Download - Exponential Lesson Plan 2
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Specific Learning Objective: Graph and analyze exponential functions
and solving problems involving exponential equations.
Lesson and Activity Schedule
Lesson72 min
Skill/Concept Activity Material TechTools
Lesson-1 Activating prior knowledge
about exponential functions
and understanding the
concept of exponential
growth and decay through
Mathematical Modeling.
Group
Activity: Group
Assignment on
modeling of
exponential
functions, Class
discussion
Mathematical
modeling
assignment
MS-Excel
smart-board
activity as an
activation
strategy
Lesson-2 Sketching the graphs of
exponential functions by
applying a set of
transformations through
investigation with the help of
technology
Investigative
group
assignment for
students using
graphing
calculator, Class
discussion,
home-work
1.Assignment
2.Matching
activity for
home
Computer lab
with 2d-
graphing
calculator
software
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Check Agree or Disagree beside each statement Compare your choice and explanation with a partner
Statement Agree Disagree
. Each of the following is a function
i. y =2 x2-5
i i . y = x/4 + 7
iii . y = 3x
iv. 2x+ 3y - 5 = 0
. The base of y = 2x is x
. y = 3x is the same as y = x3 .
. The area, y , of a square floor with one side
measuring x can be modelled by the equation
y = 2x
. For the function on the grid below, the x-intercept is -3
and the y-intercept is 1
. y = (1/5) x is an exponential function
. The domain of y = 2 x is R
. The range of y = 10 x is y > 0.
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You have won the grand pr ize in a contest and have two options for receiving your cash payment:
(a ) The $100-a -Day plan, in which you receive $1,000 immediately, plus $100 per day for 30
days; (b) The Double Your Money plan, in which you receive $0.01 immediately, and your
winnings double every day for 30 days.
:
1. Which payment plan seems like the better deal? Why?
2. Does the information in the table change your choice for which payment plan is the better deal?
Why or why not?
3. How much money will you have collected by day 30, with?
a) The $100-a-Day plan?
b) With the Double Your Money plan?
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A scientist places one bacterium in a Petri dish at 9:00 am. The bacterium can reproduce at a rate that doubles its
population every minute. The scientist observes that the Petri dish is completely full at 10:00 am.
1. Complete the following table using the information above:
0
1
12
2
3
4
5
6
2. Fit a function to the growth of the bacterium where time represents independent variable (t)
and number of bacteria represent dependent variable (n)
3. At what value of t does the value of n become 2 60
4. Sketch the function on a graph paper
5. Locate the x and y intercepts on the graph
6. State the domain, range and asymptote for the function
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Suppose Mario has been sprayed with a shrinking spray and he is getting reduced to half his size
every minute. Let us say we start at 9:00 am being t=0, 9:01am being t=1 and so on.
1. Complete the table if dependent variable y (size of Mario) = 1 at t=0.
-2
-1
0 1
1
2
3
2. Write an exponential function fitting the situation
3. Sketch a graph for the function
4. Locate the x and y intercepts on the graph
5. Write the domain, range and asymptote of the function
6. What would be Marios size after 10 minutes?
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General form of exponential function y = b x , y= a. b x , difference between exponential growth and
decay, and graphs for both the situations are discussed in the class. Students are given text-book
assignments for home.
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Plan-2
Objectives:
Determining through investigation with graphing calculator the
impact of varying values and signs of bases and exponents on
exponential functions
Demonstrating an understanding of the effects of horizontal and
vertical compressions, stretches and shifts
Applying the prior knowledge of shifts, stretches, compressions
and reflections to sketch variations in exponential function
Procedure
1. Investigative assignment consisting of four parts(in groups) asActivating and Acquiring strategy
2. Class Discussion3. Application and reinforcement (selected text questions
discussed and assigned for home-work together with a matching
activity)
1. Investigative AssignmentStudents are divided into groups of four and each student in the
group gets all four parts of the investigative assignment
Instructionsfor the assignments:
Each of the equations is in the form: y= bx
For each part of the investigation graph all the given
equations on the grid provided.
Complete the chart that follows
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Group Activity: Draw the graph for each equation and complete the table following it. Discuss the
results with your group members.
y= 2x y= 4x y=10x
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
Discuss:
KQ1. What these graphs have in common
Equations
1. y= 2x
2. y= 4x
3. y=10x
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KQ2. The impact of value of the base, on the graph of the function
Group Activity: Draw the graph for each equation and complete the table following it. Discuss the
results with your group members.
y=(1/2)x y=(1/4)x y=(1/10)x
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
Discuss:
KQ1. What these graphs have in common
Equations
1. y=(1/2)x
2. y=(1/4)x
3. y=(1/10)x
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KQ2. The impact of changing the value of the base on the graph
Group Activity: Draw the graph for each equation and complete the table following it. Discuss the
results with your group members.
y= 2 -x y= 4 -x y=10- x
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
Discuss:
KQ1. The impact of changing the signs of exponents on the graph of an exponential function
Equations
1. y= 2 - x
2. y= 4 - x
3. y=10 -x
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Group Activity: Draw the graph for each equation and complete the table following it. Discuss the
results with your group members.
y=-2x y=-4x
y intercept is
x intercept is
asymptote is
domain is
range is
y intercept is
x intercept is
asymptote is
domain is
range is
KQ1. Figure out the impact of a negative sign upfront on the graph of an exponential function
Equations
1. y= 2x
2. y=-2x
3. y= 4x
4. y=-4x
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a) How does i) increase in b (b>1) affect the graph of y = b x
i i) decrease in b (b
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Match each graph with an equation that best represents the relationship. For each graph, state the x-
intercept, y-intercept, domain, range, and asymptote.
i) y = 3 - x ii) y =(1/4)x
iii) y = 5 -x iv) y= (2.2)x
v) y = 5x vi) y= 3x
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***********Not included in the lesson******
Plan-3
Objective: Overview and Reinforcement of graphs of exponential
functions and their characteristics in the form of a brain-storming
quiz.
Quiz
Consider f(x)=Bx as the standard function and g(x)=a.B b(x-c)+d as
the transformation of it
Given function is the transformed function g in each case
For each of the listed functions :
1. g(x)=3(x+1)-2
2. g(x)=-5.2(x-3)+2
3. g(x)=(1/2)x Hint: write g(x) as 2 -x
Bonus question
4. g(x)=23x
Find and state
a) The standard function f for each case
b) domain and range of g
c) horizontal asymptotes of the graph of g
d) x and y intercepts of the graph of g
e) left or right shift (shift along x-axis)
f) up or down shift (shift along y-axis)
g) reflection of g in x-axis (if any)
h) reflection in y-axis(if any)
i) sketch the graph of g on the graph paper provided
j) change in co-ordinates from f to g
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Plan-3
Objectives:
Linking exponential growth and decay to real-life situations by
solving a variety of real-life problems involving the
application of exponential equations to loans, populations,
investments or radioactivity
Solving a variety of Math problems in multiple and sequential
steps
Solving problems in a sequence of steps
Students take notes as teacher explains the sequential steps to
solving exponential problems.
Q1. The population, P million, of Alberta can be modeled by the
equation P=2.28(1.014)n, where n is the number of years since 1981.
Assume that this pattern continues. Determine when the population of
Alberta might become 4 million.
Step1. Write the equation
Step2. Substitute P for 4
Step3. Solve the equation for n
Q2. In 1995, Canadas population was 29.6 million, and was growing at
about 1.24% per year. Estimate the doubling time for Canadas
population growth.
Step1. Write the equation, P= 29.6 (1.0124) n
Step2. Substitute 229.6 for P in the equation
Step3. Solve the equation for n Ans. 56 years
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Formula for calculating compound interest:
where,
P = principal amount (initial investment)
r = annual nominal interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
A = amount after time t
Q3: An amount of $1500.00 is deposited in a bank paying an annual interest
rate of 4.3%, compounded quarterly. Find the balance after 6 years.
Solution:
Step1. Write the formula
Step2. Insert P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6
Step3. Solve for A
A= 1500( 1+ 0.043/4)46=1938.84
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Student Reflection (Re-enforcement) Sheet:
Q1. Consider the equation P=100(0.87) n that models the percent of
caffeine in your body n hours after consumption. Write this equation
as an exponential function with as the base instead of 0.87.
Q2. What does represent in this question?
Q3. Rewrite the equation in Q2 with 2 as base instead of 1.0124.
(Ans. P= 29.6 2 n56).
Q4. The population of a swarm of insects can multiply fivefold in
four weeks. Let Po represent the population at t=0.
Write an expression to represent the population after
i) 5 weeks
ii) 7 weeks
Ans)
i) P=55/ 4Po
ii) P=57/ 4Po
Have the students apply the formulas and steps above to the selected
questions from the text which are discussed and also explained by the
teacher in the class.
Home-work: Students are assigned selected questions for home-work.
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Plan-4
Objective:
Students will work on a mini project on exponential models
(growth or decay). This will act as a wrap up to the lessons
and as a review.
Procedure:
Students are divided into groups of four and they start working on
their projects which is due next week. Computer lab is booked in
advance for the project which is due next week.
1. Project Assignment
Instructions
Think of one real situation that involves exponential growth and that
involves exponential decay. Project has to be in the form of a power-point
presentation with one copy of printed version of it for the teacher. Your
project should include the following:
1. Introduction - Briefly explain the situation. You may make up your own
information, but make it realistic. Include the facts needed to write an
equation.
2. Equation - Model the situation with an exponential equation.
3. Graph - Make a graph of the exponential function. Be sure to label
what each axis represent and use an appropriate scale.
Note: Relevant hints and ideas will be provided by the teacher, if
required. Project is to be submitted same day next week.
Assessment: Projects will be considered for grades
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Plan-5:
Objectives:
Re-visiting the concept of geometric sequences
Investigating the concept of geometric geries in relation to
exponential functions
Deriving the formula for the sum of a geometric series.
Recognizing and be able to solve the real-life situations and
problems where geometric series arise, in a sequential manner
Procedure
1. Activating: Exploration sheet to investigate and conclude theexpression for the sum of finite geometric series.
2. Teacher-directed activity: Teacher explains the differencebetween a finite and infinite geometric sequence and series and
their sums. Selected questions from the text are explained on
board by the teacher and students take their notes.
3. Application and home-work: Students are assigned selectedquestions for home-work.
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1. Exploration Sheet: Complete the table and answer the questionson the next page
Considering r to be a real number, expand and simplify Answers
1. (r-1) (r+1) Example
r2-1
2. (r-1) (r2+r+1)
3. (r-1) (r3+r 2+r+1)
4. (r-1) (r4+r 3+r 2+r+1)
If (r-1)(r+1) = r2-1 implies 1+r = (r2-1)/r-1
Find an expression for each of the following:
1. 1+r+r2
2. 1+r+r2+r3
3. 1+r+r2+r3+r4
4. 1+r+r2+r3+..+rn
5. a+ar+ar 2+a r3+.+arn where a is a real number
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Answer the following questions:
Q1. Write the expression for the following sums:
a) 1 + r + r2 + r3 + r4 +..+ rn
b) a + ar + ar2 + ar 3 +..+ an
Q2. Write the expression for a geometric sequence, sum of whose
first n terms is represented by:
a) 1 + r + r2 + r3 + r4 +..+ rn
b) a + ar + ar 2 + ar 3 +..+ an
Q3. If r represents the common ratio of the sequence
a, ar , ar2 , ar3, fill in the blanks:
r= a3/a 2 = a4/_ = a5/_ = a2/_
Q4. For the following geometric sequences
a) 2, 4, 8
b) 5, 1/2, 1/20
state
i) a, the first term of the sequence
ii) r, ratio for the geometric sequences
iii) S10, the sum of first 10 terms
Q5. Can you find the sum of infinite number of terms of the
following geometric series:
a) 2, 6, 18, 54,
b) 1, 1/2, 1/4, 1/8,
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2. Teacher-directed activity: Teacher explains the differencebetween a finite and infinite geometric sequence and series and
their sums. Selected questions from the text are explained on
board by the teacher and students take their notes.
3. Application and home-work: Students are assigned selectedquestions for home-work.
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Plan-6
Objective
Exponential Functions Review Quiz will serve as a wrap-up to
the topic, help students recapture and relate the different
concepts, practise their problem-solving skills, and prepare
for the Unit test.
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Exponential Functions Review Assignment
1. Sketch each of the following functions and state its domain,range, intercepts and equation of asymptote:
a) y=2x
-4
b) y=|3x-1|
c) y=22- x
d) y=-3.e2x
2. Solvea) 27x=9 2x-1
b) 42x-1=64
c) 3(5 x+1)=15
d) 81/ 4(1/4)x/2=163/ 4
e) (5256)/( 664)=2 x
f) 27x(9 2x-1)=3x+4
g) 5x- 1=2.3x
3. The first three terms of a geometric sequence are -6, (5-x), -50/3, . Find all possible values of the 2 nd term.
4. The sum of the 3rd and 4th term of a geometric sequence is 36and the sum of the fourth and 5 th term is 108. Determine the
geometric sequence.
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5. Solve the following problems:Investment
a)Find the amount of money you will have after 10 years if
$15,000 is invested in accounts paying 6% interest
compounded:
1. Annually 2. Quarterly 3. Monthly 4. Daily
Population
b)If the world population is about 6 billion people now and if
the population grows continuously at an annual rate of 1.7%,
what will the population be in 10 years?
Radiology and Half-life
c)In 2 minutes, a sample of Radium-221 decays to 6.25% of its
original amount. Find its half-life.
Oil Industry
d)An oil well produces 25,000 barrels of oil during the first
month of production. Suppose its production drops by 5% each
month, estimate the total production before the well runs
dry.
Business
e)In 2008, the Pennrose Gazette, the local newspaper for
Pennrose County, counted 20,000 readers. The publisher
predicts that its readership will contract by 10% each year.
Use this information to answer the following questions:
1. Write a function describing the diminishing number of
Pennrose Gazette readers.
2. Based on the function, predict the number of readers in
2015.
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Answers
1. a) Domain: RRange: (-4,)
x intercept: -2
y intercept: -3
Asymptote: y=-4
b) Domain: R
Range: (0,)
x intercept: 0
y intercept: 0
no asymptote
c) Domain: R
Range: (0,)
no x-intercept
y-intercept: 4
Asymptote: x-axis or y=0
d) Domain: R
Range: (-,0)
no x-intercept
y-intercept: -3
Asymptote: y=0
2. a) x=2
b) x=1.5055 approx
c) x=0.86 approx
d) x=-2.25
e) x=0.599 approx
f) x=1
g) x=4.508
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3. a2=-10 or a2=10
4.The sequence is 1,3,9,27,.
5. a) 1) $26,862.72
2) $27, 210.28
3) $27,290.95
4) $27,330.43
b) 7.1 billion
c) 30 seconds
d) 500,000 barrels
e) 1) y=20,000(.90) x
2) 9,565 people
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A ball dropped off a roof reaches a bounce height equal to one-half the height before the bounce
(for example, a ball dropped from a height of 500 feet reaches a height of 250 feet
after the first bounce).
Determine whether the data in each table represents a linear function or an exponential function.
Explain your answer.
1.
x Y
0
1 2
2 8
3 32
4 128
5 5122.
x Y
0 7
1 13
2 19
3 25
4 31
5 37