2.718
n
As n en
Example 1: Simplifying the Natural Base Expressions
a) 3 4e e b) 3
2
10
5
e
e c) 4 2(3 )xe d) 5 2(2 )xe
Example 2: Use a calculator to Evaluate the Expressions
a) 6e b) 0.12e
Graphing e:
Since 2 < e < 3, the graph of y = ex is between the graphs
of y = 2x and y = 3x
11
n
n
93
The irrational number e, is called the natural base. The function xf x e is called the natural
exponential function.
Graph:
( ) xf x e
Graph: ( ) xf x e
x y
-1
0
1
x y
-1
0
1
a. State the domain and range. D: _________ R:________
b. Give the equation of the horizontal asymptote. HA:_____________
x-10 -5 5 10
y
-10
-5
5
10
a. State the domain and range. D: _________ R:________
b. Give the equation of the horizontal asymptote. HA:_____________
x-10 -5 5 10
y
-10
-5
5
10
94
Graph 1 3xy e
a. State the domain and range. D: _________ R:________ b. Give the equation of the horizontal asymptote. HA:_____________
Graph 2xy e
a. State the domain and range. D: _________ R:________ b. Give the equation of the horizontal asymptote. HA:_____________
x y
-1
0
1
x y
x y
x y
-1
0
1
x y
x y
x-10 -5 5 10
y
-10
-5
5
10
x-10 -5 5 10
y
-10
-5
5
10
95
Continuous Compound Interest
Example:
John invests $1750 into a bank account that pays at a rate of 1.65% compounded continuously. Assuming
he does not add any more money to the account, how much money will be in his account in 15 years? How
much after 30 years?
15 years 30 years
Exponential Growth
The function 0( ) , where 0ktA t A e k can model many kinds of population growths.
Where: 0A = population at time 0, t = amount of time,
( )A t = population after time, k = exponential growth rate.
The growth rate unit must be the same as the time unit.
Exponential Decay
Decay, or decline, is represented by the function 0( ) , where 0ktA t A e k
Where: 0A = initial amount of the substance, t = amount of time,
( )A t = amount of the substance left after time, k = exponential decay rate.
The growth rate unit must be the same as the time unit.
The half-life is the amount of time it takes for half of an amount of substance to decay.
96
Exponential Model for the Half-Life of a substance:
Use the graph to determine the half-life and the exponential model for Iodine-131.
The inverse function of xy e is the natural log and is a log of base 𝒆. It can be written as 𝒍𝒐𝒈𝒆𝒙 or simply
as 𝒍𝒏𝒙.
[ANYTIME 𝑙𝑛 𝑎𝑛𝑑 𝑒 ARE NEXT TO EACH OTHER, THEY WILL CANCEL THEMSELVES OUT.]
𝑙𝑛𝑒 = 1
𝑙𝑛𝑒𝑥 = 𝑥𝑙𝑛𝑒
Use the properties above to solve the following equations.
a. 2 2.5 20xe b. 1 30xe c. 2
5 7.2 9.1x
e
d. 26 12x e. 2 54 6x f. 3 12 3x x
97
Continuous Interest:
Example 1: An initial investment of $100 is now valued at $149.18. The interest rate is 8% compounded
continuously. How long has the money been invested?
Example 2: An initial investment of $200 is now valued at $315.24 after compounding continuously for 7
years. Find the interest rate.
Example 3: What interest rate do you need for a $5000 investment to double in 10 years?
Example 4: Carbon-14 is used to determine the age of artifacts in carbon dating. It has a half-life of 5730
years. Write the exponential decay function for a 24 mg sample and then find the amount of carbon-14
remaining after 30 millennia. (1 millennium = 1000 years.)
98
Sections 4.4 -4.6 Logarithmic Functions
The inverse function to an exponential function is the logarithm. If you are trying to solve an exponential
and you get stuck, change it to a logarithm and vice versa.
𝑥 = 𝑙𝑜𝑔𝑏𝑦 is read as “The log, base b, of y. The y-part of a log is called the argument of the log. The
expression 𝑙𝑜𝑔𝑏 represents the exponent to which the base “b” must be raised in order to obtain y.
The domain of a logarithm can be found by setting the argument greater than zero and solving.
Find the domain:
a. 3( ) log (2 5)f x x b. 2( ) ln( 4)f x x
Changing a log to an exponential: just remember that THE BASE IS THE BASE IS THE BASE... Remove the log
operator and “switch” the x and y’s position. Changing an exponential to a log still remember that THE
BASE IS THE BASE IS THE BASE. Introduce the log operator and make the base of the exponential the base
of the log, then “switch” the x and y’s position.
Exponential Form Logarithmic Form 23 9
5
1log 2
25
12
log 2 1
110 10
8log 1 0
A Logarithm to the base of a positive number y is defined as follows:
Exponential form Logarithmic Form
xy b is the same as logb y x
Special Properties of logs:
a. log 1 0b b. log 1b b c. log p
b b p d. logb p
b p
99
Evaluate
a. 4log 64 b. 2
1log
8 c. 1
4
log 256 d. 3log 11
3 e. 7log 7 f. 300log 1
A common log is a log of base 10. It can be written as 𝒍𝒐𝒈𝟏𝟎𝒙 or simply as 𝒍𝒐𝒈𝒙 because the base of a log
that has no base showing is “understood” to be base 10.
A natural log is a log of base 𝑒. It can be written as 𝒍𝒐𝒈𝒆𝒙 or simply as 𝒍𝒏𝒙.
Evaluate using the Change of base formula. Did you get the same answers as earlier?
a. 4log 64 b. 2
1log
8 c. 1
4
log 256 d. 7log 7 f. 300log 1
Use 9log 5 .732 and 9log 11 1.091 to approximate the following without a calculator.
a. 9
5log
11 b. 9log 55 c. 9log 25
Expand:
a. 6
5log 2x b. 7
3log
y
x
100
Condense:
a. log6 2log2 log3 b. 8 8 8
12log log 5 log
3x y
Solving equations (Solutions can be EXTRANEOUS)
a. 4log ( 3) 2x b. 6 6log ( 5) log 2x x c. 1
5125
x d. 5 18x
e. ln(3 9) 21x f. 4 4log ( 3) log (8 17)x x g. 4 12 32x x h. 5 48 10 35x
i. 2 54 6x j.
3 12 3x x
Exponential word problem (revisited):
Recall the car depreciation problem from earlier in the unit: 24,000(1 .16)ty . We used the calculator
intersect to solve the problem of when the car would be worth $12,000. Let’s solve it now without a
calculator using what we have learned about solving exponential equations.
101
Word Problems
1. The slope s of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach by this equation 0.159 0.118logs d . Find the slope of the beach if the average
diameter of the sand particle is 0.25 mm.
2. The Richter magnitude M of an earthquake is based on the intensity I of the earthquake and the
intensity 0I of an earthquake that can be barely felt. One formula used is 0
logI
MI
. If the
intensity of the Los Angeles earthquake in 1994 was 6.810 times 0I , what was the magnitude of
the earthquake? What magnitude on the Richter scale does an earthquake have if its intensity is 100 times the intensity of a barely felt earthquake?
3. The moment magnitude M of an earthquake that releases energy E (in Ergs) can be modeled by the equation 0.29ln 1.17M E . If the earthquake in Prince William Sound in 1964 had a moment magnitude of 8.6, how much energy did it release?
102
Section 4.4 Graphing and Modeling
Exponential and logarithmic functions are inverses of each other. The graph of y = 2x is shown in red. The
graph of its inverse is found by reflecting the graph across the line y = x.
LOGARITHMIC FUNCTION 𝒇(𝒙) = 𝒍𝒐𝒈𝒃𝒙
Domain: (0, )
Range: (– , )
(𝑥) = 𝑙𝑜𝑔𝑏𝑥, 𝑎 > 1, is increasing and continuous
on its entire domain, (0, ) .
The y-axis is a vertical asymptote as x 0 from the right.
The graph passes through the points (1
𝑏, −1) , (1,0), 𝑎𝑛𝑑 (𝑏, 1)
Example:
Graph the function
3logy x
x y
-1
0
1 x-10 -5 5 10
y
-10
-5
5
10
103
You can translate the graph of logarithms using the rule previously discussed applied to the equation
log ( )by a x h k .
How does a affect the graph? ____________________________________
How does h affect the graph? ____________________________________
How does k affect the graph? ____________________________________
Now graph:
3log ( 1)y x
a. State the domain and range.
D: _________ R:________ b. Give the equation of the vertical asymptote.
VA:_____________
3log ( 6) 4y x
a. State the domain and range. D: _________ R:________
b. Give the equation of the vertical asymptote. VA:_____________
x y
-1
0
1
x y
x y
x y
-1
0
1
x y
x y
x-10 -5 5 10
y
-10
-5
5
10
x-10 -5 5 10
y
-10
-5
5
10
104
2log ( 3)y x
a. State the domain and range. D: _________ R:________ b. Give the equation of the vertical asymptote. VA:_____________
You can also graph log base 10 and log base e (ln) in your graphing calculator. Show how you could use the
Change of Base formula to graph the same functions in your calculator.
a. 3logy x b. 3log ( 1)y x c. 3log ( 6) 4y x d. 2log ( 3)y x
a. ___________y b. ___________y c. ___________y d. ___________y
Now graph letter d in your calculator. Is it the same graph you drew by hand earlier? Y N
x y
-1
0
1
x y
x y
x-10 -5 5 10
y
-10
-5
5
10
105
Logistic Growth
A logistic growth model is a function of the form 𝑓(𝑥) =𝑐
1+𝑎𝑒−𝑟𝑥 where 𝑎, 𝑟, 𝑎𝑛𝑑 𝑐 are positive constants.
The number 𝑐 is called the carrying capacity. Logistic functions, like exponential functions grow quickly at the beginning, but because of restrictions that place limits on the size of the underlying population, eventually grow slowly and then level off.
Domain: ( , ) Range: (0, )c
Example:
106
Evaluating a Logistic Growth Function
Example: Evaluate the function 0.3
5( )
1 xf x
e
for the given value of x.
a. 0f b. 4f c. 1f
Solving a Logistic Growth Equation:
Example: Solve the equations.
a. 25
201 2 xe
b. 4
41
1 xe
c.
3
10050
1 5 xe
Using a Logistical Growth Model:
107
Regression Revisited:
1. A population of single-celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table shown. a. Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth. b. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 18th hour.
2. The given data shows the average growth rates of 12 Weeping Higan cherry trees planted in Washington, D.C. At the time of planting, the trees were one year old and were all 6 feet in height.
a. Determine a logarithmic regression model equation to represent this data.
b. What was the average height of the trees at one and one-half years of age? (to the nearest tenth of a foot)
c. If the height of a tree is 25 feet, what is the age of the tree to the nearest tenth of a year?