Integrated Algebra
Chapter 11: Exponential Functions & Radicals
Name:______________________________
Teacher:____________________________
Pd: _______
Table of Contents
Chapter 11-3 (Day 1): SWBAT: Solve problems involving exponential growth,
exponential decay.
Pgs: 3 – 8
HW: Pgs 9-11
Chapter 11-3 (Day 2): SWBAT: Solve problems involving exponential growth,
exponential decay and half-life.
Pgs: 12 – 16
HW: Pg 17
Chapter 11-6 & 11-7 (Day 3): SWBAT: Add, Subtract and simplify radical
expressions Pgs: 18 - 22
HW: Pgs 23 – 24
Chapter 11 – 6 & 11-8 (Day 4): SWBAT: Multiply and Divide radical expressions
Pgs: 25 – 30
HW: Pg 31
Review: SWBAT Solve problems involving exponential growth, exponential decay. SWBAT Add, Subtract, Multiply, Divide, and simplify radical expressions Pgs: 32 – 39
o CHAPTER 11 EXAM
3
Day 1: Exponential Growth and Exponential Decay
SWBAT: Solve problems involving exponential growth, exponential decay
Warm-up:
Exponential growth occurs when a quantity increases by the same rate r in each period t.
When this happens, the value of the quantity at any given time can be calculated as a function of
the rate and the original amount.
Exponential decay occurs when a quantity decreases by the same rate r in each time period t.
Just like exponential growth, the value of the quantity at any given time can be calculated by
using the rate and the original amount.
Explain:
4
Example 1:
The original value of a painting is $9,000 and the value increases by 7% each year. Write an exponential growth
function to model this situation. Then find the painting’s value in 15 years.
Example 2: The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Write an
exponential decay function to model this situation, and then find the population in 2012.
Practice:
1) A sculpture is increasing in value at a rate of 8% per year, and its value in 2000 was $1200. Write an
exponential growth function to model this situation, and then find the sculpture’s value in 2006.
Answer: ______________
Step 1: Write the exponential growth function for this situation
Step 2: Find the value in 15 years.
Step 1: Write the exponential growth function for this situation
Step 2: Find the value in 2006.
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
Step 1 Write the exponential decay function for this situation
Step 2 Find the value in 12 years.
y = _________ ( 1 ______ ) ____
y = _________ ( _____ ) ____
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2) The number of employees at a certain company is 1440 and is increasing at a rate of 1.5% per year.
Write an exponential growth function to model this situation. Then find the number of employees in the
company after 9 years.
Answer: ______________ employees
3) The fish population in a local stream is decreasing at a rate of 3% per year. The original population was
48,000. Write an exponential decay function to model this situation. Then find the population after
7 years.
Answer: ______________
4) The deer population of a game preserve is decreasing by 2% per year. The original population was 1850.
Write an exponential decay function to model the situation. Then find the population after 4 years.
Answer: ______________
Step 1 Write the exponential decay function for this situation
Step 2 Find the value in 7 years.
Step 1: Write the exponential growth function for this situation
Step 2: Find the number of employees in the company after 9 years.
.
y = __________ ( 1 ______ ) ____
y = __________ ( _____ ) ____
y = ____________ ( 1 ______ ) ____
y = _____________ ( _____ ) ____
Step 2 Find the value in 7 years.
y = _______ ( 1 ______ ) ____
y = _______ ( _____ ) ____
Step 1 Write the exponential decay function for this situation
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Regents Questions:
Example 3: A realtor estimates that a certain new house worth $500,000 will gain value at a rate of 6% per
year since 2009. Make a table of values to approximate the number of years it will take the house to gain a
value of 3 million dollars.
What is the real-world meaning of year 0?
Which type of model best represents the data in your table? Explain. Write a function for the data.
Example 4:
Use the information in the table to predict the number of termites in the termite colony after one year.
Termite Colony Population
Time (months) Number of Termites
0 20
1 80
2 320
3 1,280
1) 5,120 termites 3) 16,777,216 termites
2) 335,544,320 termites 4) 9,920 termites
7
Example 5:
Is the equation A = 1500 (1 – 0.14)t
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1) exponential growth and 14%
2) exponential growth and 86%
3) exponential decay and 14%
4) exponential decay and 86%
Example 6:
Is the equation A = 5000 (1 + .04)t
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1) exponential growth and 4%
2) exponential growth and 96%
3) exponential decay and 4%
4) exponential decay and 96%
Example 7:
The fish population of Lake Collins is decreasing at a rate of 4% per year. In 2002 there were about 1,250 fish.
Determine whether this model is an exponential growth or exponential decay, and which equation can be used
to find the population in 2008?
1) exponential growth ; y = 1250(0.96)6
2) exponential growth ; y = 1250(1.04)6
3) exponential decay ; y = 1250(1.04)6
4) exponential decay ; y = 1250(0.96)6
Example 8:
The value of a gold coin picturing the head of the Roman Emperor Vespasian is increasing at the rate of 5 per
year. The coin is worth $105 now. Determine whether this model is an exponential growth or exponential
decay, and which equation can be used to find what the coin will be worth in 11 years?
1) exponential growth ; y = 105(0.95)11
2) exponential growth ; y = 105(1.05)11
3) exponential decay ; y = 105(1.05)11
Explain here!
Explain here!
Explain here!
Explain here!
8
Challenge
Summary:
Exit Ticket:
1)
2) The value of a gold coin picturing the head of the Roman Emperor Marcus Aurelius is increasing at the rate
of 7 per year. If the coin is worth $145 now, what will it be worth in 14 years?
1) $308.44 3) $373.89
2) $287.10 4) $243.00
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Homework
Write an exponential growth function to model each situation. Then find the value of the function after
the given number of years.
1)
2)
3)
Write an exponential decay function to model each situation. Then find the value of the function after
the given number of years.
4)
5)
6)
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7) Is the equation A = 3200 (1 – 0.30)t
a model of exponential growth or exponential decay, and what is the
rate (percent) of change per time period?
1) exponential growth and 30%
2) exponential growth and 70%
3) exponential decay and 30%
4) exponential decay and 70%
8) Is the equation A = 1756 (1 + .17)
t a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1) exponential growth and 17%
2) exponential growth and 83%
3) exponential decay and 17%
4) exponential decay and 83%
9) Is the equation A = 10,000 (0.45)t
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1) exponential growth and 45%
2) exponential growth and 55%
3) exponential decay and 45%
4) exponential decay and 55%
10) Is the equation A = 5400 (1.07)t
a model of exponential growth or exponential decay, and what is the rate
(percent) of change per time period?
1) exponential growth and 7%
2) exponential growth and 93%
3) exponential decay and 7%
4) exponential decay and 93%
Explain here!
Explain here!
Explain here!
Explain here!
11
11)
12)
13)
Hint: Make a Table of Values
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D a y 2 : M o r e W i t h E x p o n e n t i a l F u n c t i o n s
SWBAT: Solve problems involving exponential growth, exponential decay
Warm-Up
1) fdf 2)
1)
13
2)
3)
4)
5)
14
6)
7)
8)
9)
15
10)
11)
16
Challenge
SUMMARY
Exit Ticket
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Day 2 – HW
4.
5.
6.
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Day 3 - Radicals
SWBAT: Add, Subtract and simplify radical expressions
Warm – Up
QUIZ
Example 1: Simplifying Square-Root Expressions
Simplify each expression.
A. B. C.
Practice # 1
Simplify each expression.
1) 2) 3)
36 49 100
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Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Example 2: Simplest radical form
Simplify. All variables represent nonnegative numbers.
A. B. C.
Practice # 2
Simplify. All variables represent nonnegative numbers.
1.) 2.) 3.)
Example 3: Simplest radical form Simplify. All variables represent nonnegative numbers.
A. B.
Practice # 3 Simplify. All variables represent nonnegative numbers.
1.) 2.) 3.)
8 18
45 72
4 27 -3 20
5 28 2 75 5 8
48
80
20
Example 4: Adding and Subtracting Square-Root Expressions
Add or subtract.
Practice
Add or subtract.
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Example 5: Simplify Before Adding or Subtracting
Simplify each expression.
A. B.
Practice
Add or subtract.
A. B.
A.
a. b.
21
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Example 6: Simplify Before Adding and Subtracting
Simplify each expression.
A. B.
Practice
Add or Subtract.
A. B. C.
Challenge Problem:
Find the perimeter of the triangle. Give the answer as a radical expression in simplest form.
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Summary:
Exit Ticket:
23
Homework
Simplify. All variables represent nonnegative numbers.
1.) 2.) 3.)
4.) 5.) 6.)
7.) 8.) 9.)
10.) 11.) 12.)
13.) 14.) 15.)
-3 98
81 180
125 52 + 56
169
2 12
4 24 20
27 3 45 28
48 2 32 18
24
Use addition or subtraction to combine the following square roots that have the same radicands.
16. 3 10 9 10 17. 8 5 3 5 18. 14 7 7 7
For problems 19 through 27, combine each of the following expressions by first simplifying the square roots
and then combining like radicands. Express each answer in simplest radical form.
19. 8 5 2 20. 3 18 4 2 21. 3 20 2 45
22. 28 5 7 23. 2 54 7 24 24. 50 200
25. 7 45 80 26. 48 27 27. 200 2 18
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Day 4: (Multiplying and Dividing Radicals)
SWBAT: Multiply and Divide radical expressions
Warm-Up 1) 2) Simplify.
26
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Section 1: Simplifying Radical Review
a) √ b) √
Section 2: Adding and Subtracting Radicals
√ + √ d) √ √
e) Gfg f)
DIVIDING RADICALS
Example 1: Using the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
A) B)
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Practice: Using the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
1. 2.
Dividing Radical Expressions
Example 2: Using the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
A) B) C)
Practice: Using the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
1. 2. 3.
32
124
33
9615
When dividing radicals, you must divide the numbers outside the radicals and then
divide the numbers inside the radicals.
28
Multiplying Radical Expressions
Example 3: Multiplying Square Roots
Multiply. Write the product in simplest form.
Practice
Multiply. Write the product in simplest form.
Example 4: Using the Distributive Property
Multiply. Write each product in simplest form.
A. B.
A. B.
A.
When multiplying radicals, you must multiply the numbers outside the radicals and
then multiply the numbers inside the radicals.
29
Practice
Multiply. Write each product in simplest form.
Challenge Problem: Multiply. Write the product in simplest form.
Summary:
A. 10624
5354
30
SUMMARY….CONTINUED
Exit Ticket:
31
Homework
Simplify each radical expression.
(1) 8
32 (2)
2
98 (3)
5
245
(4) 2
100 (5)
4
72 (6)
64
20
(7) 2
80 (8)
42
203 (9)
25
1820
Multiply. Write each product in simplest form.
10) 11) 4 5 2 5 12)
13) 14) 15)
16) 17) 18)
3832 341053
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Chapter 11 Review
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Chapter 11 Review
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Chapter 11 Review
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Chapter 11 Review
49. Which function represents an exponential decay?
Explain your answer below.
52.
50. The value of a gold coin picturing the head of the Roman Emperor Vespasian is increasing at the rate of 5% per year. If the coin is worth $105 now, what will it be worth in 11 years?
a. $169.79 b. $160.00 c. $179.59 d. $162.75
53. The function f(x) = 300(0.85)x models the number of landlocked salmon in the lake x months after the lake was stocked. If the lake was stocked with fish in early April, which is the best estimate of the number of landlocked salmon in early July? A. 157 B. 184 C. 217 D. 255
51.
54. Use the data in the table to describe how the restaurant’s sales are changing. Then write a function that models the data. Use your function to predict the amount of sales after 8 years.
Restaurant Sales Year 0 1 2 3 Sales ($)
15,000 15,900 16,854 17,865.24
Function Rule: ____________________________ Answer: _____________________
52.
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53.
54.
55.
56.