Integrated Algebra - White Plains Public Schools

Integrated Algebra

Chapter 11: Exponential Functions & Radicals

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Table of Contents

Chapter 11-2 (Day 1): SWBAT: Graph exponential

functions Pgs: 1 – 4

HW: 5

Chapter 11-3 (Day 2): SWBAT: Solve problems involving

exponential growth, exponential decay and half-life

Pgs: 6 – 9

HW: 10 - 11

Chapter 11-6 & 11-7 (Day 3): SWBAT: Add, Subtract

and simplify radical expressions Pgs: 12 - 16

HW: 17 – 18

Chapter 11 – 6 & 11-8 (Day 4): SWBAT: Multiply and

Divide radical expressions

Pgs: 19 – 22

HW: 23

o Chapter 11 - Practice Test: Review using E-Clickers

o CHAPTER 11 EXAM

1

Chapter 11- 2 - Graphing Exponential Functions

SWBAT: Graph exponential functions Warm – Up

Example 1: Graphing y = abx with a > 0 and b > 1

Graph: y = 0.5(2)x Practice

1) Graph: y = 2x 2) Graph: y = 0.2(5)x

2

Example 2: Graphing y = abx with a < 0 and b > 1

Practice

3) Graph y = –6x 4) Graph y = –3(3)x

3

Example 3: Graphing y = abx with 0 < b < 1 Graph: y = 4(0.6)x

Example 4: Graphing y = abx with 0 < b < 1

Graph :

4

Challenge Problem

Summary: The box summarizes the general shapes of exponential function graphs. For y = abx, if b > 1, then the For y = abx, if 0 < b < 1, then the graph will have one of these graph will have one of these shapes. shapes.

Exit Ticket

5

Homework 1) Graph: y =3x 2) Graph: y = -2(3)x 3) Graph: 4) Graph:

5. Which equation represents a quadratic function?

6

Chapter 11-3

Exponential Growth and Exponential Decay

SWBAT: Solve problems involving exponential growth, exponential decay and half – life

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.

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.

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.

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 6 years.

7

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.

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.

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) 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.

Step 1 Write the exponential decay

function for this situation

Step 2 Find the value in 12 years.

Step 1 Write the exponential decay

function for this situation

Step 2 Find the value in 7 years.

8

2) 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.

A common application of exponential decay is half-life of a substance is the time it takes

for one-half of the substance to decay into another substance.

Example 1: Science Application

Astatine-218 has a half-life of 2 seconds. Find the amount left from a 500 gram sample of astatine-218 after 10

seconds.

Practice: Science Applications

1) Astatine-218 has a half-life of 2 seconds. Find the amount left from a 500-gram sample of astatine-218 after

30 seconds.

2) Cesium-137 has a half-life of 30 years. Find the amount of cesium-137 left from a 100 milligram sample

after 180 years.

Step 1 Find t, the number of half-

lives in the given time period.

Step 2 Set up the half-life function and

solve for the final amount.

Step 1 Find t, the number of half-

lives in the given time period.

Step 2 Set up the half-life function and

solve for the final amount.

9

Challenge Problem:

Summary:

Exit Ticket:

10

Homework

1)

Write an exponential growth function to model each situation. Then find the value of the function after

the given number of years.

2)

3)

4)

Write an exponential decay function to model each situation. Then find the value of the function after

the given number of years.

5)

6)

11

7)

8)

Solve the following half-life problems:

9)

10)

11)

12

Chapter 11 – 6 & 11 – 7 (Radicals) SWBAT: Add, Subtract and simplify radical expressions

Warm Up

Identify the perfect square in each set.

1. 45 81 27 111

2. 156 99 8 25

3. 256 84 12 1000

4. 35 216 196 72

Write each number as a product of prime numbers.

5. 36 6. 24

13

Example 1: Simplifying Square-Root Expressions

Simplify each expression.

A. B. C.

Practice # 1

Simplify each expression.

1) 2) 3)

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.)

36 49 100

8 18

45 72

48

80

14

Example 3: Simplest radical form Simplify. All variables represent nonnegative numbers.

A. B.

Practice # 3 Simplify. All variables represent nonnegative numbers.

1.) 2.) 3.)

Example 4: Adding and Subtracting Square-Root Expressions

Add or subtract.

Practice

Add or subtract.

4 27 -3 20

5 28 2 75 5 8

15

Example 5: Simplify Before Adding or Subtracting

Simplify each expression.

A. B.

Practice

Add or subtract.

A. B.

Example 6: Simplify Before Adding and Subtracting

Simplify each expression.

A. B.

Practice

Add or Subtract.

A. B. C.

Challenge Problem:

16

Find the perimeter of the triangle. Give the answer as a radical expression in simplest form.

Summary:

Exit Ticket:

17

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

18

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

19

Chapter 11 – 6 & 11- 8 (Multiplying and Dividing Radicals) SWBAT: Multiply and Divide radical expressions

Warm Up

Example 1: 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.

20

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.

Multiplying Radical Expressions

Example 3: Multiplying Square Roots

Multiply. Write the product in simplest form.

32

124

33

9615

When multiplying radicals, you must multiply the numbers outside the radicals and

then multiply the numbers inside the radicals.

1585432

When dividing radicals, you must divide the numbers outside the radicals and then

divide the numbers inside the radicals.

6252

304

21

Practice

Multiply. Write the product in simplest form.

Example 4: Using the Distributive Property

Multiply. Write each product in simplest form.

Practice

Multiply. Write each product in simplest form.

Challenge Problem: Multiply. Write the product in simplest form.

10624

5354

22

Summary:

Exit Ticket:

23

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