PreCalculus
2nd Nine-Weeks
Scope and Sequence
Topic 2: Polynomial, Power, and Rational Functions (40 – 45 days)
(Continued from 1st Nine-Weeks)
A) Determines the characteristics of the polynomial functions of any degree, general shape,
number of real and nonreal (real and nonreal), domain and range, and end behavior, and
finds real and nonreal zeros.
B) Identifies power functions and direct and inverse variation.
C) Describes and compares the characteristics of rational functions; e.g., general shape, number
of zeros (real and nonreal), domain and range, asymptotic behavior, and end behavior.
D) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and
make predictions.
Topic 3: Exponential, Logarithmic, and Logistic Functions (25 – 30 days)
(Continued in 3rd Nine-Weeks)
A) Identify exponential, logarithmic, and logistic functions.
B) Describe and compare the characteristics of exponential, logarithmic, and logistic functions:
e.g. general shape, number of roots, domain and range, asymptotic and end behavior,
extrema, local and global behavior.
C) Solve exponential, logarithmic, and logistic equations graphically and algebraically.
D) Create a scatterplot of bivariate data and identify an exponential, logarithmic, or logistic
function to model the data and make predictions.
COLUMBUS PUBLIC SCHOOLS
MATHEMATICS CURRICULUM GUIDE
GRADE LEVEL
PreCalculus
STATE STANDARD 4 and 5
Patterns, Functions, and Algebra
Data Analysis and Probability
TIME RANGE
25-30 days
GRADING
PERIOD
2-3
MATHEMATICAL TOPIC 3
Exponential, Logarithmic, and Logistic Functions
A)
B)
C)
D)
CPS LEARNING GOALS
Identifies exponential, logarithmic, and logistic functions.
Describes and compares the characteristics of exponential, logarithmic, and logistic
functions: e.g. general shape, number of roots, domain and range, asymptotic and end
behavior, extrema, local and global behavior.
Solves exponential, logarithmic, and logistic equations graphically and algebraically.
Creates a scatterplot of bivariate data and identifies an exponential, logarithmic, or logistic
function to model the data and make predictions.
COURSE LEVEL INDICATORS
Course Level (i.e., How does a student demonstrate mastery?):
9 Describes how a change in the value of a constant in an exponential, logarithmic, or logistic
equation affects the graph of the equation. Math A:11-A:11
9 Identifies exponential and logarithmic functions to any base including e and natural log.
Math A:12-A:03
9 Identifies logarithmic and exponential equations as inverses and uses the relationship to solve
equations. Math A:12-A:04
x
9 Recognizes e as lim⎛⎜1+ 1 ⎞⎟ . Math A:12-A:10
x →∞
⎝
x⎠
9 Sketches the graph of exponential, logistic, and logarithmic functions (including base e and
natural log) and their transformations. Math A:12-A:03
9 Models real world data with exponential, logarithmic, and logistic functions.
Math D:11-A:04
9 Identifies the inverse of exponential and logarithmic functions including e and natural log.
Math A:12-A:04
9 Solves problems involving compound interest, annuities, growth, and decay.
Math D:11-A:03 and Math A:11-A:04
9 Solves exponential and logarithmic equations to any base including e and natural log.
Math A:11-A:03
9 Applies the rules of logarithms. Math A:11-A:03
9 Transforms bivariate data so it can be modeled by a function; e.g., uses logarithms to allow
nonlinear relationships to be modeled by linear functions. Math D:12-A:02
Previous Level:
9 Performs computations using the rules of logarithms and exponents. Math N:11-C:08
9 Describes the characteristics of quadratics with complex roots. Math A:11-B:03
9 Identifies the inverse of exponential and logarithmic functions. Math A:11-A:06
9 Applies the rules of exponents. Math N:11-C:08
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 1 of 131
Columbus Public Schools 7/20/05
The description from the state, for the Patterns, Functions, and Algebra Standard says:
Students use patterns, relations, and functions to model, represent, and analyze problem
situations that involve variable quantities. Students analyze, model and solve problems using
various representations such as tables, graphs, and equations.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local
and global behavior.
The description from the state, for the Data Analysis and Probability Standard says:
Students pose questions and collect, organize represent, interpret and analyze data to answer
those questions. Students develop and evaluate inferences, predictions and arguments that are
abased on data.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Create and analyze tabular and graphical displays of data using appropriate tools, including
spreadsheets and graphing calculators.
The description from the state, for the Mathematical Processes Standard says:
Students use mathematical processes and knowledge to solve problems. Students apply
problem-solving and decision-making techniques, and communicate mathematical ideas.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
H. Use formal mathematical language and notation to represent ideas, to demonstrate
relationships within and among representation systems, and to formulate generalizations.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 2 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - A
Which of the following gives the equation of an exponential function?
A. y = 2 x5 − 7 x 2
B. y = −
1
x3
C. y = 2 − 7 x
D. y = 0.32 x
Which of the following gives the equation of a logarithmic function?
A. f ( x) = ln(− x)
B. f ( x) = 6 x
C. f ( x) = cos( x)
D. f ( x) = 2 x + 1
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 3 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - A
Answers/Rubrics
Low Complexity
Which of the following gives the equation of an exponential function?
A. y = 2 x 5 − 7 x 2
B. y = −
1
x3
C. y = 2 − 7 x
D. y = 0.32 x
Answer: D
Moderate Complexity
Which of the following gives the equation of a logarithmic function?
A. f ( x) = ln(− x)
B. f ( x) = 6 x
C. f ( x) = cos( x)
D. f ( x) = 2 x + 1
Answer: A
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 4 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - A
Which of the following graphs represents a logistic function?
A.
-10 -8
B.
10
8
-6
-4
C.
10
8
D.
10
8
10
8
6
6
6
6
4
4
4
4
2
2
2
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
2
4
6
8
10
-10 -8
-6
-4
-2
2
-2
-2
-2
-2
-4
-4
-4
-4
-6
-6
-6
-6
-8
-8
-8
-8
-10
-10
-10
-10
4
6
8
10
Describe the difference between the graph of an exponential function and the graph of a logistic
function.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 5 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - A
Answers/Rubrics
High Complexity
Which of the following graphs represents a logistic function?
A.
-10 -8
B.
10
-6
-4
C.
10
D.
10
10
8
8
8
8
6
6
6
6
4
4
4
4
2
2
2
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
2
4
6
8
10
-10 -8
-6
-4
-2
2
-2
-2
-2
-2
-4
-4
-4
-4
-6
-6
-6
-6
-8
-8
-8
-8
-10
-10
-10
-10
4
6
8
10
Answer: A
Short Answer/Extended Response
Describe the difference between the graph of an exponential function and the graph of a logistic
function.
Answer: An exponential graph has a single horizontal asymptote while a logistic graph
has two horizontal asymptotes.
A 2-point response provides a clear description of the difference between the two graphs.
A 1-point response provides a description of either the exponential graph or the logistic
graph but does not describe the difference between the two graphs.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 6 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
How does the graph of y = 3e x +1 − 2 compare to the graph of y = e x ?
A. y = 3e x +1 − 2 is steeper, moved one unit to the left and two units down.
B. y = 3e x +1 − 2 is steeper, moved one unit to the right and two units down.
C. y = 3e x +1 − 2 is less steep, moved one unit to the left and two units down.
D. y = 3e x +1 − 2 is less steep, moved one unit to the right and two units down.
How does the graph of y = a x compare to the graph of y = b x if 0 < a < b?
A. y = a x is steeper and has the same y-intercept.
B. y = a x is steeper and has y-intercept a + b.
C. y = a x is less steep and has the same y-intercept.
D. y = a x is less steep and has y-intercept a – b
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 7 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
Answers/Rubrics
Low Complexity
How does the graph of y = 3e x +1 − 2 compare to the graph of y = e x ?
A. y = 3e x +1 − 2 is steeper, moved one unit to the left and two units down.
B. y = 3e x +1 − 2 is steeper, moved one unit to the right and two units down.
C. y = 3e x +1 − 2 is less steep, moved one unit to the left and two units down.
D. y = 3e x +1 − 2 is less steep, moved one unit to the right and two units down.
Answer: A
Moderate Complexity
How does the graph of y = a x compare to the graph of y = b x if 0 < a < b?
A. y = a x is steeper and has the same y-intercept.
B. y = log a x is steeper and has y-intercept a + b.
C. y = a x is less steep and has the same y-intercept.
D. y = a x is less steep and has y-intercept a – b
Answer: C
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 8 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
How does the graph of y = log a x compare to the graph of y = log 3a x for a > 0 ?
A. y = log a x is steeper and has the same x-intercept.
B. y = log a x is steeper and has x-intercept a + b.
C. y = log a x is less steep and has the same x-intercept.
D. y = log a x is less steep and has x-intercept a – b.
Find the y-intercept and horizontal asymptotes of f (x) =
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 9 of 131
8
. Justify your answers.
1 + 3(.7)x
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
Answers/Rubrics
High Complexity
How does the graph of y = log a x compare to the graph of y = log 3a x for a > 0 ?
A. y = log a x is steeper and has the same y-intercept.
B. y = log a x is steeper and has y-intercept a + b.
C. y = log a x is less steep and has the same y-intercept.
D. y = log a x is less steep and has y-intercept a – b.
Answer: A
Short Answer/Extended Response
Find the y-intercept and horizontal asymptotes of f (x) =
8
. Justify your answers.
1 + 3(.7)x
Answer: The horizontal asymptote is y = 8, because the numerator of the logistic is 8.
8
8
The y-intercept is found evaluating f(0). f (0 ) =
= = 2 . The point is (0, 2).
0
1 + 3(.7 )
4
A 2-point response contains the correct y-intercept and horizontal asymptote and the
supporting justification for each statement.
A 1-point response contains at least two of the four components above.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 10 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
Given f (x) = e 7 − x , which statement describes the function and its end behavior?
A. Exponential growth function; lim f (x) = ∞ ; lim f (x) = 0
x→−∞
x→∞
B. Exponential growth function; lim f (x) = 0 ; lim f (x) = ∞
x→−∞
x→∞
C. Exponential decay function; lim f (x) = ∞ ; lim f (x) = 0
x→−∞
x→∞
D. Exponential decay function; lim f (x) = 0 ; lim f (x) = ∞
x→−∞
x→∞
⎛6 x⎞
Which statement is equivalent to log 7 ⎜
⎟?
⎝ y ⎠
A.
.5(log 7 6)log 7 x
log 7 y
B. log 7 x − .5 log 7 6 + log 7 y
C. log 7 6 + .5 log x7 − log 7 y
D. .5 log 7 (6x) − log 7 y
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 11 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
Answers/Rubrics
Low Complexity
Given f (x) = e7 − x , which statement describes the function and its end behavior?
A. Exponential growth function; lim f (x) = ∞ ; lim f (x) = 0
x→−∞
x→∞
B. Exponential growth function; lim f (x) = 0 ; lim f (x) = ∞
x→−∞
x→∞
C. Exponential decay function; lim f (x) = ∞ ; lim f (x) = 0
x→−∞
x→∞
D. Exponential decay function; lim f (x) = 0 ; lim f (x) = ∞
x→−∞
x→∞
Answer: C
Moderate Complexity
⎛6 x⎞
Which statement is equivalent to log 7 ⎜
⎟?
⎝ y ⎠
A.
.5(log 7 6)log 7 x
log 7 y
B. log 7 x − .5 log 7 6 + log 7 y
C. log 7 6 + .5 log x7 − log 7 y
D. .5 log 7 (6x) − log 7 y
Answer: C
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 12 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
⎛ x⎞
Which answer contains the correct way to enter y = log8 ⎜ ⎟ into the Y= menu of the calculator
⎝ 4⎠
in order to graph it?
A.
log x
4 log 8
B.
log x
log 32
C.
8 log x
log 4
D.
log x − log 4
log 8
8
. Give the domain, range, asymptote(s), any symmetry
1 + 3(.7)x
displayed, continuity, and end behavior.
Use the function f (x) =
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 13 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - B
Answers/Rubrics
High Complexity
⎛ x⎞
Which answer contains the correct way to enter y = log8 ⎜ ⎟ into the Y= menu of the
⎝ 4⎠
calculator in order to graph it?
log x
A.
4 log 8
B.
log x
log 32
C.
8 log x
log 4
D.
log x − log 4
log 8
Answer: D
Short Answer/Extended Response
8
. Give the domain, range, asymptote(s), any symmetry
1 + 3(.7)x
displayed, continuity, and end behavior.
Use the function f (x) =
Answer: The domain is the set of all real numbers, the range is (0, 8), and y = 0, and y = 8
are horizontal asymptotes. The graph is symmetric about the point (0, .5) and is always
continuous. The end behavior is described by lim f ( x ) = 0 and lim f ( x ) = 8 .
x →−∞
x →∞
A 2-point response correctly gives all of the answers.
A 1-point response correctly gives at least four of the answers.
A 0-point response gives less than four correct answers.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 14 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - C
What is the solution to the equation log x 256 = 8 ?
A. 2
B. 5
C. 8
D. 32
The number of bacteria in a Petri dish doubles every 3 hours. If there are originally 10 bacteria
in the dish, after how many hours will the number of bacteria be equal to 200?
A. 4.322 hours
B. 12.966 hours
C. 22.932 hours
D. 30.048 hours
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 15 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - C
Answers/Rubrics
Low Complexity
What is the solution to the equation log x 256 = 8 ?
A. 2
B. 5
C. 8
D. 32
Answer: A
Moderate Complexity
The number of bacteria in a Petri dish doubles every 3 hours. If there are originally 10 bacteria
in the dish, after how many hours will the number of bacteria be equal to 200?
A. 4.322 hours
B. 12.966 hours
C. 22.932 hours
D. 30.048 hours
Answer: B
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 16 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - C
Some especially sour vinegar has a pH of 2.4 and a box of Leg and Sickle baking soda has a pH
of 8.4. How many times greater is the hydrogen-ion concentration of the vinegar than that of the
baking soda? Hint: pH is based on a logarithmic scale.
A. 3.98
B. 6
C. 103.98
D. 106
A hard-boiled egg at temperature 96°C is placed in 16°C water to cool. Four minutes later the
temperature of the egg is 45°C. Use Newton's Law of Cooling to determine when the
temperature of the egg will be 20°C. Show your solution.
Newton’s Law of Cooling: T (t ) = Tm + (T0 − Tm )e− kt
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 17 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - C
Answers/Rubrics
High Complexity
Some especially sour vinegar has a pH of 2.4 and a box of Leg and Sickle baking soda has a pH
of 8.4. How many times greater is the hydrogen-ion concentration of the vinegar than that of the
baking soda? Hint: pH is based on a logarithmic scale.
A. 3.98
B. 6
C. 103.98
D. 106
Answer: D
Short Answer/Extended Response
A hard-boiled egg at temperature 96°C is placed in 16°C water to cool. Four minutes later the
temperature of the egg is 45°C. Use Newton's Law of Cooling to determine when the
temperature of the egg will be 20°C. Show your solution.
Newton’s Law of Cooling: T (t ) = Tm + (T0 − Tm )e− kt
Answer: T0=96 and TM=16, T0–TM=80 and T (t ) = Tm + (T0 − Tm )e − kt = 16 + 80e − kt .
So,
20 = 16 + 80e − kt
4
45 = 16 + 80e − kt
= 4e − kt
80
29
ln
= −4k − 4
4
80
ln
= − kt
80
29
ln
k = − 80
ln 4
4
t = − 80 ≈ 11.81
k
A 4-point response correctly applies Newton's Law of Cooling, solves for k, and finds the
correct amount of time.
A 3-point response correctly applies Newton's Law of Cooling, solves for k, and finds a time
making computational errors.
A 2-point response correctly applies Newton's Law of Cooling and sets up the two
equations to be solved.
A 1-point response correctly applies Newton's Law of Cooling.
A 0-point response demonstrates no mathematical understanding.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 18 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - D
Use the scatterplot below:
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
Which statement best describes a function that would model the data?
A. The model should be of the form y = ax b , where a and b are positive.
B. The model should be of the form y = ax b , where a is positive and b is negative.
.
C. The model should be of the form y = a log b x , where a and b are positive.
D. The model should be of the form y = a log b x , where a is positive and b is negative.
E
Given the table below:
x
y
1
0.5
2
2.6
3
3.6
4
4.5
5
4.9
6
5.4
7
5.7
Which equation is a logarithmic model for the data?
A. y = 0.86(1.38)x
B. y = 0.79(1.45)x
C. y = 0.63 + 2.67ln x
D. y = 1.63 + 2.06ln x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 19 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - D
Answers/Rubrics
Low Complexity
Use the scatterplot below:
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
Which statement best describes a function that would model the data?
A. The model should be of the form y = ax b , where a and b are positive.
B. The model should be of the form y = ax b , where a is positive and b is negative.
.
C. The model should be of the form y = a log b x , where a and b are positive.
D. The model should be of the form y = a log b x , where a is positive and b is negative.
Answer: A
Moderate Complexity
Given the table below:
1
2
3
4
x
0.5
2.6
3.6
4.5
y
Which equation is a logarithmic model for the data?
5
4.9
6
5.4
7
5.7
A. y = 0.86(1.38)x
B. y = 0.79(1.45)x
C. y = 0.63 + 2.67ln x
D. y = 1.63 + 2.06ln x
Answer: C
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 20 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - D
Given the table below showing the US population (in millions) for the decades from 1900 to
2000:
Year
1900
Population 76.2
1910
92.2
1920
106
1930 1940 1950 1960 1970 1980 1990 2000
123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4
If the trend continued, what is the best estimate of the population in the year 2020?
A. 306.8 million
B. 343.2 million
C. 346.0 million
D. 374.5 million
The graph below shows the graphs of one exponential function and one logistic function,
modeling the growth of bacteria in a Petri dish over a ten-hour period.
Number of Bacteria
1400
y=g(x)
1200
y=f(x)
1000
800
600
400
200
1 2 3 4 5 6 7 8 9 10
Hour
Identify which function (f(x) or g(x)) is logistic and which is exponential. Discuss the rates of
growth of each function and give examples of situations that would cause the two different
models of growth of bacteria over ten hours.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 21 of 131
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Exponent, Log, & Logistic Fctn - D
Answers/Rubrics
High Complexity
Given the table below showing the US population (in millions) for the decades from 1900 to
2000:
Year
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Population 76.2 92.2 106
123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4
If the trend continued, what is the best estimate of the population in the year 2020?
A. 306.8 million
B. 343.2 million
C. 346.0 million
D. 374.5 million
Answer: D
Short Answer/Extended Response
The graph below shows the graphs of one exponential function and one logistic function,
modeling the growth of bacteria in a Petri dish over a ten-hour period.
Number of Bacteria
1400
1200
y=f(x)
1000
800
600
400
y=g(x)
200
1
2
3
4
5
Hour
6
7
8
9 10
Identify which function (f(x) or g(x)) is logistic and which is exponential. Discuss the rates of
growth of each function and give examples of situations that would cause the two different
models of growth of bacteria over ten hours.
Answer: g(x) is exponential and f(x) is logistic. Although the growth is approximately
the same for the first few hours, the growth of the logistic function begins to slow and
eventually stops, with the population stabilizing at about 1200. In the exponential
population, the bacteria had not experienced a shortage of food or space over the first
ten hours. In the logistic population, the population conditions would not support such
growth and the growth leveled off.
A 2-point response correctly identifies both graphs and provides a logical explanation.
A 1-point response correctly identifies both graphs but does not provide an explanation.
A 0-point response demonstrates no mathematical understanding.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Teacher Introduction
Exponential, Logarithmic, and Logistic Functions
The nature of the topic requires that the learning goals be integrated. The pacing guide and
correlations demonstrate this.
The students will probably have had limited exposure to exponential functions and no exposure
to either logarithmic or logistic functions.
The strategies and activities section of learning goal A refer to teacher notes (included in this
Curriculum Guide) that provide you, the teacher, with a method of introducing these three
functions.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Columbus Public Schools 7/20/05
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: logarithm, logistic function, exponential function, logarithmic function,
scatterplot, asymptotes, domain, range, root, end behavior, extrema, asymptotic
behavior, Newton’s Law of Cooling, local behavior, global behavior, natural
logarithm, growth, decay, base e, compound interest, annuity.
Core:
Learning Goal A: Identify exponential, logarithmic, and logistic functions.
1. Introduce an exponential function by having students complete the “Million Dollar Mission”
(included in this Curriculum Guide).
2. Continue the introduction of exponential functions by using the teacher notes (included in
this Curriculum Guide).
3. Students will learn the behaviors of different types of exponential bases by completing the
activity “Comparing Exponential Effects” (included in this Curriculum Guide).
4. Develop the concept of fractional exponents by comparing values obtained by raising
numbers to fractional exponents versus whole number exponents. This activity clarifies how
fractional exponents behave. Specifically, students will make observations such as 22.6 lies
somewhere between 22 and 23. Students complete the “What Are Fractional Exponents
Really?” activity (included in this Curriculum Guide), answering all questions. Regroup as a
whole class to discuss the questions at the end of each section.
5. Have the students explore, using the calculator, what happens to the exponential function,
f(x) = Nacx, as the values of N, a, and c change. The students should explore changing one
value at a time and write down their conclusion. Once they have written the effect of each
value, have the students share their findings with a partner. If their conclusions do not agree,
encourage them to explore more to come to a clearer understanding. Wrap up this
exploration by summarizing the basic features of an exponential function. See teacher notes
for summarization of the key features of exponential functions (included in this Curriculum
Guide).
6. Have students complete the activity “Constant Effects?” (included in this Curriculum Guide)
to have the students discover how adding or subtracting a number to the exponent and adding
or subtracting a number from the function will affect the graph of the function.
7. Introduce logarithmic functions by using the teacher notes (included in this Curriculum
Guide).
Learning Goal B: Describe and compare the characteristics of exponential, logarithmic and
logistic functions: e.g. general shape, number of roots, domain and range, asymptotic and end
behavior, extrema, local and global behavior.
1. Have the students complete the “Paper Folding Activity” (included in this Curriculum Guide)
as a starting point for describing the characteristics of exponential graphs.
2. Students will complete the “M&M’s activity” (included in this Curriculum Guide) as a
follow-up to the Paper Folding activity and to further solidify the exponential graph
characteristics.
3. Relating negative exponents to what they know already about positive exponents, students
will take expressions like 2 −3 x and ( 13 ) and convert them to expressions without negative
exponents. They will describe and sketch the resulting function found in the activity.
“Negative and Positive Exponential Function Graphs” (included in this Curriculum Guide).
−2 x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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4. Apply the concept of rate of change to different exponential functions. Students complete
the activity “Rate of Change of Exponential Functions” (included in this Curriculum Guide)
in order to find that a larger growth factor does translate to a greater rate of change.
5. Students will be able to see the inverse relationship between a logarithmic and an exponential
function by completing “Let’s Graph Logarithmic Functions” (included in this Curriculum
Guide).
Learning Goal C: Solve exponential, logarithmic, and logistic equations graphically and
algebraically.
1. After introducing exponential growth and decay, discuss the ways to solve exponential
functions. The student will be able to complete the worksheet “Exponential Growth and
Decay” (included in this Curriculum Guide).
2. Introduce the generalizations necessary to be able to simplify logarithmic expressions.
Students should already know how to evaluate logarithms on their calculators (including
logarithms of bases other than 10 or e using the “change of base rule”). They will be able to
use their calculators to carry out the computations in the worksheet “Discovering
Logarithmic Identities” (included in this Curriculum Guide) in order to discover some of the
more basic logarithm combination laws. Upon completion, students should compare their
conclusions with each other.
3. Introduce the three fundamental logarithm combination laws (log of a product, log of a
quotient, log of a power) by observing numerical patterns. “Logarithm Combination Laws”
(included in this Curriculum Guide). Follow up with lots of practice combining and taking
apart logarithmic expressions.
4. Have students practice solving logarithmic equations by using the “Logarithmic cut-out
puzzle” (included in this Curriculum Guide).
5. Students will develop a better understanding of how to solve logarithmic equations by
completing the “Properties of Logarithms” (included in this Curriculum Guide).
6. Student can proceed from solving a logarithmic equation in basic definition form to solving
logarithmic equations which require more logical thought using the worksheet “Exercises
with Logarithms” (included in this Curriculum Guide).
7. Extend knowledge of exponential curves to an application where students derive information
from the graph itself. The activity “Time Estimation” (included in this Curriculum Guide)
has students use the trace function on the TI-83 graphing calculator to find an x-value (time)
which produces a given y-value (amount).
8. Practice solving the exponential equations by using logarithms. Students complete the
activity “Why Settle for Time Estimation When You can be Exact” (included in this
Curriculum Guide) which is built off of the Time Estimation Activity.
9. Another application of using logarithms to solve an equation is done by using the activity,
“The Rule of 72” (included in this Curriculum Guide).
Learning Goal D: Create a scatterplot of bivariate data and identify an exponential,
logarithmic, or logistic function to model the data and make predictions.
1. Complete the activity – “USA TODAY Snapshot – More of U.S.” (included in this
Curriculum Guide) to show a set of data that will provide an exponential equation.
2. Use a graph to solve an exponential equation where the variable to solve for is in the
exponent. Students complete the activity “Determining the Half-Life of Hydrogen-3”
(included in this Curriculum Guide) using the substitutions to obtain a graph which they use
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 25 of 131
Columbus Public Schools 7/20/05
3.
4.
5.
6.
7.
8.
to estimate the half-life. (This involves the use of logarithms to solve for a variable in the
exponent).
Have students complete the activity "Not Just a Good Idea, It's the Law" (included in this
Curriculum Guide). Here students trace the temperature of a microwaved potato as it gets
closer and closer to room temperature. The measuring time will be approximately 200
minutes, depending on the size of the potato and other factors, so you need to plan
accordingly. Students get to use Newton's Law of Cooling in order to create an exponential
equation to model their data. They also get to consider some of the limits of the regression
function of the TI-83 calculator and how to get around these.
In “Ball Bounce Revisited” (included in this Curriculum Guide) the students will have the
opportunity to see that data that was collected and analyzed using parabolas can be used to
produce an exponential function as well.
In the “Population Growth Investigation” (included in this Curriculum Guide,) students
summarize their knowledge of exponential growth and take a first look at logistic growth.
The teacher notes preceding the activity describe the use of the activity for extending
exponential growth into an intuitive understanding of logistic growth. They also describe the
next day discussion of the activity which is essential for students to understand logistic
functions.
The in-class simulation “Rumors,” described in the teacher notes of this Curriculum Guide is
the culminating activity fro the Population Growth Investigation. Students are actively
involved in creating and modeling a logistic scenario.
Complete the activity – “Breast Cancer Risks” (included in this Curriculum Guide) to show a
set of data that will provide a logistic equation.
The student will complete the “Comparison of Curve-Fittings” (included in this Curriculum
Guide) as a way to compare the different types of curves that fit a given set of data.
Reteach:
1. To illustrate the relationship between decimal exponents and integer exponents, students will
consider the continuity aspect of exponential functions; ( i.e., there are points between any
two points found on a graph). Students will examine various pieces of an exponential
graph by completing the activity “Decimal Exponents” (included in this Curriculum Guide).
2. Apply concepts of exponential decay to the real-world situation of depreciation. Students are
guided through this process in the “Car Value Depreciation” worksheet (included in this
Curriculum Guide). Students work individually and compare answers at the end. The
question may also arise of how much error is acceptable, and how do we check our final
answer?
3. The student will use their calculator to reinforce the relationships of common logarithms in
“Exploring Common Logarithms” (included in this Curriculum Guide).
4. Additional practice in the relationship between logarithms and exponentials is provided by
the worksheet “Logarithmic & Exponential Form” (included in this Curriculum Guide).
5. Investigating “Compound Interest” (included in this Curriculum Guide) will allow the
student to explore the difference between simple and compound interest.
6. Some students will have a difficult time understanding the base of natural logarithms, e.
Their understanding can be increased by using “A Number Called e” (included in this
Curriculum Guide).
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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RESOURCES
Learning Goal A:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 276-289; 300309.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource
Manual pp. 55-56
Learning Goal B:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 276-319.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource
Manual pp. 55-60
Learning Goal C:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 320-333.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource
Manual pp. 61-64
Learning Goal D:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 290-299; 334345.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource
Manual pp. 57-58; 65-66.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 27 of 131
Columbus Public Schools 7/20/05
The Million Dollar Mission
Exponent, Log, & Logistic Fctn - A
Name
You’re sitting in math class, minding your own business, when in walks a Bill Gates kind of guy
- the real success story of your school. He's made it big, and now he has a job offer for you.
He doesn't give too many details, mumbles something about the possibility of danger. He's
going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?)
And you've got to make sure your passport is current. (Get real, Bill, this isn’t Paris). But do
you ever sit up at the next thing he says:
You'll have your choice of two payment options:
1.
2.
One cent on the first day, two cents on the second day, and double your salary every day
thereafter for the thirty days; or
Exactly $1,000,000. (That's one million dollars!)
You jump up out of your seat at that. You've got your man, Bill, right here. You'll take that
million. You are there. And off you go on this dangerous million-dollar mission.
So how smart was this guy? Did you make the best choice? Before we decide for sure, let's
investigate the first payment option. Complete the table for the first week's work.
First Week – First Option
Day No.
1
2
3
4
5
6
7
Pay for that Day
.01
.02
Total Pay (In Dollars)
.01
.03
So, after a whole week you would have only made
.
That's pretty awful, all right. There's no way to make a million in a month at this rate. Right?
Let's check out the second week. Complete the second table.
Second Week – First Option
Day No.
8
9
10
11
12
13
14
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Pay for that Day
Total Pay (In Dollars)
Page 28 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - A
Well, you would make a little more the second week; at least you would have made
. But there's still a big difference between this salary and $1,000,000.
What about the third week?
Third Week – First Option
Day No.
15
16
17
18
19
20
21
Pay for that Day
Total Pay (In Dollars)
We're getting into some serious money here now, but still nowhere even close to a
million. And there's only 10 days left. So it looks like the million dollars is the best deal. Of
course, we suspected that all along.
Fourth Week – First Option
Day No.
22
23
24
25
26
27
28
Pay for that Day
Total Pay (In Dollars)
Hold it! Look what has happened. What's going on here? This can't be right. This is
amazing. Look how fast this pay is growing. Let's keep going. I can't wait to see what the
total will be.
Last 2 Days – First Option
Day No.
29
30
Pay for that Day
Total Pay (In Dollars)
In 30 days, it increases from 1 penny to over
is absolutely amazing.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 29 of 131
dollars. That
Columbus Public Schools 7/20/05
The Million Dollar Mission
Answer Key
Exponent, Log, & Logistic Fctn - A
You're sitting in math class, minding your own business, when in walks a Bill Gates kind of guy
- the real success story of your school. He's made it big, and now he has a job offer for you.
He doesn't give too many details, mumbles something about the possibility of danger. He's
going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?)
And you've got to make sure your passport is current. (Get real, Bill, this isn’t Paris). But do
you ever sit up at the next thing he says:
You'll have your choice of two payment options:
1.
2.
One cent on the first day, two cents on the second day, and double your salary every day
thereafter for the thirty days; or
Exactly $1,000,000. (That's one million dollars!)
You jump up out of your seat at that. You've got your man, Bill, right here. You'll take that
million. You are there. And off you go on this dangerous million-dollar mission.
So how smart was this guy? Did you make the best choice? Before we decide for sure, let's
investigate the first payment option. Complete the table for the first week's work.
First Week – First Option
Day No.
1
2
3
4
5
6
7
Pay for that Day
.01
.02
.04
.08
.16
.32
.64
Total Pay (In Dollars)
.01
.03
.07
.15
.31
.63
1.27
So after a whole week you would have only made $1.27 .
That's pretty awful, all right. There's no way to make a million in a month at this rate. Right?
Let's check out the second week. Complete the second table.
Second Week – First Option
Day No.
8
9
10
11
12
13
14
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Pay for that Day
1.28
2.56
5.12
10.24
20.48
40.96
81.92
Total Pay (In Dollars)
2.55
5.11
10.23
20.47
40.95
81.91
163.83
Page 30 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - A
Well, you would make a little more the second week; at least you would have made
$163.83
. But there's still a big difference between this salary and $1,000,000.
What about the third week?
Third Week – First Option
Day No.
15
16
17
18
19
20
21
Pay for that Day
163.84
327.68
655.36
1,310.72
2,621.44
5,242.88
10,485.76
Total Pay (In Dollars)
327.67
655.35
1,310.71
2,621.43
5,242.87
10,485.75
20,971.51
We're getting into some serious money here now, but still nowhere even close to a million.
And there's only 10 days left. So it looks like the million dollars is the best deal. Of course, we
suspected that all along.
Fourth Week – First Option
Day No.
22
23
24
25
26
27
28
Pay for that Day
20,971.52
41,943.04
83,886.08
167,772.16
335,544.32
671,088.64
1,342,177.28
Total Pay (In Dollars)
41,943.03
83,886.07
167,772.15
335,544.31
671,088.63
1,342,177.27
2,684,354.55
Hold it! Look what has happened. What's going on here? This can't be right. This is amazing.
Look how fast this pay is growing. Let's keep going. I can't wait to see what the total will be.
Last 2 Days – First Option
Day No.
29
30
Pay for that Day
2,684,354.56
5,368,709.12
In 30 days, it increases from 1 penny to over
amazing.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
10 million
Page 31 of 131
Total Pay (In Dollars)
5,368,709.11
10,737,418.23
dollars. That is absolutely
Columbus Public Schools 7/20/05
Introduction to the Exponential Function
Teacher Notes - A
Think of a number, any number.
Double it.
Double it again.
Now think about how fast your number grew at each stage. The larger your number
grew, the faster it grew, right? Do it again, or keep doubling it a few more times, to get the
feel of this fact.
The student can see this process very quickly by using the calculator.
The student inputs their numbers into the calculator times 2, and then
press ENTER. Continuing to use the calculator, the student would input
times 2 (Ans*2) so that the student has taken the previous answer times 2.
At this point, the student can continue hitting the ENTER key to see how
the number has increased.
This is the crucial property that makes an exponential function different from any other
function: it increases (or decreases) faster if its value is larger because its growth rate is
directly proportional to its value.
Going back to your doubling exercise, let's say you chose the number 5.
Then doubling it gives 5 x 2 = 10.
Doubling again gives 5 x 2 x 2 = 20.
If you doubled again you'd get 5 x 2 x 2 x 2 = 40.
So, for the first doubling, when the value was 5, the process of doubling increased it by 5.
But later when its value had increased to 20, the same process of doubling increased it by 20.
This shows the growth rate depending on the value.
Now looking back at the equations above, we see that we can write them more briefly.
At the first step we have 5 x 2 = 10.
At the second step we have 5 x 22 = 20.
And then at the third step we have 5 x 23 = 40.
Can you see a general rule?
Of course we can do the same thing with tripling instead of doubling, or we could halve
the number each time instead (i.e. multiply by a half). In fact we can choose any number
(integer, fraction, decimal, irrational .... any number!) and just keep multiplying by that
number.
Suppose we call this number p. Then what we have found above is that after the nth step
the value we get is the original number, multiplied by pn.
In this way we are looking at our result as a function of n (the step). If we know n we can
use the above formula to work out the result.
Finally, let's go back to the "doubling 5" example and suppose we want to increase our
number gradually, continuously, instead of in jumps or steps.
This is more useful in some real problems, for example if we know something is growing
steadily so that it doubles every minute, but we want to know how much it has grown after a
minute and a half.
Then between the first and second steps we'd want to get a result that's bigger than the
first result, 5 x 2 = 10, but not as big as the second result, 5 x 22 = 20.
So instead of 5 x 21 (after 1 minute) or 5 x 22 (after 2 minutes) we just use 5 x 21.5 to find
the value after 1.5 minutes.
Therefore, we can have any number as a power, so in this way we can calculate the value
at any time t as 5 x 2t.
This is called an exponential function of t, because the t is the exponent (which means the
power).
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 32 of 131
Columbus Public Schools 7/20/05
Comparing Exponential Effects
Exponent, Log, & Logistic Fctn - A
Name
Directions: Using a calculator, fill in the chart below. For the observations column, indicate if
the numbers in that row appear to be increasing, decreasing, or no change.
n
n2
n3
n4
n5
Observations
3
5
0.4
0
1
0.1
1.2
0.9
1.003
How can you predict whether the numbers in each row are increasing or decreasing?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 33 of 131
Columbus Public Schools 7/20/05
Comparing Exponential Effects
Answer Key
Exponent, Log, & Logistic Fctn - A
Directions: Using a calculator, fill in the chart below. For the observations column, indicate if
the numbers in that row appear to be increasing, decreasing, or no change.
Observations
n
n2
n3
n4
n5
3
9
27
81
243
Increasing
5
25
125
625
3125
Increasing
0.4
.16
.064
.0256
.01024
Decreasing
0
0
0
0
0
No change
1
1
1
1
1
No change
0.1
.01
.001
.0001
.00001
Decreasing
1.2
1.44
1.728
2.0736
2.48832
Increasing
0.9
.81
.729
.6561
.59049
Decreasing
1.003
1.006009
1.009027027 1.012054108 1.01509027
Increasing
How can you predict whether the numbers in each row are increasing or decreasing?
0 < n < 1 means numbers get smaller or are decreasing.
n > 1 means the numbers get bigger or are increasing.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 34 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - A
What Are Fractional Exponents Really?
Name
1. In the space between each pair of integers of the top row (1&2, 2&3, 3&4), write a mixed
number (in fraction or decimal form) whose value lies somewhere between them.
2. Use your calculator to raise the given integer (in the leftmost column) to the power given
along the top. Write your result in the grid.
x=
2x
3x
4x
5x
1
2
3
4
3. What do you observe about the numbers in each row as you move from left-to-right?
4. What does this tell you about the nature of your “in-between” exponents?
5. The following fractions all lie somewhere between 0 and 1. Order them from least to
greatest, then place them in the first row of the table below, in least-to-greatest order:
1
1
2
1
1
3
3
2
3
3
4
5
4
5
Then complete the table as above.
x=
2x
3x
4x
5x
0
2 =1
30 = 1
40 = 1
50 = 1
1
2 =2
31 = 3
41 = 4
51 = 5
0
1
2
Hint: To type 5 3 into the TI-83 graphing calculator, you will need to treat the fractional
exponent as division and enclose it in parentheses. This example would look like 5^(2 / 3).
6. What do you notice about the values in each row as you go from left to right?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 35 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - A
What Are Fractional Exponents Really?
Name
1. In the space between each pair of integers of the top row (1&2, 2&3, 3&4), write a mixed
number (in fraction or decimal form) whose value lies somewhere between them.
2. Use your calculator to raise the given integer (in the leftmost column) to the power given
along the top. Write your result in the grid.
Note: Answers to the blank columns will vary with student input values for x.
x=
2x
3x
4x
5x
1
2
3
4
5
2
4
9
16
25
3
8
27
64
125
4
16
81
256
625
3. What do you observe about the numbers in each row as you move from left-to-right?
The numbers get progressively larger.
4. What does this tell you about the nature of your “in-between” exponents?
They give values ‘in-between’ the values from the whole number exponents.
5. The following fractions all lie somewhere between 0 and 1. Order them from least to
greatest, then place them in the first row of the table below, in least-to-greatest order:
1
1
2
1
1
3
3
2
3
3
4
5
4
5
1
1
2
= .5
= .33
= .66
2
3
3
Then complete the table as above.
Note:
x=
2x
3x
4x
5x
0
2 =1
30 = 1
40 = 1
50 = 1
.2
1.1487
1.2457
1.3195
1.3797
0
.25
1.1892
1.3161
1.4953
1.4953
1
1
= .25
= .2
4
5
3
= .75
4
3
= .6
5
.33
.5
.6
.66
.75
1.2599 1.4142 1.5157 1.5874 1.6818
1.4422 1.7321 1.9332 2.0801 2.2795
1.5874
2
2.2974 2.5198 2.8284
1.7100 2.2361 2.6265 2.9240 3.3437
1
2 =2
31 = 3
41 = 4
51 = 5
1
2
Hint: To type 5 3 into the TI-83 graphing calculator, you will need to treat the fractional
exponent as division and enclose it in parentheses. This example would look like 5^(2 / 3).
6. What do you notice about the values in each row as you go from left to right?
They get progressively larger.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 36 of 131
Columbus Public Schools 7/20/05
Basic Features of Exponential Functions
Teacher Notes - A
f(x) = Nacx
When x = 0, a0 = 1.
The function ax is never zero for any value of x when a ≠ 0.
For a > 1, the exponential function will increase.
For 0 < a < 1, the exponential function will decrease, but will never reaches zero.
As c > 1, the exponent enhances the growth property, making the growth faster.
As 0 < c < 1, the exponent inhibits the growth property, making the growth slower.
For c < 0, the graph is reflected about the y-axis.
For a > 0, the choice of a affects how rapidly the function grows or decays as you leave
x = 0: for larger values of a, the growth or decay is faster.
9. The constant N multiplies the function by N everywhere, in particular it gives f (0) = N.
10. For N > 0, the value of N represents the starting value.
11. For N < 0, the graph is a reflection about the x-axis.
1.
2.
3.
4.
5.
6.
7.
8.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 37 of 131
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Constant Effects?
Exponent, Log, & Logistic Fctn - A
Name
In this activity, you will explore how the constants h and k affect the graph of the exponential
function
f ( x) = b x − h + k .
1. Sketch the graph of
f ( x) = 3x below.
2. Sketch the graph of g ( x) = 3
transformed from the graph of
on the same coordinate plane above. How is this graph
f ( x) = 3x ? Is this what you had expected?
3. Sketch the graph of h( x) = 3
transformed from the graph of
x −3
x+2
on the same coordinate plane above. How is this graph
f ( x) = 3x ? Is this what you had expected?
4. In the exponential function f ( x) = b
x−h
, how does the constant h affect the graph of
f ( x) = b ? Be specific in your answer.
x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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5. Sketch the graph of
f ( x) = 4 x below.
6. Sketch the graph of g ( x) = 4
Exponent, Log, & Logistic Fctn - A
+ 8 on the same coordinate plane above. How is this graph
x
transformed from the graph of f ( x) = 4 ? Is this what you had expected?
7. Sketch the graph of h( x) = 4
x
− 2 on the same coordinate plane above. How is this graph
x
transformed from the graph of f ( x) = 4 ? Is this what you had expected?
x
8. In the exponential function f ( x) = b + k , how does the constant k affect the graph of
x
f ( x) = b x ? Be specific in your answer.
9. How do you think the graph of k ( x) = 2
x+2
+ 5 is transformed from the graph of
f ( x) = 2 x ?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 39 of 131
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Constant Effects?
Answer Key
Exponent, Log, & Logistic Fctn - A
In this activity, you will explore how the constants h and k affect the graph of the exponential
function
f ( x) = b x − h + k .
1. Sketch the graph of
f ( x) = 3x below.
2. Sketch the graph of g ( x) = 3
x −3
on the same coordinate plane above. How is this graph
transformed from the graph of f ( x) = 3 ? Is this what you had expected?
This is a shift right 3 units.
x
3. Sketch the graph of h( x) = 3
x+2
on the same coordinate plane above. How is this graph
transformed from the graph of f ( x) = 3 ? Is this what you had expected?
This is a shift left 2 units.
x
4. In the exponential function f ( x) = b
x−h
, how does the constant h affect the graph of
f ( x) = b ? Be specific in your answer.
x
If h is positive then the graph will shift right h units. If h is negative then the graph
will shift left h units.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 40 of 131
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5. Sketch the graph of
f ( x) = 4 x below.
6. Sketch the graph of g ( x) = 4
Exponent, Log, & Logistic Fctn - A
+ 8 on the same coordinate plane above. How is this graph
x
transformed from the graph of f ( x) = 4 ? Is this what you had expected?
x
It is a shift up 8 units.
7. Sketch the graph of h( x) = 4
− 2 on the same coordinate plane above. How is this graph
x
transformed from the graph of f ( x) = 4 ? Is this what you had expected?
x
It is a shift down 2 units.
8. In the exponential function f ( x) = b + k , how does the constant k affect the graph of
x
f ( x) = b x ? Be specific in your answer.
If k is positive then the graph will shift up k units. If k is negative then the graph
will shift down k units.
9. How do you think the graph of k ( x) = 2
x+2
+ 5 is transformed from the graph of
f ( x) = 2 x ?
The graph of k(x) will shift left 2 units and up 5 units from the graph of f(x).
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 41 of 131
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Introduction to the Logarithm Function
Teacher Notes - A
The value of the log function for an input x is the power you have to raise something to, to
get x. The "something" is called the base and it can be any number you choose. If we choose
the base to be 2, then the log function is called the "log base 2" function. So the value of log
base 2 for input x, is the power you have to raise 2 to, to get x. Just remember that the log is the
power, the index, the exponent, the little number up in the air.
It is inconvenient to write log base 2 all the time, so we write it more concisely as below, with
the base written as a subscript.
log2(x)
If there's no base shown as a subscript, the function is just log(x), this means that the base is 10.
Another common base is "e". This turns out to be such a useful base that the function "log-tobase-e" is known as the "natural log" function. It is very often written as ln(x), rather than
loge(x).
The "base" is the number that's raised to the power. The other number is what we want to get
when we raise the base to the power.
So the expression "log base 5 of 125" means: "the power we have to raise 5 to, to get 125" or
log5125 = 3.
Although logarithms may seem awkward, they turn a problem that involves powers into a
problem that does not involve powers, thus easier to solve. The second purpose for logs is to
look at different scales at the same time.
The main thing to remember when trying to find the log of something: it's the power that we're
looking for.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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The Paper Folding Activity
Exponent, Log, & Logistic Fctn - B
Name
Part I. Number of Sections
1. Fold an 8.5 x 11” sheet of paper in half and determine the number of sections the paper has
after you have made the fold.
2. Record the data in the table and continue in the same manner until it becomes too hard to fold
the paper.
Number of Folds Number of Sections
3. Make a scatterplot of your data.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - B
4. Determine a mathematical model that represents this data by examining the patterns in the
table.
5. What might be different if you tried this experiment with an 8.5 x 11” sheet of wax paper or
tissue paper?
Part II: Area of Smallest Section
6. Fold an 8.5 x 11” sheet of paper in half and determine the fractional part of the smallest
section after you have made the fold.
7. Record the data in the table and continue in the same manner until it becomes too hard to fold
the paper.
Number of Folds Fractional Part
of Smallest
Section
0
1
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - B
8. Make a scatterplot of your data.
9. Determine a mathematical model that represents the data by examining the patterns in the
table.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 45 of 131
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The Paper Folding Activity
Answer Key
Exponent, Log, & Logistic Fctn - B
Part I. Number of Sections
1. Fold an 8.5 x 11” sheet of paper in half and determine the number of sections the paper has
after you have made the fold.
2. Record the data in the table and continue in the same manner until it becomes too hard to fold
the paper.
Number of Folds Number of Sections
0
1
1
2
2
4
3
8
4
16
5
32
6
64
3. Make a scatterplot of your data.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - B
4. Determine a mathematical model that represents this data by examining the patterns in the
table.
y = 2x
5. What might be different if you tried this experiment with an 8.5 x 11” sheet of wax paper or
tissue paper?
The results would be the same, but you would be able to make more folds and
collect more data because the paper would be thinner.
Part II: Area of Smallest Section
6. Fold an 8.5 x 11” sheet of paper in half and determine the fractional part of the smallest
section after you have made the fold.
7. Record the data in the table and continue in the same manner until it becomes too hard to fold
the paper.
Number of Folds Fractional Part
of Smallest
Section
0
1
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
1
1/2
2
1/4
3
1/8
4
1/16
5
1/32
6
1/64
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Exponent, Log, & Logistic Fctn - B
8. Make a scatterplot of your data.
Area of Smallest
Section
9. Determine a mathematical model that represents the data by examining the patterns in the
table.
y = (1/2)x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 48 of 131
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The “M&M” Investigation
Exponent, Log, & Logistic Fctn - B
Name
Part I: Collecting Data
1. Empty your bag of M&M’s and count them. Then place them back in the bag, and mix them
well. Pour them out on the desk, count the number that show an “m” and place these back in
the bag. The others may be eaten or removed. Record the number that show an “m” in your
data table then repeat this procedure. Continue until the number of M&M’s remaining is less
than 5 but greater than 0. Record your data in the data table.
Trial Number Number of M&M’s
Remaining
0
1
2
3
4
5
6
7
8
Part II: Graphing and Determining the Exponential Model
2. Use a graphing calculator to make a scatterplot of your data. Copy your scatterplot onto the
grid below. Then use the graphing calculator to determine an exponential model, and graph
the equation. Sketch in the graph, and write your exponential equation.
Equation:
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 49 of 131
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Exponent, Log, & Logistic Fctn - B
Part III: Interpreting the Data
3. In your model, y = a(b)x, what value do you have for a?
What does that seem to relate to when you consider your data?
When x = 0, what is your function value?
Compare this to the values in your data table.
4. What is the value for b in your exponential model?
How does this value relate to the data collection process?
5. How does the M&M experiment compare to the paper folding activity?
How are they alike and how are they different?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 50 of 131
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The “M&M” Investigation
Answer Key
Exponent, Log, & Logistic Fctn - B
Part I: Collecting Data
1. Empty your bag of M&M’s and count them. Then place them back in the bag, and mix them
well. Pour them out on the desk, count the number that show an “m” and place these back in
the bag. The others may be eaten or removed. Record the number that show an “m” in your
data table then repeat this procedure. Continue until the number of M&M’s remaining is less
than 5 but greater than 0. Record your data in the data table.
Answers will vary. A sample set of data has been included on this sheet for your reference.
Trial Number Number of M&M’s
Remaining
0
140
1
76
2
39
3
22
4
12
5
8
6
3
Part II: Graphing and Determining the Exponential Model
2. Use a graphing calculator to make a scatterplot of your data. Copy your scatterplot onto the
grid below. Then use the graphing calculator to determine an exponential model, and graph
the equation. Sketch in the graph, and write your exponential equation.
y = 140.1(0.54)x
Number Remaining
Equation:
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 51 of 131
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Exponent, Log, & Logistic Fctn - B
Part III: Interpreting the Data
3. In your model, y = a(b)x, what value do you have for a? 140.1
What does that seem to relate to when you consider your data? initial number of m & m’s
When x = 0, what is your function value?
140.1
Compare this to the values in your data table.
Very close to the original number of M&M’s in the bag
4. What is the value for b in your exponential model?
b = 0.54
How does this value relate to the data collection process?
This is close to the probability that an M&M candy will land with the “m” showing.
As the data is collected, the number of M&M’s decreases by almost half.
5. How does the M&M experiment compare to the paper folding activity?
The M&M data pattern is close to the data pattern for the paper folding activity when
the student recorded the trial number and the area of the smallest section.
How are they alike and how are they different?
The starting numbers are different, but the pattern of decrease is almost the same each term is about half of the previous term. in the case of the paper, it is exactly half.
In the case of the candy, it is approximately half.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - B
Negative and Positive Exponential Function Graphs
Name
Function
Function Rewritten with
Positive Exponents
Graph of Function
y = 2 −3 x
⎛ ⎞
y = ⎜1⎟
⎝ 3⎠
⎛ ⎞
y =⎜ 2⎟
⎝5⎠
−x
−2 x
y = 1.25− x
y = 6− x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 53 of 131
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Exponent, Log, & Logistic Fctn - B
Negative and Positive Exponential Function Graphs
Answer Key
Function
Function Rewritten with
Positive Exponents
y = 2−3 x
⎛ ⎞
y =⎜1⎟
⎝ 2⎠
3x
Graph of Function
10
or y = 13 x
2
8
6
4
2
-5 -4
-3 -2 -1
1
2
3
4
5
-2
-4
-6
-8
-10
⎛ ⎞
y = ⎜1⎟
⎝ 3⎠
−x
y=3
10
x
8
6
4
2
-5 -4 -3 -2 -1
1
2
3
4
5
-2
-4
-6
-8
-10
⎛ ⎞
y =⎜ 2⎟
⎝5⎠
−2 x
⎛ 5⎞
y=⎜ ⎟
⎝ 2⎠
2x
⎛ 25 ⎞
or y = ⎜ ⎟
⎝ 4 ⎠
10
x
8
6
4
2
-5 -4 -3 -2 -1
1
2
3
4
5
-2
-4
-6
-8
-10
y = 1.25− x
x
⎛ 1 ⎞ ⎛ 4⎞
y=⎜
⎟ =⎜ ⎟
⎝ 1.25 ⎠ ⎝ 5 ⎠
10
x
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
y = 6− x
⎛1⎞
y=⎜ ⎟
⎝ 6⎠
10
x
8
6
4
2
-5 -4 -3 -2 -1
1
2
3
4
5
-2
-4
-6
-8
-10
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 54 of 131
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Exponent, Log, & Logistic Fctn - B
Rate of Change of Exponential Functions
Name
Let’s compare the properties of some exponential function graphs. Fill in the following table.
Use your calculator if you need to:
1. f ( x ) = 3x
f(0) = ___
(y-intercept)
f(1) = ___
f(2) = ___
f(3) = ___
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
3. h( x ) = 2 x
(y-intercept)
h(2) = ___
h(3) = ___
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
10
8
6
4
2
g(0) = ___
(y-intercept)
2 4 6 8 10
g(1) = ___
g(2) = ___
g(3) = ___
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
4. j ( x ) = 4 x
10
8
6
4
2
h(0) = ___
h(1) = ___
2. g ( x ) = 3.5x
10
8
6
4
2
10
8
6
4
2
j(0) = ___
(y-intercept)
2 4 6 8 10
j(1) = ___
j(2) = ___
j(3) = ___
2 4 6 8 10
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
2 4 6 8 10
To compare the rate of change of these functions, we will find the slopes between points on these
graphs as follows:
Average Rate of
Average Rate of
Average Rate of
Change From 0 To 1 Change From 1 To 2 Change From 2 To 3
For f:
f (1) − f (0)
=
1− 0
f (2) − f (1)
=
2 −1
f (3) − f (2)
=
3− 2
For g:
g (1) − g (0)
=
1− 0
g (2) − g (1)
=
2 −1
g (3) − g (2)
=
3− 2
For h:
h (1) − h (0)
=
1− 0
h (2) − h (1)
=
2 −1
h (3) − h (2)
=
3−2
For j:
j (1) − j (0)
=
1− 0
j (2) − j (1)
=
2 −1
j (3) − j (2)
=
3−2
Arrange the four functions from steepest (increasing most quickly) to shallowest.
Would you expect these patterns to continue for higher values of x? Why or why not?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 55 of 131
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Exponent, Log, & Logistic Fctn - B
Rate of Change of Exponential Functions
Answer Key
Let’s compare the properties of some exponential function graphs. Fill in the following table.
Use your calculator if you need to:
1. f ( x ) = 3x
f(0) = 1
(y-intercept)
f(1) = 3
f(2) = 9
f(3) = 27
3. h( x ) = 2 x
h(0) = 1
(y-intercept)
h(1) = 2
h(2) = 4
h(3) = 8
2. g ( x ) = 3.5x
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
g(0) = 1
(y-intercept)
2 4 6 8 10
g(1) = 3.5
g(2) = 12.25
g(3) = 42.875
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
4. j ( x ) = 4 x
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
10
8
6
4
2
10
8
6
4
2
j(0) = 1
(y-intercept)
2 4 6 8 10
2 4 6 8 10
j(1) = 4
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
j(2) = 16
j(3) = 64
2 4 6 8 10
To compare the rate of change of these functions, we will find the slopes between points on these
graphs as follows:
Average Rate of
Average Rate of
Average Rate of
Change From 0 To 1 Change From 1 To 2 Change From 2 To 3
For f:
f (1) − f (0)
=2
1− 0
f (2) − f (1)
=6
2 −1
f (3) − f (2)
= 18
3− 2
For g:
g (1) − g (0)
= 2.5
1− 0
g (2) − g (1)
= 8.75
2 −1
g (3) − g (2)
= 30.625
3− 2
For h:
h (1) − h (0)
=1
1− 0
h (2) − h (1)
=2
2 −1
h (3) − h (2)
=4
3−2
For j:
j (1) − j (0)
=3
1− 0
j (2) − j (1)
= 12
2 −1
j (3) − j (2)
= 48
3−2
Arrange the four functions from steepest (increasing most quickly) to shallowest.
j, g, f, h
Would you expect these patterns to continue for higher values of x? Why or why not?
Yes, because you are multiplying bases of numbers greater than 1.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - B
Let’s Graph Logarithmic Functions
Name
1. Given the function f ( x ) = 2 ,
a) Complete the table below for f(x).
x
x
-1
0
1
2
3
b) Graph f(x) using the ordered pairs from (a).
y
c) Graph the inverse of f(x) on the same
coordinate plane as f(x). Write down the
five new ordered pairs on the inverse graph.
x
y
The inverse of the exponential function that you have just graphed is called a Logarithmic
Function. We will now find the equation for this inverse of f(x).
x
d) Using y = 2 , what is the first thing you have to do to find the inverse? Write down your
new equation.
Write this new equation in logarithmic form. This equation is the inverse of f(x). Use the
ordered pairs from part (c) to check that your equation is correct.
e) What is the domain of f(x)?
What is the range of f(x)?
What is the domain of the inverse of f(x)?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 57 of 131
What is the range of the inverse of f(x)?
Columbus Public Schools 7/20/05
2. Given the function f ( x ) = 3 ,
a) Complete the table below for f(x).
x
x
-1
0
1
2
Exponent, Log, & Logistic Fctn - B
b) Graph f(x) using the ordered pairs from (a).
y
c) Graph the inverse of f(x) on the same
coordinate plane as f(x). Write down the
four new ordered pairs on the inverse graph.
x
y
x
d) Using y = 3 , what is the first thing you have to do to find the inverse? Write down your
new equation.
Write this new exponential equation in logarithmic form. Use the ordered pairs from part (c)
to check that your equation is correct.
e) What is the domain of f(x)?
What is the range of f(x)?
What is the domain of the inverse of f(x)?
What is the range of the inverse of f(x)?
x
3. Based on #1 and #2, if the exponential function f(x) = b is given, what is the inverse
function of f(x)?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 58 of 131
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Exponent, Log, & Logistic Fctn - B
Let’s Graph Logarithmic Functions
Answer Key
1. Given the function f ( x ) = 2 ,
a) Complete the table below for f(x).
x
x
-1
0
1
2
3
b) Graph f(x) using the ordered pairs from (a).
y
½
1
2
4
8
c) Graph the inverse of f(x) on the same
coordinate plane as f(x). Write down the
five new ordered pairs on the inverse graph.
x
½
1
2
4
8
y
-1
0
1
2
3
The inverse of the exponential function that you have just graphed is called a Logarithmic
Function. We will now find the equation for this inverse of f(x).
x
d) Using y = 2 , what is the first thing you have to do to find the inverse? Write down your
new equation.
Switch x and y.
x = 2y
Write this new exponential equation in logarithmic form. This equation is the inverse of f(x).
Use the ordered pairs from part (c) to check that your equation is correct.
y = log2x
e) What is the domain of f(x)?
What is the range of f(x)?
(0, ∞ )
(-∞ , ∞ )
What is the domain of the inverse of f(x)?
(0, ∞ )
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
What is the range of the inverse of f(x)?
(-∞ , ∞ )
Page 59 of 131
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2. Given the function f ( x ) = 3 ,
a) Complete the table below for f(x).
x
x
-1
0
1
2
Exponent, Log, & Logistic Fctn - B
b) Graph f(x) using the ordered pairs from (a).
y
1/3
1
3
9
c) Graph the inverse of f(x) on the same
coordinate plane as f(x). Write down the
four new ordered pairs on the inverse graph.
x
1/3
1
3
9
y
-1
0
1
2
x
d) Using y = 3 , what is the first thing you have to do to find the inverse? Write down your
new equation.
Switch x and y.
x = 3y
Write this new equation in logarithmic form. Use the ordered pairs from part (c) to check
that your equation is correct.
y = log3x
e) What is the domain of f(x)?
What is the range of f(x)?
(0, ∞)
(-∞, ∞ )
What is the domain of the inverse of f(x)?
(0, ∞)
What is the range of the inverse of f(x)?
(-∞, ∞ )
x
3. Based on #1 and #2, if the exponential function f(x) = b is given, what is the inverse
function of f(x)?
y =f -1(x) = logbx
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponential Growth and Decay Exponent, Log, & Logistic Fctn - C
Name
The processes of uninhibited population growth and radioactive decay both follow the
exponential model P(t) = Po(ekt) where Po is the initial amount and P(t) is the amount after time t,
k is positive for processes which grow over time, and is negative for processes which decay.
Complete the following of growth or decay:
1. If a city has a population of 340 people, and if the population grows continuously at an
annual rate of 2.3%, what will the population be in 6 years?
2. A certain radioactive isotope has a half-life of 37 years. How many years will it take for 100
grams to decay to 64 grams?
3. If a small island has a population of 420 people, and if the population doubles every 9 years,
what will the population be in 7 years?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 61 of 131
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Exponent, Log, & Logistic Fctn - C
4. A certain radioactive isotope decays from 70 grams to 54 grams in 27 years. What is the
half-life of the isotope?
5. A certain radioactive isotope has a half-life of 16 days. If one starts with 15 grams of the
isotope, how much is left after 4 days?
6. A certain radioactive isotope has a half-life of 14 years. If 19 grams of the isotope are left
after 5 years, how much was present at the beginning?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 62 of 131
Columbus Public Schools 7/20/05
Exponential Growth and Decay Exponent, Log, & Logistic Fctn - C
Answer Key
The processes of uninhibited population growth and radioactive decay both follow the
exponential model P(t) = Po(ekt) where Po is the initial amount and P(t) is the amount after time t,
k is positive for processes which grow over time, and is negative for processes which decay.
Complete the following of growth or decay:
1. If a city has a population of 340 people, and if the population grows continuously at an
annual rate of 2.3%, what will the population be in 6 years?
P(t) = Po(ekt)
P(6) = 340(e0.023(6))
Po = 340
P(6) = 390 people
k = 0.023
t=6
2. A certain radioactive isotope has a half-life of 37 years. How many years will it take for 100
grams to decay to 64 grams?
P(t) = Po(ekt)
Po
= Po ( e 37 k )
2
1
= e 37 k
2
⎛1⎞
ln ⎜ ⎟ = ln ( e 37 k )
⎝ 2⎠
k=
1 ⎛1⎞
ln ⎜ ⎟
37 ⎝ 2 ⎠
P(t) = Po(ekt)
64 = 100(ekt)
64
= e kt
100
⎛ 64 ⎞
kt
ln ⎜
⎟ = ln ( e )
⎝ 100 ⎠
⎛ 64 ⎞
37 ln ⎜
⎟
1 ⎛ 64 ⎞
⎝ 100 ⎠ = 23.82 years
t = ln ⎜
=
k ⎝ 100 ⎟⎠
⎛1⎞
ln ⎜ ⎟
⎝ 2⎠
3. If a small island has a population of 420 people, and if the population doubles every 9 years,
what will the population be in 7 years?
P(t) = Po(ekt)
P(t) = Po(ekt)
840 = 420(e9k)
P(7) = 420(e7k)
2 = e9k
ln 2
k= 9
P(7) = 720 people
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 63 of 131
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Exponent, Log, & Logistic Fctn - C
4. A certain radioactive isotope decays from 70 grams to 54 grams in 27 years. What is the halflife of the isotope?
1
54 = 70e 27 k
= e kt
2
1 ⎛ 54 ⎞
⎛1⎞
ln ⎜ ⎟
ln ⎜ ⎟ = ln ( e kt )
k=
27 ⎝ 70 ⎠
⎝ 2⎠
⎛1⎞
kt = ln ⎜ ⎟
⎝ 2⎠
t = 72.12 years
5. A certain radioactive isotope has a half-life of 16 days. If one starts with 15 grams of the
isotope, how much is left after 4 days?
15
= 15e16 k
2
ln 2
k=−
16
P(4) = 15e4k = 12.61 grams
6. A certain radioactive isotope has a half-life of 14 years. If 19 grams of the isotope are left
after 5 years, how much was present at the beginning?
1
= e 14 k
2
ln 2
k=−
14
19 = Poe5k
Po = 24.34 grams
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 64 of 131
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Exponential/Logarithmic Functions – C
Discovering Logarithmic Identities
Name
Each of the following equations can be solved by turning the logarithmic equation into an
equivalent exponential equation. In each set, see if you can determine what the general rule is
supposed to be before you get to the end:
A. log 2 2 =
log 7 7 =
log 5 5 =
log 8 8 =
General Rule: log a a =
B. log 4 1 =
log 3 1 =
log 5 1 =
log 2 1 =
General Rule: log b 1 =
C. log 3 (32 ) =
log5 (54 ) =
log 2 (27 ) =
log 8 (8 3 ) =
log 14 14 30 =
log(105 ) =
General Rule: log b (b g ) =
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 65 of 131
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Discovering Logarithmic Identities
Answer Key
Exponent, Log, & Logistic Fctn - C
Each of the following equations can be solved by turning the logarithmic equation into an
equivalent exponential equation. In each set, see if you can determine what the general rule is
supposed to be before you get to the end:
A. log 2 2 = x
2x = 2
x=1
log 7 7 = x
7x = 7
x=1
log 5 5 = x
5x = 5
x=1
log 8 8 = x
8x = 8
x=1
General Rule: log a a = 1
ax = a
B. log 4 1 = x
4x = 1
x=0
log 3 1 = x
3x = 1
x=0
log 5 1 = x
5x = 1
x=0
log 2 1 = x
2x = 1
x=0
General Rule: log b 1 = x
bx = 1
C. log 3 (32 ) = x
3x = 32
x=2
log5 (54 ) = x
5x = 54
x=4
log 2 (27 ) = x
2x = 27
x=7
log 8 (8 3 ) = x
8x = 83
x=3
log 14 14 30 = x
14x = 1430
x = 30
log(105 ) = x
10x = 105
x=5
General Rule: log b (b g ) = x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
bx = bg
Page 66 of 131
x=1
x=0
x=g
Columbus Public Schools 7/20/05
Logarithm Combination Laws
Exponent, Log, & Logistic Fctn - C
Name
Using the TI-83 graphing calculator, answer all parts for each of the following:
A. If 21 = 7 ⋅ x , then what does x have to be?
log(21) = _____
log(7) = _____ log(x) = _____
What is the relationship between these last three answers?
B. If 30 = y ⋅ 5 , then what does y have to be?
log(30) = _____
log(y) = _____ log(5) = _____
What is the relationship between these last three answers?
C. If 72 ÷ 8 = x , then what is x?
log(72)= _____ log(8)= _____ log(x) = _____
What is the relationship between these three last answers?
D. If 243 = 3x, then what value of x will make this true?
log(243) = _____ log(3) = _____
What relationship do you see between these last two answers and x?
E. If 7 = 49 x , then what is x?
log(7) = _____ log(49) = _____
What relationship do we see now? (Your answer should be related to part D above).
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 67 of 131
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Logarithm Combination Laws
Answer Key
Exponent, Log, & Logistic Fctn - C
Using the TI-83 graphing calculator, answer all parts for each of the following.
A. If 21 = 7 ⋅ x , then what does x have to be? x = 3
log(21) = 1.32
log(7) = .85 log(3) = .47
What is the relationship between these last three answers?
log 21 = log 7 + log 3
B. If 30 = y ⋅ 5 , then what does y have to be? y = 6
log(30) = 1.4771 log(6) = .7782 log(5) = .6990
What is the relationship between these last three answers?
log 30 = log 6 + log 5
C. If 72 ÷ 8 = x , then what is x? x = 9
log(72) = 1.8573 log(8) = .9030 log(9) = .9542
What is the relationship between these three last answers?
log 72 = log 8 + log 9 or log 72 – log 8 = log 9
D. If 243 = 3 x , then what value of x will make this true? x = 5
log(243) = 2.38560 log (3) = 0.47712
What relationship do you see between these last two answers and x?
The first log is 5 times the second.
E. If 7 = 49 x , then what is x? x =
1
2
log(7) = .8451 log(49) = 1.6902
What relationship do we see now? (Your answer should be related to part D above).
The second is double the first.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 68 of 131
Columbus Public Schools 7/20/05
Log Cut Out Puzzle
Exponent, Log, & Logistic Fctn - C
Name
All bases are positive.
Cut out the squares.
Arrange them so that touching
edges are equivalent equations.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 69 of 131
Columbus Public Schools 7/20/05
Log Cut Out Puzzle
Answer Key
Exponent, Log, & Logistic Fctn - C
Note: This is only one possible answer. There may be others.
1
2
3
4
8
5
6
7
9
10
11
12
13
14
15
16
2
12
3
15
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
1
6
9
8
7
10
13
16
11
14
5
4
Page 70 of 131
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Properties of Logarithms
Exponent, Log, & Logistic Fctn - C
Name
Part I:
1. Evaluate log28 + log24.
2. Think of a logarithm expression with base 2 whose value is the same as the value in #1.
3. Evaluate log33 + log39.
4. Think of a logarithm expression with base 3 whose value is the same as the value in #3.
5. Evaluate log1 + log1000.
6. Think of a common logarithm expression whose value is the same as the value in #5.
7. What pattern do you see in the problems above?
How can you write logbu + logbv as a single logarithm?
Part II:
8. Evaluate log264 – log28.
9. Think of a logarithm expression with base 2 whose value is the same as the value in #8.
10. Evaluate log381 – log327.
11. Think of a logarithm expression with base 3 whose value is the same as the value in #10.
12. Evaluate log10,000 – log10.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 71 of 131
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Exponent, Log, & Logistic Fctn - C
13. Think of a common logarithm expression whose value is the same as the value in #12.
14. What pattern do you see in the problems above?
How can you write logbu – logbv as a single logarithm?
Part III:
15. Evaluate log243
16. Think of a logarithm expression with base 2 (besides log264) whose value is the same as the
value in #15.
17. Evaluate log3272.
18. Think of a logarithm expression with base 3 (besides log3729) whose value is the same as the
value in #17.
19. Evaluate log104.
20. Think of a common logarithm expression (besides log10,000) whose value is the same as the
value in #12.
21. What pattern do you see in the problems above?
k
How else can you write logbu ?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 72 of 131
Columbus Public Schools 7/20/05
Properties of Logarithms
Answer Key
Exponent, Log, & Logistic Fctn - C
Name
Part I:
1. Evaluate log28 + log24.
3+2=5
2. Think of a logarithm expression with base 2 whose value is the same as the value in #1.
log232
3. Evaluate log33 + log39.
1+2=3
4. Think of a logarithm expression with base 3 whose value is the same as the value in #3.
log327
5. Evaluate log1 + log1000.
0+3=3
6. Think of a common logarithm expression whose value is the same as the value in #5.
log1000
7. What pattern do you see in the problems above?
The logarithm of a product is the sum of the logarithms of its factors.
How can you write logbu + logbv as a single logarithm?
logb(uv) = logbu + logbv
Part II:
8. Evaluate log264 – log28.
6–3=3
9. Think of a logarithm expression with base 2 whose value is the same as the value in #8.
log28
10. Evaluate log381 – log327.
4–3=1
11. Think of a logarithm expression with base 3 whose value is the same as the value in #10.
log33
12. Evaluate log10,000 – log10.
4–1=3
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 73 of 131
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Exponent, Log, & Logistic Fctn - C
13. Think of a common logarithm expression whose value is the same as the value in #12.
log1000
14. What pattern do you see in the problems above?
The logarithm of a quotient is the difference of the logarithms of the numerator and
the denominator.
How can you write logbu – logbv as a single logarithm?
u
logb ⎛⎜ ⎞⎟ = logbu - logbv
⎝v⎠
Part III:
15. Evaluate log243
6
16. Think of a logarithm expression with base 2 (besides log264) whose value is the same as the
value in #15.
3log24
17. Evaluate log3272.
6
18. Think of a logarithm expression with base 3 (besides log3729) whose value is the same as the
value in #17.
2log327
19. Evaluate log104.
4
20. Think of a common logarithm expression (besides log10,000) whose value is the same as the
value in #12.
4log10
21. What pattern do you see in the problems above?
The exponent can be written as a coefficient of the logarithm expression. The
logarithm of a power is the product of the logarithm and the exponent.
k
How else can you write logbu ?
klogbu
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 74 of 131
Columbus Public Schools 7/20/05
Exercises with Logarithms
Exponent, Log, & Logistic Fctn - C
Name
1. Which numbers x satisfy the equation: (log3x)(logx5) = log35 ?
2.
Suppose that the Canadian dollar loses 5% of its value each year. How many years are
needed in order that the Canadian dollar to lose 90% of its value? (That is, the future value
of the dollar to become the present value of a dime.)
3. Simplify the product: P = (log23)(log34)(log45) ... (log3132)
4. If p =
log b (log a a 2 )
log b a
find ap.
5. If log b (xy) = 11 and log b (x/y) = 5, what is log b x?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 75 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - C
6. Positive integers A, B, and C, with no common factor greater than 1, exist such that
A log 200 5 + B log 200 2 = C. What is A + B + C?
7. What is the value of 25log5
8.
2
?
A computer manufacturer finds that when x millions of dollars are spent on research, the
profit, P(x), in millions of dollars, is given by P ( x) = 20 + 5log 3 ( x + 3) . How much should
be spent on research to make a profit of 40 million dollars?
9. Solve the system of equations y = log 2 2 x and y = log 4 x for all x.
10. Solve the equation log3 (x - 2) + log3 10 = log3 (x2 + 3x – 10)
11. log2 (9 - 2x) = 3 – x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 76 of 131
Columbus Public Schools 7/20/05
Exercises with Logarithms
Answer Key
Exponent, Log, & Logistic Fctn - C
1. Which numbers x satisfy the equation: (log3x)(logx5) = log35 ?
All x > 0, x ≠ 1
2.
Suppose that the Canadian dollar loses 5% of its value each year. How many years are
needed in order that the Canadian dollar to lose 90% of its value? (That is, the future value
of the dollar to become the present value of a dime.)
About 45 years
3. Simplify the product: P = (log23)(log34)(log45) ... (log3132)
P=5
log b (log a a 2 )
4. If p =
log b a
find ap.
ap = 2
5. If log b (xy) = 11 and log b (x/y) = 5, what is log b x?
8
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 77 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - C
6. Positive integers A, B, and C, with no common factor greater than 1, exist such that
A log 200 5 + B log 200 2 = C. What is A + B + C?
6
7. What is the value of 25log5
2
?
2
8.
A computer manufacturer finds that when x millions of dollars are spent on research, the
profit, P(x), in millions of dollars, is given by P ( x) = 20 + 5log 3 ( x + 3) . How much should
be spent on research to make a profit of 40 million dollars?
78 million
9. Solve the system of equations y = log 2 2 x and y = log 4 x .
(.25, -1)
10. Solve the equation log3 (x - 2) + log3 10 = log3 (x2 + 3x – 10)
5
11. log2 (9 - 2x) = 3 – x
0, 3
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 78 of 131
Columbus Public Schools 7/20/05
Time Estimation
Exponent, Log, & Logistic Fctn - C
Name
For this activity, we will be looking at the compound interest formula from another angle.
Recall the compound interest formula:
A = P(1+ nr ) n⋅t
1. Determine what the compound interest formula would look like if we are compounding
annually at 11% interest. Assume $100 to start. You should have a function for A in terms
of t.
2. Put this formula into the TI-83 graphing calculator (you’ll be using x instead of t for this.
What x- and y-windows would make sense here? Decide how many years you wish to
consider, and sketch the graph you get.
3. Using the trace function on the calculator, determine approximately how many years it would
take for our investment to:
a) double (this happens when y reaches 200)
b) triple
c) quadruple
d) multiply by 10
4. If our interest is 8% instead of 11%, would you expect it to take more or less time? Graph
the relevant amount-vs.-time function, check your hypothesis, and justify your answer.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 79 of 131
Columbus Public Schools 7/20/05
Time Estimation
Answer Key
Exponent, Log, & Logistic Fctn - C
For this activity, we will be looking at the compound interest formula from another angle.
Recall the compound interest formula:
A = P(1+ nr ) n⋅t
1. Determine what the compound interest formula would look like if we are compounding
annually at 11% interest. Assume $100 to start. You should have a function for A in terms
of t.
A = 100(1 + .11)t
2. Put this formula into the TI-83 graphing calculator (you’ll be using x instead of t for this.
What x- and y-windows would make sense here? Decide how many years you wish to
consider, and sketch the graph you get.
500
450
400
350
300
250
200
150
100
50
5
10
15
20
3. Using the trace function on the calculator, determine approximately how many years it would
take for our investment to:
a) double (this happens when y reaches 200)
6.64 years
b) triple
10.53 years
c) quadruple
13.28 years
d) multiply by 10
22.06 years
4. If our interest is 8% instead of 11%, would you expect it to take more or less time? Graph
the relevant amount-vs.-time function, check your hypothesis, and justify your answer.
500
More time, the graph is not as steep,
the values increase more slowly.
450
400
350
300
250
200
150
100
50
5
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 80 of 131
10
15
20
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - C
Why Settle for Time Estimation When You Can Be
Exact?
Name
Luckily, logarithms allow us to solve for the t-variable in the compound interest formula. Let’s
see how we can use this fact to find out how long it takes any amount of money to double, triple,
etc.
1. Rewrite the compound interest formula with the appropriate substitutions to indicate 11%
interest, $100 to start, and annual compounding. Don’t forget to substitute the correct value
for A also (how much should it be if we’re doubling our investment?). What does the
formula now look like?
2. Solve this formula for t. See how this compares with your approximation in exercise 3a of
the last activity.
3. Now create and solve an appropriate equation to represent tripling of our investment.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 81 of 131
Columbus Public Schools 7/20/05
4. Do the same for quadrupling.
Exponent, Log, & Logistic Fctn - C
5. How long, in exact terms, should it take to multiply our initial investment by 10?
6. Observe the relationship between each problem above and the exact answer (in terms of
logarithms). Without going through all the steps of solving an exponential equation, how
long should it take our investment to double if the interest rate were 8%? How should our
numerical answer compare with number 2 above? Be sure to justify your answers.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 82 of 131
Columbus Public Schools 7/20/05
Exponent, Log, & Logistic Fctn - C
Why Settle for Time Estimation When You Can Be
Exact?
Answer Key
Luckily, logarithms allow us to solve for the t-variable in the compound interest formula.. Let’s
see how we can use this fact to find out how long it takes any amount of money to double, triple,
etc.
1. Rewrite the compound interest formula with the appropriate substitutions to indicate 11%
interest, $100 to start, and annual compounding. Don’t forget to substitute the correct value
for A also (how much should it be if we’re doubling our investment?). What does the
formula now look like?
t
Answer: 200 = 100 (1 + .11)
2. Solve this formula for t. See how this compares with your approximation in exercise 3a of
the last activity.
Answer:
t
200 = 100 (1 + .11)
2 = (1.11)
t
log 2 = t log1.11
log 2
log1.11
This evaluates to 6.64 years, just like the graph.
t=
3. Now create and solve an appropriate equation to represent tripling of our investment.
Answer:
t
300 = 100 (1 + .11)
3 = (1.11)
t
log 3 = log1.11
t=
log 3
log1.11
This evaluates to 10.53 years, just like the graph.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 83 of 131
Columbus Public Schools 7/20/05
4. Do the same for quadrupling.
Exponent, Log, & Logistic Fctn - C
Answer:
t
400 = 100 (1 + .11)
4 = (1.11)
t
log 4 = t log1.11
t=
log 4
log1.11
This evaluates to 13.28 years, just like the graph.
5. How long, in exact terms, should it take to multiply our initial investment by 10?
Answer:
t
1000 = 100 (1 + .11)
10 = (1.11)
t
log10 = t log1.11
t=
log10
log1.11
This evaluates to 22.06 years, just like the graph.
6. Observe the relationship between each problem above and the exact answer (in terms of
logarithms). Without going through all the steps of solving an exponential equation, how
long should it take our investment to double if the interest rate were 8%? How should our
numerical answer compare with number 2 above? Be sure to justify your answers.
Answer: t =
log 2
log1.08
Since log 1.08 is less than log 1.11, t should be greater.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 84 of 131
Columbus Public Schools 7/20/05
The Rule of 72
Exponent, Log, & Logistic Fctn - C
Name
Directions: For each row of the chart below, we are trying to double our investment. To do this,
you will need to calculate the final amount A (double our investment) and then use the formula
rt
for continuous compound interest A = Pe to find the missing variable (round to 4 decimal
places).
P
$500
A
r
.06
$2000
.08
$40
.13
t
$70
12 years
$900
7 years
$13,000
10 years
$50,000
9 years
$250,000
5 years
What is the relationship you observe between our values for r and t?
Solve the continuous compound interest formula for the expression rt. How does this compare
with our last observation?
This relationship is referred to as the "Rule of 72". Why do you think the number 72 is used
here?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 85 of 131
Columbus Public Schools 7/20/05
The Rule of 72
Answer Key
Exponent, Log, & Logistic Fctn - C
Directions: For each row of the chart below, we are trying to double our investment. To do this,
you will need to calculate the final amount A (double our investment) and then use the formula
rt
for continuous compound interest A = Pe to find the missing variable (round to 4 decimal
places).
P
$500
A
$1000
r
.06
t
11.5525 years
$2000
$4000
.08
8.6643 years
$40
$80
.13
5.3319 years
$70
$140
.0578
12 years
$900
$1800
.099
7 years
$6500
$13,000
.0693
10 years
$25,000
$50,000
.0770
9 years
$125,000
$250,000
.1386
5 years
What is the relationship you observe between our values for r and t?
They multiply to 0.693.
Solve the continuous compound interest formula for the expression rt. How does this compare
with our last observation?
A = Pert
A
= ert 2 = ert ln 2 = rt
P
This agrees because ln (2) ≈ 0.693
This relationship is referred to as the "Rule of 72". Why do you think the number 72 is used?
72 has many more factors than 69, and it gives approximate results, though not as accurate
as if we said rate times time.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 86 of 131
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USA TODAY Snapshot – More of U.S.
Name
The USA TODAY Snapshot - "More of U.S." shows the population (in millions) of the United
States from 1940 through 1997. You will use an exponential function to model the growth of the
population over time. Population growth often is restricted while the exponential growth model
is not restricted. However, the population data often behaves as an exponential function over a
limited time period. You will use the model to make a prediction about the population for a
known year and then compare this value with the actual population. Finally, you will be given a
population and determine the year this population figure was attained.
MATH TODAY STUDENT EDITION
Focus Questions:
1. Assume that the population of the United States is growing exponentially. What is the
exponential function that best models the data provided?
2. What is the projected population in this Snapshot for 1997? What is the percent error in the
estimated population compared to the actual population?
3. Determine when the U.S. population reached 100 million.
MATH TODAY STUDENT EDITION PAGE 2
Data Source:
U.S. Census Bureau
Activity 1: Assume that the population of the United States is growing exponentially. What
is the exponential function that best models the data provided?
A. Use the 1900-2000 data in the table below to create a scatterplot on the handheld. Let 0
represent 1900, 10 represent 1910, and so forth.
Year
0 10 20 30 40 50 60 70 80 90 100
Pop.
U.S.
76 92 106 123 132 151 179 203 227 249 281
(millions)
Population
(millions)
Year
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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B. Use the regression capabilities of the handheld to determine the exponential regression model
for the data set.
Exponential regression model: __________________________________
Activity 2: What is the projected population for 1997? What is the percent error in the
estimated population compared to the actual population?
A. Graph the scatterplot and the exponential regression model in the same window.
Population
(millions)
Year
B. Trace the regression model to find the projected population for 1997.
Projected population for 1997:
C. Compare the projected population to the listed population in the Snapshot "More of U.S." for
1997. What is the percent error in your projected population? Note: The population at the end
of 1997 was 269 million.
___________________________________________________________
Activity 3: Determine when the U.S. population reached 100 million.
When did the U.S. population reach 100 million?
U.S. Population will reach 100 million in:
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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USA TODAY Snapshot – More of U.S.
Answer Key
THE NATION’S NEWSPAPER
The USA TODAY Snapshot - "More of U.S." shows the population (in millions) of the United
States from 1940 through 1997. You will use an exponential function to model the growth of the
population over time. Population growth often is restricted while the exponential growth model
is not restricted. However, the population data often behaves as an exponential function over a
limited time period. You will use the model to make a prediction about the population for a
known year and then compare this value with the actual population. Finally, you will be given a
population and determine the year this population figure was attained.
MATH TODAY STUDENT EDITION
Focus Questions:
1. Assume that the population of the United States is growing exponentially. What is the
exponential function that best models the data provided?
2. What is the projected population in this Snapshot for 1997? What is the percent error in the
estimated population compared to the actual population?
3. Determine when the U.S. population reached 100 million.
MATH TODAY STUDENT EDITION PAGE 2
Data Source:
U.S. Census Bureau
Activity 1: Assume that the population of the United States is growing exponentially. What
is the exponential function that best models the data provided?
A. Use the 1900-2000 data in the table below to create a scatterplot on the handheld. Let 0
represent 1900, 10 represent 1910, and so forth.
Year
0 10 20 30 40 50 60 70 80 90 100
Pop.
U.S.
76 92 106 123 132 151 179 203 227 249 281
(millions)
Population
(Millions)
Years
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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B. Use the regression capabilities of the handheld to determine the exponential regression model
for the data set.
Exponential regression model: y = 80.3741(1.01291^x)
Activity 2: What is the projected population for 1997? What is the percent error in the
estimated population compared to the actual population?
A. Graph the scatterplot and the exponential regression model in the same window.
Population
(Millions)
Years
B. Trace the regression model to find the projected population for 1997.
Projected population for 1997: 278.97086 million
C. Compare the projected population to the listed population in the Snapshot "More of U.S." for
1997. What is the percent error in your projected population? Note: The population at the end
of 1997 was 269 million.
3.7 %
Activity 3: Determine when the U.S. population reached 100 million.
When did the U.S. population reach 100 million?
U.S. Population will reach 100 million in: 1917
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Determining the Half-Life of Hydrogen-3
Name
1. Hydrogen-3, an isotope of Hydrogen, decays at the rate of 5.59% per year. If we start with
10 grams of Hydrogen-3, how much would be left at the end of one year? What percent of
the original amount remains at the end of one year? Record your answers for years one
through six in the table below.
Number of Years
0
1
2
3
4
5
6
Amount of Hydrogen-3
Remaining (in grams)
10
Percent of Original
Hydrogen-3 Amount
Remaining
100.00%
2. The time required for half of a radioactive substance to decay is called the half-life of that
substance. Add rows to the table above until the percentage of the original amount goes
below 50%. Complete the extended table. Approximately how many years did it take to
reach this 50% mark? This is our approximate half-life for hydrogen-3.
3. The general formula for using half-life to determine the amount of radioactive decay is:
t
S = I ⋅ (0.5) h where I = the initial amount of hydrogen-3, S = the remaining amount of
hydrogen-3, and t = amount of time. Use one of your rows of data from the table above to
substitute values (number of years, amount remaining, and our initial amount of 10g) until
there is only one variable left, h.
4. Graph the equation on a TI-83 graphing calculator. Use your graph to determine what
x-value would represent the half-life of hydrogen-3. Explain why you chose this particular
place on the graph, and what that point represents.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Determining the Half-Life of Hydrogen-3
Answer Key
1. Hydrogen-3, an isotope of Hydrogen, decays at the rate of 5.59% per year. If we start with
10 grams of Hydrogen-3, how much would be left at the end of one year? What percent of
the original amount remains at the end of one year? Record your answers for years one
through six in the table below.
Percent of Original
Amount of Hydrogen-3
Number of Years
Hydrogen-3 Amount
Remaining (in grams)
Remaining
0
1
2
3
4
5
6
10
9.4410
8.9133
8.4150
7.9446
7.5005
7.0812
100.00%
94.41%
89.13%
84.15%
79.45%
75.01%
70.81%
2. The time required for half of a radioactive substance to decay is called the half-life of that
substance. Add rows to the table above until the percentage of the original amount goes
below 50%. Complete the extended table. Approximately how many years did it take to
reach this 50% mark? This is our approximate half-life for hydrogen-3.
12 years
3. The general formula for using half-life to determine the amount of radioactive decay is:
t
S = I • (0.5) h where I = the initial amount of hydrogen-3, S = the remaining amount of
hydrogen-3, and t = amount of time. Use one of your rows of data from the table above to
substitute values (number of years, amount remaining, and our initial amount of 10g) until
there is only one variable left, h.
Student answers will vary depending on which line of the table the student uses. For
example, when t = 2 years, you get S = 10(.5)(2/h).
4. Graph the equation on a TI-83 graphing calculator. Use your graph to determine what
x-value would represent the half-life of hydrogen-3. Explain why you chose this particular
10
place on the graph, and what that point represents.
x = 12.05
Using y = 10(.5)(2/h) and the trace function when
y = 8.91325, then x is our half-life.
5
10
20
30
40
50
-5
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Not Just a Good Idea—It's The Law!
Name
Directions:
1. Use a fork to stick holes in your potato (4 should be enough). Make one hole large enough to
be able to insert a thermometer later.
2. Microwave the potato on high for 5 minutes.
3. Remove the potato and start the stopwatch or timer.
4. Immediately insert the thermometer and record the temperature, as well as the time, in
minutes.
5. Every 5 minutes, re-record the temperature and time on the recording sheet.
6. Keep repeating step 5 until the you get the same temperature readings for 2 consecutive
readings.
7. Determine the room temperature and make a note of it.
time (minutes)
temperature (Fahrenheit or Celsius)
8. Use your TI-83 to graph these points (put time in L1, temperature in L2). Sketch that graph
on the axis below, then try to get an appropriate regression equation to fit the data. (First
determine which type of regression might be the most appropriate).
9. Now for the fun part. Using the information you gathered and Newton’s Law of Cooling,
derive an equation which should model this data.
10. Graph both of the equations from questions 8 and 9 along with a scatterplot of the data.
Which function more closely represents our data? Can you figure out why this is the case?
11. How might you get the exponential regression to work better?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Not Just a Good Idea—It's The Law!
Answer Key
Directions:
1. Use a fork to stick holes in your potato (4 should be enough). Make one hole large enough to
be able to insert a thermometer later.
2. Microwave the potato on high for 5 minutes.
3. Remove the potato and start the stopwatch or timer.
4. Immediately insert the thermometer and record the temperature, as well as the time, in
minutes.
5. Every 5 minutes, re-record the temperature and time on the recording sheet.
6. Keep repeating step 5 until the you get the same temperature readings for 2 consecutive
readings.
7. Determine the room temperature and make a note of it.
Answers will vary. This is one possible result.
time (minutes)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
80
90
100
110
120
130
140
150
160
170
180
190
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
temperature (oFahrenheit or Celsius)
208 ºF
193.4
178.8
168.8
159.4
151.1
144.6
138.3
132.6
127.9
123.4
119.4
115.5
112.4
109.2
104
99.6
95.9
92.6
89.9
87
85.4
83.6
82
80.9
79.8
78.9
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8. Use your TI-83 to graph these points (put time in L1, temperature in L2). Sketch that graph
on the axis below, then try to get an appropriate regression equation to fit the data. (First
determine which type of regression might be the most appropriate).
Exponential Regression Equation:
y = 168.255(.995231x)
9. Now for the fun part. Using the information you gathered and Newton’s Law of Cooling,
derive an equation which should model this data.
Tt = Tm + (T0 − Tm )e − kt
Tt = 74 + (208 − 74)e − kt
Tt = 74 + 134e − kt
use the fact that at t=20 minutes, T=161:
161 = 74 + 134e −20 k
87
= e −20 k
134
⎛ 87 ⎞
ln ⎜
⎝ 134 ⎟⎠
k=
−20
k ≈ 0.02159658
This makes our equation (courtesy of Newton's Law):
Tt = 74 + 134e −0.02159658 t
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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10. Graph both of the equations from questions 8 and 9 along with a scatterplot of the data.
Which function more closely represents our data? Can you figure out why this is the case?
This equation gives us a much closer fit than the calculator's attempt at regression because
bx
the TI doesn't do vertical shifts. It will only give you exponentials of the form a .
11. How might you get the exponential regression to work better?
Here's one way:
In your Stats editor, define L3 to have the values of whatever your room temperature is
taken away from L2's values.
Then do an exponential regression of L1 versus L3; put the results into Y1(X).
Translate L1 upward by whatever your room temperature is (add you room temperature
to the results from your regression).
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Ball Bounce Revisited
Exponent, Log, & Logistic Fctn - D
Name
In the Ball Bounce activity, you found that the graph of
time after the ball released vs. the height of the ball
from the ground formed a series of parabolas. Your
graph should have looked something like this.
In this activity, we are going to investigate how the
maximum height of each parabola changes from bounce
to bounce.
From your ball bounce data, complete the chart, giving
the maximum height of each bounce. (Ignore any extra
spaces; not every group got the same number of
bounces.
Make a scatterplot of the data in your chart. What kind of function
Bounce Height (in ft.)
do you think would model this data? Complete the appropriate
Number
regression, and add the curve to your scatterplot. Give your
1
regression equation and sketch the graph below.
2
3
4
5
6
7
8
You probably found an exponential regression, an equation of the form y = abx. What might a
stand for in the real world situation? What might b stand for in the real-world situation?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Ball Bounce Revisited
Answer Key
Exponent, Log, & Logistic Fctn - D
In the Ball Bounce activity, you found that the graph of time after the
ball released vs. the height of the ball from the ground formed a series
of parabolas. Your graph should have looked something like this.
In this activity, we are going to investigate how the maximum
height of each parabola changes from bounce to bounce.
From your ball bounce data, complete the chart, giving the maximum
height of each bounce. (Ignore any extra spaces; not every group
got the same number of bounces.
Make a scatterplot of the data in your chart. What kind of function
Bounce Height (in ft.)
do you think would model this data? Complete the appropriate
Number
regression, and add the curve to your scatterplot. Give your
1
3.13013
regression equation and sketch the graph below.
2
2.45672
This data is from the sample data given in the Ball Bounce
3
1.95853
Activity in Topic 2-Polynomial, Power, and Rational Functions.
4
1.56214
5
1.24908
6
1.01891
7
8
You probably found an exponential regression, an equation of the form y = abx. Compare your
values with those of other groups. What might a stand for in the real world situation? What
might b stand for in the real-world situation?
The variable, a, could stand for the initial height from which the ball was dropped. The
variable, b, is probably some sort of measure of elasticity or “bounciness.”
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Logistic Growth
Teacher Notes - D
The logistic growth investigation will probably take three classes. On the first day, the class
should visit the computer lab to explore the website and collect the data. This activity has
usually been available on the website http://www.otherwise.com/population/exponent.html, but
has occasionally moved. If it is not there, a Google search using the parameters habitat fish
"logistic growth" applet should find it. It is easily recognizable because clicking “Run applet”
produces a screen that simulates the number of fish in pond with pictures of fish.
The first part of the activity reinforces the students’ previous knowledge of exponential growth.
The second part is an introduction to logistic growth. Each part requires that the teacher assign
each student a particular value to record data for. It is necessary for students to complete the
entire exploration and collect their assigned data for the lesson to be a success. The internet
portion of the investigation should be completed in one class. This is possible so long as the
students remain on task.
The next day, the class should discuss logistic behavior and the constraints that caused the
behavior exhibited. These should include food supply, oxygen supply, disease, predators, and
probably many other factors the students will come up with.
Spend some time looking at the graph of the 1.5 birth rate graph. Discuss the concavity of the
graph, where the population is increasing rapidly, and where the rate of increase begins to slow
down. Estimate the generation at which the change occurs. Complete a logistic regression on
1008.56
, which is of the form
the data. The calculator should give approximately y =
1 + 1331.19e −491234 x
c
y=
. While this is the classic form of a logistic equation in calculus, it isn’t very
1 + ae − kx
helpful for PreCalculus students to gain insight into the problem.
c
. Either of these equations can be obtained
1 + b − mx
from the other, but the second allows the students to approximate the equation without regression
by identifying important parts of the problem situation.
Another way to look at the equation is y =
After looking at the calculator regression, most students will guess that c should be a number
close the carrying capacity. Also, because the graph appears to be exponential at the beginning,
the use of 1.5, the birth rate for b. When y = 1000− x is graphed with data, the graph looks like
1 + 1.5
the graph below.
Looking at the graph and remembering that the concavity
changed at approximately the 15th generation, a horizontal
shift of about 15 seems appropriate.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Teacher Notes - D
1000
1 + 1.5−( x −15 )
By adjusting the equation slightly, you can come up with a
1000
.
good fit with y =
1 + 1.5−1.3( x −14.5 )
y=
The next day, the class should follow up with the “Rumors” activity, the next section in the
teacher notes.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - D
Population Growth Investigation
Name
Go to http://www.otherwise.com/population/exponent.html and select Exponential Growth
from the buttons at the top of the page. Click Reset All to clear any past data. If this website is
not available, do a Google search using habitat fish "logistic growth" applet.
Run the simulation for birth rates 1, 1.2, 1.4, 1.6, 1.8, and 2 for 20 generations . Click Reset (not
Reset all) between each simulation. Your teacher will assign you one of the birth rates to keep
track of the data on the chart. Be sure to record the data for your assigned birth rate on the charts
provided. We will share the data on calculators later. Sketch the composite graph below.
Explain how each the graphs of each birth rate are different or similar. How does increasing the
birth rate change the graph?
Generation
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Population
Now click Reset All to clear any past data. Set the
average birth rate to be 1.5 and Step the population through 15 generations. Now, without
clicking either reset button, change the birth rate to 0.8. This change means that each individual
is now only producing (on average) 0.8 individuals in the next generation. Predict what the
graph will look like up through 30 generations.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Sketch the actual graph below.
Exponent, Log, & Logistic Fctn - D
What must be true of the birth rate for the population to remain constant to increase? to
decrease? How would you change the birth rate to make the population increase very rapidly?
decrease very rapidly? Test your theories using the fish website.
Use the exponential regression function of your calculator to model a function that describes the
growth of the fish population as a function of time for the birth rate you were assigned. Collect
the results from the class for the other birth rates. Describe the results in terms of the general
form of an exponential equation f(x)=abx?
Birth Rate
Function
1
1.2
1.4
1.6
1.8
2.0
What equation would model a population that begins with 1000 fish and whose birth rate is 0.8?
Graph it on your calculator and sketch it here.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - D
Return to the website to http://www.otherwise.com/population/exponent.html and select Logistic
Growth from the buttons at the top of the page.
Population
Click Reset All to clear any past data. Make sure the birth Generation
0
rate is set to 1.5. Now do a series of simulations using
1
carrying capacities of 200, 400, 600, 800, and 1000. Step
2
the population for 30 generations with each value of
3
carrying capacity. Click the Reset (Not the Reset All)
4
button between each simulation. Sketch your composite
5
graph below. Describe the general shape of these graphs.
6
Do they resemble the exponential functions in any way?
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
`
22
23
Now set the carrying capacity at 1000 and use these birth
24
rates: 1.5, 2, 2.5, 3.5, and 4 Click the Reset (Not the Reset 25
All) button between each simulation. Your teacher will
26
assign you a growth rate to record. Be sure to record the
data for your assigned birth rate on the charts provided. Sketch the composite graph below.
Describe the general shape of these graphs. How did changing the birth rates change the graphs?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Repeat the simulation using birth rates of 2, 4, and 6 and a carrying capacity of 1000. Sketch
your graphs below.
Why do these birth rates create such different graphs? Explain this in terms of what could be
happening in the pond. Can you find the minimum birth rate which will cause the population to
become extinct?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 104 of 131
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Exponent, Log, & Logistic Fctn - D
Population Growth Investigation
Answer Key
Go to http://www.otherwise.com/population/exponent.html and select Exponential Growth
from the buttons at the top of the page. Click Reset All to clear any past data. If this website is
not available, do a Google search using habitat fish "logistic growth" applet.
Run the simulation for birth rates 1, 1.2, 1.4, 1.6, 1.8, and 2 for 20 generations. Click Reset (not
Reset all) between each simulation. Your teacher will assign you one of the birth rates to keep
track of the data on the chart. Be sure to record the data for your assigned birth rate on the charts
provided. We will share the data on calculators later. Sketch the composite graph below.
Explain how each the graphs of each birth rate are different or similar. How does increasing the
birth rate change the graph?
Generation
0ANSWERS
1WILL
2VARY
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Population
Now click Reset All to clear any past data. Set the average birth rate to be 1.5 and Step the
population through 15 generations. Now, without clicking either reset button, change the birth
rate to 0.8. This change means that each individual is now only producing (on average) 0.8
individuals in the next generation. Predict what the graph will look like up through 30
generations.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 105 of 131
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Sketch the actual graph below.
Exponent, Log, & Logistic Fctn - D
What must be true of the birth rate for the population to remain constant to increase? to
decrease? How would you change the birth rate to make the population increase very rapidly?
decrease very rapidly? Test your theories using the fish website. Answers will vary.
Constant: Birth rate = 1; Decrease: Birth rate between 0 and 1; Increase Birth rate;
Decrease Birth rate
Use the exponential regression function of your calculator to model a function that describes the
growth of the fish population as a function of time for the birth rate you were assigned. Collect
the results from the class for the other birth rates. Describe the results in terms of the general
form of an exponential equation f(x)=abx?
Birth Rate
Function
1
y=2
1.2
y=2(1.2)x
1.4
y=2(1.4)x
1.6
y=2(1.6)x
1.8
y=2(1.8)x
2.0
y=2(2.0)x
What equation would model a population that begins with 1000 fish and whose birth rate is 0.8?
Graph it on your calculator and sketch it here.
y=1000(.8)x
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 106 of 131
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Return to the website to
http://www.otherwise.com/population/exponent.html and
select Logistic Growth from the buttons at the top of the
page. Click Reset All to clear any past data. Make sure
the birth rate is set to 1.5. Now do a series of simulations
using carrying capacities of 200, 400, 600, 800, and 1000.
Step the population for 30 generations with each value of
carrying capacity. Click the Reset (Not the Reset All)
button between each simulation. Sketch your composite
graph below. Describe the general shape of these graphs.
Do they resemble the exponential functions in any way?
Exponent, Log, & Logistic Fctn - D
Generation
Population
0ANSWERS
1WILL
2VARY
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
`
22
23
Now set the carrying capacity at 1000 and use these birth
rates: 1.5, 2, 2.5, 3.5, and 4 Click the Reset (Not the Reset 24
25
All) button between each simulation. Be sure to record
26
the data for your assigned birth rate on the charts
provided. Sketch the composite graph below. Describe the general shape of these graphs. How
did changing the birth rates change the graphs?
Answers will vary.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 107 of 131
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Exponent, Log, & Logistic Fctn - D
Repeat the simulation using birth rates of 2, 4, and 6 and a carrying capacity of 1000. Sketch
your graphs below.
Why do these birth rates create such different graphs? Explain this in terms of what could be
happening in the pond. Can you find the minimum birth rate which will cause the population to
become extinct?
Answers will vary.
Food supply may run out, the oxygen may run out, etc. Birth rate greater than 4 causes
extinction.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Rumors
Teacher Notes - D
In the Rumor activity, the situation is posed that one class member knows a rumor, and comes to
class and tells one person. The next day each of those students tells another student and so on.
Eventually students who already know the rumor are told rumor again, but eventually everyone
will have heard the rumor. To be effective, there should be more than 25 students in the class. If
the class is small, each student should represent two students.
To begin assign each student a whole number between 1 and the number of students in the class.
(If it is a small class, the upper number would be twice the number of students.). To select the
original student who knows the rumor, on the calculator choose a random number between 1 and
the total number of students. For this example, the total number of students will be 30.
On the MATH menu, choose #5. The syntax is (lowest number, highest number, how many)
This gives us one random number between 1 and 30. To get the next person, repeat the
command, (2nd ENTER.) Each time, the number at the end should be the number of people who
know the rumor. To keep track of the students who know the rumor, begin with them all
standing and have them sit down when they know the rumor. (If you had to double the number
of students, use raised hands.) Keep a chart of the number of students who know the rumor and
continue until all students have heard the rumor. Remember to eliminate repeated numbers. At
the end, you should have data that is approximately logistic. It might look like:
Generation
Total number
knowing rumor
0
1
1
2
3
4
5
6
7
8
9
10
11
12
13
2
4
7
12
20
24
26
27
28
28
29
29
30
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
30
is a fairly close model
1 + 2−( x − 4.5 )
to the graph. (See population growth)
The equation y =
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Breast Cancer Risks
Exponent, Log, & Logistic Fctn - D
Name
The USA TODAY shows the risks of developing breast cancer for women at various ages.
3.43%
Breast Cancer Risks
2.54%
1.49%
0.40%
0.04%
20
30
40
Age
50
60
A woman has a 12.5% lifetime risk of developing breast cancer. This graph shows the risk of
having breast cancer in the 10 years following each age above.
1. Create a scatterplot for the data in the above graph.
2. What mathematical function should be used to best model the scatterplot?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Exponent, Log, & Logistic Fctn - D
3. Use the regression capabilities of the handheld calculator to determine the mathematical
model for the data. Record your model below:
4. Plot both the scatterplot and the regression equation on the same set of axes.
5. Use the model you found to answer the following questions:
a. For a woman, what is the risk at age 70 of developing breast cancer in the next ten years?
b. For a woman, what is the risk at age 45 of developing breast cancer in the next ten years?
c. What is the age of a woman who has a risk factor of 1.75%?
d. According to the mathematical model during what 5-year period does the risk of breast cancer
appear to be increasing fastest?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Breast Cancer Risks
Answer Key
Exponent, Log, & Logistic Fctn - D
The USA TODAY shows the risks of developing breast cancer for women at various ages.
3.43%
Breast Cancer Risks
2.54%
1.49%
0.40%
0.04%
20
30
40
Age
50
60
A woman has a 12.5% lifetime risk of developing breast cancer. This graph shows the risk of
having breast cancer in the 10 years following each age above.
1. Create a scatter plot for the data in the above graph.
2. What mathematical function should be used to best model the scatter plot? logistic function
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 112 of 131
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Exponent, Log, & Logistic Fctn - D
3. Use the regression capabilities of the handheld calculator to determine the mathematical
model for the data. Record your model below:
f(x) =
(
3.73854
1 + 479.167e −.140817 x
)
4. Plot both the scatterplot and the regression equation on the same set of axes.
5. Use the model you found to answer the following questions:
a. For a woman, what is the risk at age 70 of developing breast cancer in the next ten years?
3.65 %
b. For a woman, what is the risk at age 45 of developing breast cancer in the next ten years?
2.02 %
c. What is the age of a woman who has a risk factor of 1.75%?
43
d. According to the mathematical model during what 5-year period does the risk of breast cancer
appear to be increasing fastest?
Between 40 and 45
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Comparison of Curve-Fittings
Exponent, Log, & Logistic Fctn - D
Name
The following chart gives US census data, in millions of people, for the years indicated. Put the
data in your calculator, in the lists indicated:
Year
L1
L2
Year
L1
L2
1810
1
7.24
1900
10
75.99
1820
2
9.64
1910
11
91.97
1830
3
12.87
1920
12
105.71
1840
4
17.07
1930
13
122.78
1850
5
23.19
1940
14
131.67
1860
6
31.44
1950
15
151.33
1870
7
39.82
1960
16
179.32
1880
8
50.16
1970
17
203.21
1890
9
62.95
1980
18
226.5
Make a scatterplot of L1 versus L2. Recreate this scatterplot as accurately as possible on the
blank axes below:
Using the regression features of your calculator, attempt to make a Linear, Quadratic, Cubic,
Power, Exponential, Logarithmic, Logistic, and Sine regression curve.
Given that we are modeling population data, which type of regression do we expect to give the
most accurate results? Does that match up with what you find?
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 114 of 131
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Comparison of Curve-Fittings
Answer Key
Exponent, Log, & Logistic Fctn - D
The following chart gives US census data, in millions of people, for the years indicated. Put the
data in your calculator, in the lists indicated:
Year
L1
L2
Year
L1
L2
1810
1
7.24
1900
10
75.99
1820
2
9.64
1910
11
91.97
1830
3
12.87
1920
12
105.71
1840
4
17.07
1930
13
122.78
1850
5
23.19
1940
14
131.67
1860
6
31.44
1950
15
151.33
1870
7
39.82
1960
16
179.32
1880
8
50.16
1970
17
203.21
1890
9
62.95
1980
18
226.5
Make a scatterplot of L1 versus L2. Recreate this scatterplot as accurately as possible on the
blank axes below:
Using the regression features of your calculator, attempt to make a Linear, Quadratic, Cubic,
Power, Exponential, Logarithmic, Logistic, and Sine regression curve.
Given that we are modeling population data, which type of regression do we expect to give the
most accurate results? Does that match up with what you find?
Linear Regression: y = 12.75x – 35.42
Quadratic Regression: y = .66x2 + .19x +6.47
Cubic Regression: y = .01x3 +.49x2 + 1.53x 4.05
Power Regression: y = 3.79x1.32
Exponential Regression: y = 8.16(1.22x)
Logarithmic Regression: y = - 61.65 + 72.88lnx
Logistic Regression: y =
370
Sinusoidal Regression: Does Not Exist
−.23 x
(1 + 41.45e
)
Population behaves as a logistic function; therefore the logistic regression is the most
representative of the data given.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Decimal Exponents
Expnt., Log, & Logistic Fctn - Reteach
Name
1. Using the TI-83 graphing calculator, sketch the graph of the exponential function y = 10 x for
x values from 0 through 5. Make sure to get an appropriate y-scale to be able to draw the
graph. Answer the following questions about values on that graph.
A. Find a value of y when x is between 1 and 2.
B. Find a value of y when x is between 2 and 3.
C. Find a value of y when x is between 3 and 4.
D. Are there any values of y when x is between 0 and 1? If so, give an example.
2. A. Complete the following table. Use the table feature of the TI-83 graphing calculator
to get your values.
x
10x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
B. Sketch a graph of the data in this table. (You can use the same function as in your last
graph, just change the x-values you’re using and rescale your y-values). Draw the curve
which fits these points. 10
9
8
7
6
5
4
3
2
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C. Are there values of y when x is between 0.1 and 0.2? If so, give an example.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 116 of 131
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Expnt., Log, & Logistic Fctn - Reteach
3. A. Complete the following table. Use the table feature of the TI-83 graphing calculator to
get your values.
x
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
10x
B. Sketch a graph of the data in this table. (You can use the same function as in your last
graph, just change the x-values you’re using and re-scale your y-values). Draw the curve
which fits these points.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.1
0.12
0.14
0.16
0.18
0.2
C. Are there values of y when x is between 0.10 and 0.11? If so, give an example.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 117 of 131
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Decimal Exponents
Answer Key
Expnt., Log, & Logistic Fctn - Reteach
1. Using the TI-83 graphing calculator, sketch the graph of the exponential function y = 10 x for
x values from 0 through 5. Make sure to get an appropriate y-scale to be able to draw the
graph. Answer the following questions about values on that graph.
A. Find a value of y when x is between 1 and 2.
Answers must be greater than 10 and less than 100.
B. Find a value of y when x is between 2 and 3.
Answers must be greater than 100 and less than 1000.
C. Find a value of y when x is between 3 and 4.
Answers must be greater than 1000 and less than 10,000.
D. Are there any values of y when x is between 0 and 1? If so, give an example.
Yes. Answers should be greater than 1 and less than 10.
2. A. Complete the following table. Use the table feature of the TI-83 graphing calculator
to get your values.
x
10x
0.1
1.2589
0.2
1.5849
0.3
1.9953
0.4
2.5119
0.5
3.1623
0.6
3.9811
0.7
5.0119
0.8
6.3096
0.9
7.9433
1.0
10
B. Sketch a graph of the data in this table. (You can use the same function as in your last
graph, just change the x-values you’re using and rescale your y-values). Draw the curve
which fits these points.
C. Are there values of y when x is between 0.1 and 0.2? If so, give an example.
Yes, for example 1.3, 1.4, 1.5, etc.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Expnt., Log, & Logistic Fctn - Reteach
3. A. Complete the following table. Use the table feature of the TI-83 graphing calculator to
get your values.
x
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
10x
1.2589
1.2882
1.3183
1.349
1.3804
1.4125
1.4454
1.4791
1.5136
1.5488
1.5849
B. Sketch a graph of the data in this table. (You can use the same function as in your last
graph, just change the x-values you’re using and rescale your y-values). Draw the curve
which fits these points.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.1
0.12
0.14
0.16
0.18
0.2
C. Are there values of y when x is between 0.10 and 0.11? If so, give an example.
Yes, for example 1.26, 1.27, etc.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Car Value Depreciation
Expnt., Log, & Logistic Fctn - Reteach
Name
Introduction: The depreciation of a car’s value is an example of exponential decay. If a car
tends to lose 18% of its value every year, then we can say the value depreciates by 18%, and that
it keeps 100% – 18% or 82% of its value from year to year. A single dollar depreciating at this
rate would follow the rule of y = 0.82 x where x is the number of years and y is the final value of
that initial dollar. If we’re talking about a car whose new cost is P dollars, then the formula turns
into y = P ⋅ 0.82 x , with x and y the same as before.
1. Complete the following table showing the value and the total depreciation of a car whose
original value is $42,000 over a period of 5 years. Note that the total depreciation is defined
as the original value of the car minus the current value.
t (years)
0
Car Value ($)
42,000
Total Depreciation ($)
0
2. Make a graph for this table of data (You could also graph the formula above, as long as you
substitute the proper value for P. Alternatively, you could plot the data points from your
table). Sketch that graph.
3. Using your graph, estimate how long it would take for the car’s value to drop to $10,000.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 120 of 131
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Car Value Depreciation
Answer Key
Expnt., Log, & Logistic Fctn - Reteach
Introduction: The depreciation of a car’s value is an example of exponential decay. If a car
tends to lose 18% of its value every year, then we can say the value depreciates by 18%, and that
it keeps 100% – 18% or 82% of its value from year to year. A single dollar depreciating at this
rate would follow the rule of y = 0.82 x where x is the number of years and y is the final value of
that initial dollar. If we’re talking about a car whose new cost is P dollars, then the formula turns
into y = P ⋅ 0.82 x , with x and y the same as before.
1. Complete the following table showing the value and the total depreciation of a car whose
original value is $42,000 over a period of 5 years. Note that the total depreciation is defined
as the original value of the car minus the current value.
t (years)
0
1
2
3
4
5
Car Value ($)
42,000
34,440
28,241
23,157
18,989
15,571
Total Depreciation ($)
0
7,560
13,759
18,843
23,011
26,429
2. Make a graph for this table of data (You could also graph the formula above, as long as you
substitute the proper value for P. Alternatively, you could plot the data points from your
table). Sketch that graph.
40000
35000
30000
25000
20000
15000
10000
5000
5
10
15
20
25
30
3. Using your graph, estimate how long it would take for the car’s value to drop to $10,000.
By tracing on the graph, the answer is 7.23 years.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Expnt., Log, & Logistic Fctn - Rtch
Exploring Common Logarithms
Name
1. Using your graphing calculator, evaluate the following.
log 10 = __________
log 100 = __________
log 1000 = __________
log 10,000 = __________
log 0.1= __________
log 0.01= __________
log 0.001= __________
log 0.0001= __________
2. Based on your answers above, what can you conjecture about common logarithms? Answer
in complete sentences.
3. Without using your calculator, evaluate log 1. Use your calculator to check your answer.
4. Enter log (-10) into your calculator. Write what the calculator displays. Why do you think
the calculator displayed this answer?
5. Find x in each equation.
a) log x = 6
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
b) log
1
=x
10, 000
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Exploring Common LogarithmsExpnt., Log, & Logistic Fctn - Rtch
Answer Key
1. Using your graphing calculator, evaluate the following.
log 10 = ____1____
log 100 = _____2____
log 1000 = ____3_____
log 10,000 = ____4_____
log 0.1= ____- 1____
log 0.01= ___- 2_____
log 0.001= ____- 3____
log 0.0001= ____- 4____
2. Based on your answers above, what can you conjecture about common logarithms? Answer
in complete sentences.
The base is 10. The answer is the exponent of base 10.
3. Without using your calculator, evaluate log 1. Use your calculator to check your answer.
Zero
4. Enter log (-10) into your calculator. Write what the calculator displays. Why do you think
the calculator displayed this answer?
Error: Nonreal Answer. 10 raised to any power will never result in a -10.
5. Find x in each equation.
a) log x = 6
x = 1,000,000
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
b) log
1
=x
10, 000
x=-4
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Expnt., Log, & Logistic Fctn - Rtch
Logarithmic & Exponential Form
Name
1. Given the following logarithmic equations, what can you conclude? Explain clearly.
log 3 9 = 2
log 4 2 =
1
2
log 2 16 = 4
log 3 27 = 3
log 4 4 = 1
1
= −1
5
log 6 36 = 2
log 2
log 5
1
= −3
8
2. Evaluate log 3 81 .
3. Write
log b y = x
4. Write
an = b
in exponential form.
in logarithmic form
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
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Expnt., Log, & Logistic Fctn - Rtch
Logarithmic & Exponential Form
Answer Key
1. Given the following logarithmic equations, what can you conclude? Explain clearly.
log 3 9 = 2
log 4 2 =
1
2
log 2 16 = 4
log 3 27 = 3
log 4 4 = 1
1
= −1
5
log 6 36 = 2
log 2
log 5
1
= −3
8
The subscript is the base and the answer is the exponent of the base. For example,
in log39 = 2, 3 is the base and 2 is the exponent of 3 and 32 = 9.
2. Evaluate log 3 81 .
4
3. Write
log b y = x
in exponential form.
bx = y
4. Write
an = b
in logarithmic form
loga b = n
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 125 of 131
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Expnt., Log, & Logistic Fctn - Rtch
Investigating Compound Interest
Name
In this investigation, you will explore how compound interest is different than simple interest.
Simple interest is paid on the initial principal where as compound interest is paid on the initial
principal and from previously earned interest. In addition, you will be able to come up with the
compound interest formula.
1. You deposited $1000 into a savings account paying 6% annual interest.
a) If the interest is compounded once a year, how much will you have in your account at
the end of the first year (i.e. what is the balance)? Round your final answer to 2
decimal places.
b) What is your balance at the end of the 2nd year? At the end of the 3rd year? At the
end of the 4th year? Show work. Round your final answer to 2 decimal places.
End of 2nd year __________
End of 3rd year ____________
End of 4th year ___________
c) Think of a formula to represent how much you have at the end of the tth year.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 126 of 131
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Expnt., Log, & Logistic Fctn - Rtch
Many savings institutions offer compounding intervals other than annual (yearly) compounding.
For example, a bank that offers quarterly compounding computes interest on an account every
quarter, that is, every 3 months. Thus instead of compounding interest once each year, the
interest will be compounded 4 times each year. If a bank advertises that it is offering 6% annual
interest compounded quarterly, it does not use 6% to determine interest each quarter. Instead, it
will use 6%/4 = 1.5% each quarter. In this example, 6% is known as the nominal interest rate
and 1.5% as the quarterly interest rate.
2. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6%
compounded quarterly, how much would you have in your account after: (round to 2 decimal
places & show work)
3 months _______________
6 months _______________
9 months _______________
1 year _________________
4 years _______________
t years ____________________
3. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6%
compounded monthly, how much would you have in your account after one year?
4. What can you conclude about how the compounding periods affect the balance?
5. Come up with a formula to represent the balance, A, if you invested P dollars at a rate of r
compounded n times a year for t years.
6. Which option would you rather have?
Investing $1000 into an account paying 5% interest compounded yearly for a year
I.
OR
II.
Investing $1000 into an account paying 4.75% interest compounded monthly for a
year
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 127 of 131
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Expnt., Log, & Logistic Fctn - Rtch
Investigating Compound Interest
Answer Key
In this investigation, you will explore how compound interest is different than simple interest.
Simple interest is paid on the initial principal where as compound interest is paid on the initial
principal and from previously earned interest. In addition, you will be able to come up with the
compound interest formula.
1. You deposited $1000 into a savings account paying 6% annual interest.
a) If the interest is compounded once a year, how much will you have in your account at
the end of the first year (i.e. what is the balance)? Round your final answer to 2
decimal places.
Balance = 1000 + 1000(.06)(1) = $1060
OR
Balance = 1000(1 + 0.06) = 1000(1.06) = $1060
b) What is your balance at the end of the 2nd year? At the end of the 3rd year? At the
end of the 4th year? Show work. Round your final answer to 2 decimal places.
1060 + 1060(0.06) = 1060(1 + 0.06) = 1060(1.06)
= 1000(1.06)(1.06)
= 1000(1.06)2
End of 2nd year __$1123.60___
1123.60 + 1123.60(0.06) = 1123.60(1 + 0.06) = 1123.60(1.06)
= 1000(1.06)(1.06) (1.06)
= 1000(1.06)3
End of 3rd year _ $1191.02 __
1191.02 + 1191.02(0.06) = 1191.02(1 + 0.06) = 1191.02(1.06)
= 1000(1.06)(1.06) (1.06)(1.06)
= 1000(1.06)4
End of 4th year __$1262.48__
c) Think of a formula to represent how much you have at the end of the tth year.
1000(1 + .06)t = 1000(1.06)t
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 128 of 131
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Expnt., Log, & Logistic Fctn - Rtch
Many savings institutions offer compounding intervals other than annual (yearly) compounding.
For example, a bank that offers quarterly compounding computes interest on an account every
quarter, that is, every 3 months. Thus instead of compounding interest once each year, the
interest will be compounded 4 times each year. If a bank advertises that it is offering 6% annual
interest compounded quarterly, it does not use 6% to determine interest each quarter. Instead, it
will use 6%/4 = 1.5% each quarter. In this example, 6% is known as the nominal interest rate
and 1.5% as the quarterly interest rate.
2. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6%
compounded quarterly, how much would you have in your account after: (round to 2 decimal
places & show work)
3 months ____$1015_____
6 months ____$1030.23___
9 months ____$1045.68___
1 year ____$1061.37_____
4 years ____$1268.99____
t years __1000(1+0.06/4)4t___
3. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6%
compounded monthly, how much would you have in your account after one year?
12(1)
⎛ 0.06 ⎞
1000⎜1+
⎟
12 ⎠
⎝
= $1061.68
4. What can you conclude about how the compounding periods affect the balance?
The more the compound periods occur, the higher the balance will be.
5. Come up with a formula to represent the balance, A, if you invested P dollars at a rate of r
compounded n times a year for t years.
nt
⎛ r⎞
A = P ⎜1+ ⎟
⎝ n⎠
6. Which option would you rather have?
Investing $1000 into an account paying 5% interest compounded yearly for a year
I.
OR
Investing $1000 into an account paying 4.75% interest compounded monthly for a
II.
year
The best option is option I
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 129 of 131
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A Number Called e
Expnt., Log, & Logistic Fctn - Reteach
Name
nt
⎛ r⎞
Using the compound interest formula A = P ⎜1 + ⎟ , we will examine what happens to A as n,
⎝ n⎠
the compounding period, increases. In this problem we will let P = $1, the interest rate is 100%
and the time is 1 year. Complete the table below by finding the balance given each specific
compounding period. Round the balance value to three decimal places.
n
⎛ r⎞
A = P ⎜1 + ⎟
⎝ n⎠
nt
A
1
(annually)
2
(semi-annually)
4
(quarterly)
12
(monthly)
52
(weekly)
365
(daily)
8760
(hourly)
525,600
(minutely)
31,536,000
(every second)
Do you see a pattern in the balance? Explain.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 130 of 131
Columbus Public Schools 7/20/05
A Number Called e
Answer Key
Expnt., Log, & Logistic Fctn - Reteach
nt
⎛ r⎞
Using the compound interest formula A = P ⎜1 + ⎟ , we will examine what happens to A as n,
⎝ n⎠
the compounding period, increases. In this problem we will let P = $1, the interest rate is 100%
and the time is 1 year. Complete the table below by finding the balance given each specific
compounding period. Round the balance value to three decimal places.
n
⎛ r⎞
A = P ⎜1 + ⎟
⎝ n⎠
1
(annually)
⎛ 1⎞
A = 1⎜1 + ⎟
⎝ 1⎠
2
(semi-annually)
⎛ 1⎞
A = 1⎜1 + ⎟
2⎠
⎝
2(1)
4
(quarterly)
⎛ 1⎞
A = 1⎜1 + ⎟
⎝ 4⎠
4(1)
12
(monthly)
1 ⎞
⎛
A = 1⎜1 + ⎟
⎝ 12 ⎠
52
(weekly)
1 ⎞
⎛
A = 1⎜1 + ⎟
⎝ 52 ⎠
365
(daily)
1 ⎞
⎛
A = 1⎜1 +
365 ⎟⎠
⎝
8760
(hourly)
1 ⎞
⎛
A = 1⎜1 +
⎟
⎝ 8760 ⎠
525,600
(minutely)
31,536,000
(every second)
A
nt
1(1)
2
2.25
2.441
12(1)
2.613
52(1)
2.693
365(1)
2.715
8760(1)
1
⎛
⎞
A = 1⎜1 +
⎟
⎝ 525, 600 ⎠
2.718
525,600(1)
1
⎛
⎞
A = 1⎜1 +
31, 536, 000 ⎟⎠
⎝
2.718
31,536,000(1)
2.718
Do you see a pattern in the balance? Explain.
The balance goes toward the number 2.718 as n gets larger and larger.
PreCalculus Standards 4 and 5
Exponential, Logarithmic, & Logistic
Page 131 of 131
Columbus Public Schools 7/20/05