Exponential Functions & Formulas
4th Six Weeks 2017-18 Unit 10 – Exponential Functions & Formulas March 19, 2018 – April 4, 2018 MONDAY
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
March 19
20
21
22
23
EOC Warm Up
EOC Warm Up
Quiz
EOC Warm Up
EOC Warm Up
Exponential Growth & Decay
Exponential Growth & Decay
Exponential Growth & Decay Apps
Exponential Growth & Decay Apps
Exponential Growth & Decay Apps
HW: WS
HW: WS
HW: WS
HW: WS
HW: WS
26
27
28
29
30
EOC Warm Up
EOC Warm Up
Quiz
EOC Warm Up
Arithmetic & Geometric Sequences
Arithmetic & Geometric Sequences
Recursive Formulas
Recursive Formulas
HW: WS April 2
HW: WS 3
HW: WS 4
HW: WS 5
EOC Warm Up
EOC Warm Up
Recursive Formulas
Review!!!
Easter Holiday
6
TEST (Turn in Review)
HW: WS
HW: Review
Coach Schmidt: ***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM*** Ms. Martinez: ***Tutorials: Monday, Wednesday, Friday from 7:35 – 7:55 AM*** Mr. Landrum: ***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM***
0
Notes: Write and Graph Exponential Growth Functions Exponential Function Equation: _____________________ *Where “a” stands for ______________ and “b” stands for _______________.
Example 1. Sara came down with chicken pox and when she noticed them, she had 3 chicken pox. Each day after that, they doubled in amount. Fill in the table below, write the equation that would best represents the data, and graph. # of days x
0 1 2 3 4 5 6 x
# of pox y
Exponential Function:________
1
Ex 2. Describe the relationship from the tables below as either exponential or linear. Then write the equation that would best represent the data. x -2 -1 0 1 2
y 1 5 25 125 625
x -3 -2 -1 0 1 2
y .02 .2 2 20 200 2,000
x -2 -1 0 1 2
y 4 6 8 10 12
Relationship_____________
Relationship_____________
Relationship___________
Equation________________
Equation _______________
Equation______________
2
Notes: Write and Graph Exponential Decay Functions Exponential Function Equation: _____________________ *Where “a” stands for ______________ and “b” stands for _______________. A ball is dropped from a building that is 128 feet tall. The ball will bounce 1 of its previous height. Make a table of values to 4
show how high the ball is after each bounce and graph the data. # of bounces x 0
Height of ball y
Exponential Function:_______
1 2 3 4 5 6 x
3
Ex 2. Describe the relationship shown in the tables below and then write the equation that would best represent the data in the following tables. x 0 1 2 3 4
y 24 12 6 3 1.5
x
-1
0
1
y
36
12
4
2 4 3
3 4 9
Relationship_____________________
Relationship___________________
Equation________________________
Equation ______________________
4
Exponential Growth & Decay – Day 2 Draw an example of what each of the following would look like. Exponential Growth:
Exponential Decay:
Exponential Growth and Decay General Formula: ______________
Exponential Growth is when ____________________
Exponential Decay is when _____________________ Given the following formula: y = 3(½) x a) Y-intercept=________ b) Multiplier=_________ 5
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
NOTES & PRACTICE Exponential Growth Population ____________ and growth of monetary investments are examples of exponential growth.
Exponential Growth
The general equation for exponential growth is y = a(1 + 𝑟 )𝑡 . • y represents the ________ amount. • a represents the ________ amount. • r represents the ______ of change - as a ___________. • t represents _______.
Example 1: POPULATION The population of
Example 2: INVESTMENT The Garcias have
Johnson City in 2005 was 25,000. Since then, the population has grown at an average rate of 3.2% each year.
$12,000 in a savings account. The bank pays 3.5% interest on savings accounts, compounded monthly. Find the balance in 3 years.
a. Write an equation to represent the population of Johnson City since 2005.
The rate 3.5% can be written as 0.035. The special
The rate 3.2% can be written as 0.032. y = a(1 + 𝑟)𝑡 y= y=
𝑟 𝑛𝑡
equation for compound interest is A = P 1 + 𝑛 , where A represents the balance, P is the initial amount, r represents the annual rate expressed as a decimal, n represents the number of times the interest is compounded each year, and t represents the number of years the money is invested. 𝑟
b. According to the equation, what will the population of Johnson City be in 2015? In 2015 t will equal 2015 – 2005 or 10. Substitute 10 for t in the equation from part a.
A = P (1 + 𝑛 )𝑛𝑡 = ≈
y=
Exercises 1. POPULATION The population of the United States has been increasing at an average rate of 0.91%. If the population was about 303,146,000 in 2008, predict the population in 2012.
2. INVESTMENT Determine the value of an investment of $2500 if it is invested at an interest rate of 5.25% compounded monthly for 4 years.
3. POPULATION It is estimated that the population of the world is increasing at an average annual rate of 1.3%. If the 2008 population was about 6,641,000,000, predict the 2015 population.
4. INVESTMENT Determine the value of an investment of $100,000 if it is invested at an interest rate of 5.2% compounded quarterly for 12 years.
6
The general equation for exponential growth is y = a(1 − 𝑟 )𝑡 . Exponential • y represents the ________ amount. • a represents the ________ amount. Decay • r represents the ______ of change - as a ___________. • t represents _______. Example: DEPRECIATION The original price of a tractor was $45,000. The value of the tractor decreases at a steady rate of 12% per year. a. Write an equation to represent the value of the tractor since it was purchased. The rate 12% can be written as 0.12. y = a(1 − 𝑟)𝑡 y=
General equation for exponential decay
y= b. What is the value of the tractor in 5 years? y=
Exercises 1. POPULATION The population of Bulgaria has been decreasing at an annual rate of 0.89%. If the population of Bulgaria was about 7,450,349 in the year 2005, predict its population in the year 2015.
2. DEPRECIATION Mr. Gossell is a machinist. He bought some new machinery for about $125,000. He wants to calculate the value of the machinery over the next 10 years for tax purposes. If the machinery depreciates at the rate of 15% per year, what is the value of the machinery (to the nearest $100) at the end of 10 years?
3. ARCHAEOLOGY The half-life of a radioactive element is defined as the time that it takes for one-half a quantity of the element to decay. Radioactive carbon-14 is found in all living organisms and has a half-life of 5730 years. Consider a living organism with an original concentration of carbon-14 of 100 grams. a. If the organism lived 5730 years ago, what is the concentration of carbon-14 today? b. If the organism lived 11,460 years ago, determine the concentration of carbon-14 today.
4. DEPRECIATION A new car costs $32,000. It is expected to depreciate 12% each year for 4 years and then depreciate 8% each year thereafter. Find the value of the car in 6 years.
7
NOTES – Geometric Sequences A geometric sequence - each term after the first is found by _______________ the previous term by constant r called the __________________. The common ratio can be found by ____________ any term by its _______________ term. Example 1: Determine whether the sequence is arithmetic, geometric, or neither: 21, 63, 189, 567, . . . Find the ratios of the consecutive terms. If the ratios are constant, the sequence is geometric. 21
63
189
567
=
=
=
Because the ratios are _____________, the sequence is geometric. The common ratio is _____. Determine whether each sequence is arithmetic, geometric, or neither. 1. 1, 2, 4, 8, . . .
2.
2
1
1
1
, , , ,. . . 3 3 6 12
8
Example 2: Find the next three terms in this geometric sequence: – 1215, 405, –135, 45, . . . Step 1
Find the common ratio. –1215 405 –135
=
45
=
=
The value of r = Step 2
Multiply each term by the common ratio to find the next three terms. 45 ___ ___ ___ ×
1 3
×
1 3
×
1 3
The next three terms of the sequence are: Find the next three terms in each geometric sequence. 1. 648, –216, 72, . . .
1 1
7. 16, 2, 4, . . .
9
Notes – Recursive Formulas Recursive formula - allows you to find the nth term of a sequence by performing operations on one or more of the terms that precede it. Example: Find the first five terms of the sequence in which 𝑎1 = 5 and 𝑎𝑛 = –2𝑎𝑛 − 1 + 14, if n ≥ 2. The given first term is a1 = 5. Use this term and the recursive formula to find the next four terms.
𝑎2 = –2𝑎2 − 1 + 14
n=2
𝑎4 = –2𝑎4 − 1 + 14
n=4
𝑎3 = –2𝑎3 − 1 + 14
n=3
𝑎5 = –2𝑎5 − 1 + 14
n=5
The first five terms are:
Find the first five terms of the sequence. 1. 𝑎1 = –4, 𝑎𝑛 = 3𝑎𝑛 − 1, n ≥ 2
10
2. Find the recursive formula for 216, 36, 6, 1, … . A. 𝑎1 = 216, 𝑎𝑛 = 6 𝑎𝑛 − 1, n ≥ 2 1
B. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 , n ≥ 2 6
C. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 − 180, n ≥ 2 D. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 + 30, n ≥ 2
3. Find the recursive formula for -8, -3, 2, 7, … . A. 𝑎1 = -8, 𝑎𝑛 = 5 𝑎𝑛 − 1 , n ≥ 2 3
B. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1, n ≥ 2 8
C. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1 − 5, n ≥ 2 D. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1 + 5, n ≥ 2
11
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
March 19
20
21
22
23
EOC Warm Up
EOC Warm Up
Quiz
EOC Warm Up
EOC Warm Up
Exponential Growth & Decay
Exponential Growth & Decay
Exponential Growth & Decay Apps
Exponential Growth & Decay Apps
Exponential Growth & Decay Apps
HW: WS
HW: WS
HW: WS
HW: WS
HW: WS
26
27
28
29
30
EOC Warm Up
EOC Warm Up
Quiz
EOC Warm Up
Arithmetic & Geometric Sequences
Arithmetic & Geometric Sequences
Recursive Formulas
Recursive Formulas
HW: WS April 2
HW: WS 3
HW: WS 4
HW: WS 5
EOC Warm Up
EOC Warm Up
Recursive Formulas
Review!!!
Easter Holiday
6
TEST (Turn in Review)
HW: WS
HW: Review
Coach Schmidt: ***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM*** Ms. Martinez: ***Tutorials: Monday, Wednesday, Friday from 7:35 – 7:55 AM*** Mr. Landrum: ***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM***
0
Notes: Write and Graph Exponential Growth Functions Exponential Function Equation: _____________________ *Where “a” stands for ______________ and “b” stands for _______________.
Example 1. Sara came down with chicken pox and when she noticed them, she had 3 chicken pox. Each day after that, they doubled in amount. Fill in the table below, write the equation that would best represents the data, and graph. # of days x
0 1 2 3 4 5 6 x
# of pox y
Exponential Function:________
1
Ex 2. Describe the relationship from the tables below as either exponential or linear. Then write the equation that would best represent the data. x -2 -1 0 1 2
y 1 5 25 125 625
x -3 -2 -1 0 1 2
y .02 .2 2 20 200 2,000
x -2 -1 0 1 2
y 4 6 8 10 12
Relationship_____________
Relationship_____________
Relationship___________
Equation________________
Equation _______________
Equation______________
2
Notes: Write and Graph Exponential Decay Functions Exponential Function Equation: _____________________ *Where “a” stands for ______________ and “b” stands for _______________. A ball is dropped from a building that is 128 feet tall. The ball will bounce 1 of its previous height. Make a table of values to 4
show how high the ball is after each bounce and graph the data. # of bounces x 0
Height of ball y
Exponential Function:_______
1 2 3 4 5 6 x
3
Ex 2. Describe the relationship shown in the tables below and then write the equation that would best represent the data in the following tables. x 0 1 2 3 4
y 24 12 6 3 1.5
x
-1
0
1
y
36
12
4
2 4 3
3 4 9
Relationship_____________________
Relationship___________________
Equation________________________
Equation ______________________
4
Exponential Growth & Decay – Day 2 Draw an example of what each of the following would look like. Exponential Growth:
Exponential Decay:
Exponential Growth and Decay General Formula: ______________
Exponential Growth is when ____________________
Exponential Decay is when _____________________ Given the following formula: y = 3(½) x a) Y-intercept=________ b) Multiplier=_________ 5
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
NOTES & PRACTICE Exponential Growth Population ____________ and growth of monetary investments are examples of exponential growth.
Exponential Growth
The general equation for exponential growth is y = a(1 + 𝑟 )𝑡 . • y represents the ________ amount. • a represents the ________ amount. • r represents the ______ of change - as a ___________. • t represents _______.
Example 1: POPULATION The population of
Example 2: INVESTMENT The Garcias have
Johnson City in 2005 was 25,000. Since then, the population has grown at an average rate of 3.2% each year.
$12,000 in a savings account. The bank pays 3.5% interest on savings accounts, compounded monthly. Find the balance in 3 years.
a. Write an equation to represent the population of Johnson City since 2005.
The rate 3.5% can be written as 0.035. The special
The rate 3.2% can be written as 0.032. y = a(1 + 𝑟)𝑡 y= y=
𝑟 𝑛𝑡
equation for compound interest is A = P 1 + 𝑛 , where A represents the balance, P is the initial amount, r represents the annual rate expressed as a decimal, n represents the number of times the interest is compounded each year, and t represents the number of years the money is invested. 𝑟
b. According to the equation, what will the population of Johnson City be in 2015? In 2015 t will equal 2015 – 2005 or 10. Substitute 10 for t in the equation from part a.
A = P (1 + 𝑛 )𝑛𝑡 = ≈
y=
Exercises 1. POPULATION The population of the United States has been increasing at an average rate of 0.91%. If the population was about 303,146,000 in 2008, predict the population in 2012.
2. INVESTMENT Determine the value of an investment of $2500 if it is invested at an interest rate of 5.25% compounded monthly for 4 years.
3. POPULATION It is estimated that the population of the world is increasing at an average annual rate of 1.3%. If the 2008 population was about 6,641,000,000, predict the 2015 population.
4. INVESTMENT Determine the value of an investment of $100,000 if it is invested at an interest rate of 5.2% compounded quarterly for 12 years.
6
The general equation for exponential growth is y = a(1 − 𝑟 )𝑡 . Exponential • y represents the ________ amount. • a represents the ________ amount. Decay • r represents the ______ of change - as a ___________. • t represents _______. Example: DEPRECIATION The original price of a tractor was $45,000. The value of the tractor decreases at a steady rate of 12% per year. a. Write an equation to represent the value of the tractor since it was purchased. The rate 12% can be written as 0.12. y = a(1 − 𝑟)𝑡 y=
General equation for exponential decay
y= b. What is the value of the tractor in 5 years? y=
Exercises 1. POPULATION The population of Bulgaria has been decreasing at an annual rate of 0.89%. If the population of Bulgaria was about 7,450,349 in the year 2005, predict its population in the year 2015.
2. DEPRECIATION Mr. Gossell is a machinist. He bought some new machinery for about $125,000. He wants to calculate the value of the machinery over the next 10 years for tax purposes. If the machinery depreciates at the rate of 15% per year, what is the value of the machinery (to the nearest $100) at the end of 10 years?
3. ARCHAEOLOGY The half-life of a radioactive element is defined as the time that it takes for one-half a quantity of the element to decay. Radioactive carbon-14 is found in all living organisms and has a half-life of 5730 years. Consider a living organism with an original concentration of carbon-14 of 100 grams. a. If the organism lived 5730 years ago, what is the concentration of carbon-14 today? b. If the organism lived 11,460 years ago, determine the concentration of carbon-14 today.
4. DEPRECIATION A new car costs $32,000. It is expected to depreciate 12% each year for 4 years and then depreciate 8% each year thereafter. Find the value of the car in 6 years.
7
NOTES – Geometric Sequences A geometric sequence - each term after the first is found by _______________ the previous term by constant r called the __________________. The common ratio can be found by ____________ any term by its _______________ term. Example 1: Determine whether the sequence is arithmetic, geometric, or neither: 21, 63, 189, 567, . . . Find the ratios of the consecutive terms. If the ratios are constant, the sequence is geometric. 21
63
189
567
=
=
=
Because the ratios are _____________, the sequence is geometric. The common ratio is _____. Determine whether each sequence is arithmetic, geometric, or neither. 1. 1, 2, 4, 8, . . .
2.
2
1
1
1
, , , ,. . . 3 3 6 12
8
Example 2: Find the next three terms in this geometric sequence: – 1215, 405, –135, 45, . . . Step 1
Find the common ratio. –1215 405 –135
=
45
=
=
The value of r = Step 2
Multiply each term by the common ratio to find the next three terms. 45 ___ ___ ___ ×
1 3
×
1 3
×
1 3
The next three terms of the sequence are: Find the next three terms in each geometric sequence. 1. 648, –216, 72, . . .
1 1
7. 16, 2, 4, . . .
9
Notes – Recursive Formulas Recursive formula - allows you to find the nth term of a sequence by performing operations on one or more of the terms that precede it. Example: Find the first five terms of the sequence in which 𝑎1 = 5 and 𝑎𝑛 = –2𝑎𝑛 − 1 + 14, if n ≥ 2. The given first term is a1 = 5. Use this term and the recursive formula to find the next four terms.
𝑎2 = –2𝑎2 − 1 + 14
n=2
𝑎4 = –2𝑎4 − 1 + 14
n=4
𝑎3 = –2𝑎3 − 1 + 14
n=3
𝑎5 = –2𝑎5 − 1 + 14
n=5
The first five terms are:
Find the first five terms of the sequence. 1. 𝑎1 = –4, 𝑎𝑛 = 3𝑎𝑛 − 1, n ≥ 2
10
2. Find the recursive formula for 216, 36, 6, 1, … . A. 𝑎1 = 216, 𝑎𝑛 = 6 𝑎𝑛 − 1, n ≥ 2 1
B. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 , n ≥ 2 6
C. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 − 180, n ≥ 2 D. 𝑎1 = 216, 𝑎𝑛 = 𝑎𝑛 − 1 + 30, n ≥ 2
3. Find the recursive formula for -8, -3, 2, 7, … . A. 𝑎1 = -8, 𝑎𝑛 = 5 𝑎𝑛 − 1 , n ≥ 2 3
B. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1, n ≥ 2 8
C. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1 − 5, n ≥ 2 D. 𝑎1 = -8, 𝑎𝑛 = 𝑎𝑛 − 1 + 5, n ≥ 2
11
