Download Algebra 1B Assignments Exponential Functions

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Algebra 1B Assignments
Exponential Functions
(All graphs must be drawn on graph paper!)
8-6
Pages 463-465: #1-17 odd, 35, 37-40, 43, 45-47, 50, 51, 54, 55-61 odd
8-7
Pages 470-473: #1-11 odd, 12-22, 24, 25, 29, 33, 38, 40, 57-61 odd
8-8a
Pages 479-482: #3-10, 16-19, 33-39, 43, 57, 58, 60
8-8b
Pages 479-482: #20-32, 40, 41, 44-49, 52, 56, 62
Quiz
8-6 to 8-8
Worksheet: Review of Functions and Graphs
10-8
Pages 601-604: #2, 3, 5, 6, 8, 10-14, 16, 30-33, 43-47 odd
Review
Worksheet: Exponential Functions Review
Test
Exponential Functions
Section 8-6
Warm – Up:
Simplify each expression.
1.
2
3
 x2 
2.  
 x5 
4
 a 3b 0 
3. 

 c 7 
2
Use inductive reasoning to find the next two numbers in each pattern.
4. 5, 8, 11, 14,…
5. 19, 17, 15, 13,…
6. 1, 3, 9, 27,…
7. 400, 200, 100, 50,…
Objective: To differentiate between arithmetic and geometric sequences
To form geometric sequences
arithmetic sequence: A number pattern formed by adding a fixed number to each previous
term.
1, 3, 5, 7, 9,… (common difference)
(What type of graph does this make?)
geometric sequence: A number pattern formed by multiplying a fixed number to each
previous term.
1, 2, 4, 8, 16,… (common ratio)
(What type of graph does this make?)
Example #1:
Determine whether each sequence is arithmetic, geometric, or neither. State the common
difference or ratio, if applicable.
1
a) 25, 5, 1,
5
,…
c) 1, 2, 4, 7, 11,…
b) 5, 1, -3, -7,…
Example #2:
Find the common ratio and the next two terms of each sequence.
10
a) 3, -15, 75, -375,…
b) 90, 30, 10,
3
,…
Closure Question:
Explain the difference between an arithmetic and geometric sequence.
Section 8-7
Warm – Up:
Graph each function.
1. y  3 x  4
y
2. y  2 x  1
y
x
x
Simplify each expression without a calculator.
3. 5
3
4. 10  3
2
5. 7  2
1
6. 3  2
4
Objective: To evaluate and graph exponential functions
exponential function:
y  ab
x
evaluate: plug in a value for x and solve for y
domain: all possible x-values
range: all possible y-values
Example #1:
Evaluate each function.
a) y  4 x for x  2, 0, 3
x


b) f ( x )  2  3 for the domain 1,0,2
Example #2:
a) Suppose 10 rabbits are taken to an island. The rabbit population then triples every year.
Use an exponential function to find out how many rabbits there would be after 12 years.
b) Suppose two mice live in a barn. If the number of mice quadruples every 3 months, use
an exponential function to find out how many mice will be in the barn after 2 years.
Steps to graph exponential functions:
 Make a table of values with 5 points (x = -2, -1, 0, 1, 2)
 Plot the points
 Connect the points to form a smooth curve
Example #3:
Graph each function without a calculator. State the domain and range.
a) y  5
y
x
b) y  3  2
x
y
x
x
Example #4:
x
The function f ( x )  1.5 models the increase in size of an image being copied over and
y
over at 150% on a photocopier. Graph the function.
x
Closure Question:
4
Is the equation y  x an exponential function? Explain your answer.
Section 8-8a
Warm – Up:
Exponential growth and decay exploration
Objective: To model exponential growth
Explanation of Exponential Growth:
Start with 20 bacteria and grow at a rate of 30% per year.
What is the growth factor?
year 1 =
year 2 =
year 3 =
year 4 =
Exponential Growth:
y  ab
x
(examples: bacteria, population, interest)
Use the exponential rule for the bacteria problem above. Do you get the same answer?
Example #1:
In 1998 a town had a population of 13,000 people. Since 1998 the population increased 4%
per year.
a) Write an equation to model this situation.
b) Estimate the population in 2006.
Example #2:
a) Suppose you deposit $1000 in a college fund that pays 7.2% interest compounded
annually. What is the balance after 5 years?
b) Suppose you deposit $1000 in a college fund that pays 7.2% interest compounded
quarterly. What is the balance after 5 years?
Example #3:
Calculate the balance in a bank account with $3000 principal earning 3.6% compounded
monthly for 2 years.
Closure Question:
Would you rather have $600 in an account paying 5% interest compounded annually or
$500 in an account paying 6% compounded quarterly? Explain your answer.
Section 8-8b
Warm – Up:


Graph each function using the table of values 2,1,0,1,
 2 without a calculator.
y
1. y  2  4
1
2. y  3   
2
x
x
x
y
x
Suppose you deposit $1500 in an account paying 3.5% interest for 6 years.
3. Find the account balance if the money is compounded annually.
4. Find the account balance if the money is compounded quarterly.
Objective: To model exponential decay
Example #1:
The half-life of a radioactive substance is the length of time it takes for one half of the substance
to decay into another substance. To treat some forms of cancer, doctors use radioactive iodine.
The half-life of iodine-131 is 9 days. A patient receives a 32-mCi (millicuries, a measure of
radiation) treatment. How much iodine-131 is left in the patient 27 days later?
Exponential Decay:
y  ab
x
(examples: radioactive decay, endangered species, car value)
Use the exponential rule for the radioactive decay problem in example #1.
Example #2:
The population of an endangered species of animals has decreased 2.4% each year. There
were 60 of these animals in 1998.
a) Write an equation to model this situation.
b) Estimate how many animals remained in 2005.
Example #3:
You buy a used truck for $14,000. It depreciates at a rate of 15% per year. Find the value
of the truck after 6 years.
Closure Question:
Explain the difference between an exponential growth function and an exponential decay
function.
Section 10-8
Warm – Up:
Write an equation for each function.
1. A linear function that passes through the points (0, 4) and (6, 1).
2. An exponential function that passes through the points (0, 5) and (1, 15).


Graph each function using the table of values 1,2,3,4 .
3. y  2 x  1
y
4. y  1  3
x
y
x
Objective: To choose a linear or exponential model for data
Example #1:
Calculate the average rate of change for warm-up problems 3 and 4. Use the x-values from
1  2, 1  3, and 1  4. What do you notice about the average rate of change for linear
functions versus exponential functions?
x
How do you decide which model best fits the data?
1) Graph the function
2) Analyze the data
a) Linear ( y  mx  b )
Has a common difference (subtract y values)
x
b) Exponential ( y  ab )
Has a common ratio (divide y values)
Example #2:
Graph each set of points. Which model is most appropriate for each set?
y
a) (-2, 1), (0, 2), (1, 4), (2, 7)
x
b) (-2, -2), (0, 2), (1, 4), (2, 6)
Example #3:
Analyze the data to determine which kind of function best models the data in each table.
Then write an equation to model the data.
a)
x
0
1
2
3
4
y
1
1.5
2
2.5
3
b)
x
0
1
2
3
4
y
4
4.4
4.84
5.324
5.8564
Since real-life data is not always exact, you may need to find the best possible model.
Example #4:
Suppose you are studying frogs that live in a nearby wetland area. The data below were
collected by a local conservation organization. They indicate that number of frogs estimated
to be living in the wetland area over a five-year period.
a) Determine which kind of function best models the data.
Year
Estimated
Population
0
1
2
120
101
86
3
72
4
60
b) Write an equation to model the data.
Example #5:
The table below shows the population of a town in South Dakota. Let x = 0 correspond to
the year 1980.
a) Determine which kind of function best models the data.
Year
Population
0
5
10
15
20
7300
7575
7875
8200
8500
b) Write an equation to model the data.
Closure Question:
Explain how you choose a linear or exponential model for data?