Exponential Lesson Plan 2

7/31/2019 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



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



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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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

Documents

Exponential Lesson Plan 2