Algebra I Chapter 11 section 1 Geometric Sequences

Warm-up

Find the value of each expression.

1. 25

2. 2−5

3. −34

4. (−3)4

5. (0.2)3

6. 7¿

7. 15(1/3)3

8. 12(−0.4 )3

Find the next three terms

1, 3, 9, 27, 81, …

In a __________________________________. The ratio of successive terms is the same r, called the _____________________

Find the next three terms

5, -10, 20, -40, …

512, 384, 288, …

1, 4, 16, 64, …

-9, 3, -1, …

Geometric sequences can be thought of as functions. The term number, or position in the sequence is the input of the function and the term itself is the output of the function.

WORDS NUMBERS ALGEBRA1ST term 3 a₁2nd term 3(2) = 63rd term 3(2²) = 124th term 3(2)3 = 24Nth term 3(2¿¿n−1

If the first term of a geometric sequence is a₁, the nth term is an−1, and the common ratio is r, then

________________________________

The first term of a geometric sequence is 128, and the common ratio is 0.5. What is the 10 th term of the sequence?

The first term of a geometric sequence is 500 and the common ratio is 0.2 . What is the 7 th term of the sequence?

For a geometric sequence, a₁ = 8 and r = 3. Find the 5th term of this sequence.

For a geometric sequence, a₁ = 5 and r = 2. Find the 6th term of this sequence.

What is the 9th term of the sequence 2, -6, 18, -54, …?

What is the 13th term of the geometric sequence, 8, -16, 32, -64, …?

A ball is dropped from a tower. The table shows the heights of the ball’s bounces, which form a geometric sequence. What is the height of the 6th bounce?

bounce Height (cm)1 3002 1503 75

The table shows a car’s value for 3 years after it is purchased. The values form a geometric sequence. How much will the car be worth in the 10th year?

year Value ($)1 10,0002 8,0003 6,400

Homework 11.1 pg 769 #8-31, 34-39, 44-46, 55-62

Algebra I Chapter 11 section 2 Exponential Functions

Warm up

Simplify each expression. Round to the nearest whole number if necessary

1. 3²

2. 54

3. 2¿

4. 2/3¿

5. −5¿

6. -1/2(4 ¿¿3

7. 100 ¿

8. 3000 ¿

An exponential function has the form of _________________, where a ≠ 0, b ≠ 0, and b > 0

The function f ( x )=2(3)x models an insect population after x days. What will the population be on the 5 th day?

The function f ( x )=8(0.75)x models the width of a photograph in inches after it has been reduced by 25% x times. What is the width of the photograph after it has been reduced 3 times?

The function f ( x )=500 (1.035)x models the amount of money in a certificate of deposit after x years. How much money will there be in 6 years?

The function f ( x )=200,000(0.98)x , where x is the time in years, models the population of a city. What will the population be in 7 years?

Linear functions have constant first difference and quadratic functions have constant second differences. Exponential functions do not have constant differences, but they do have constant ______________.

Tell whether each set of ordered pairs satisfies an exponential function.

{(-1, 1.5), (0,3), (1, 6), (2, 12)}

{(-1, 1), (0,0), (1,1), (2, 4)}

{(-2,4), (-1,2), (0,1), (1,0.5)}

X f ( x )=2(3)x

1 62 183 544 102

X y

X y

X y

{(0,4), (1,12), (2,36), (3,108)}

{(-1,-64), (0,0), (1,64), (2,128)}

Graph y=3 (4 )x

Graph y=0.5(2)x

Graph y=−5(2)x

X y

X y

Graph y=−14

(2)x

Graph y=−1( 14)x

Graph y=4 (0.6)x

Graph of exponential functions

An accountant uses f (x)=12,330(0.869)x, where x is the time in years since the purchase, to model the value of a car. When will the car be worth $2000?

In 2000, each person in India consumed an average of 13 kg of sugar. Sugar consumption in India is projected to increase by 3.6% per year. At this growth rate, the function f (x)=13(1.036)x gives the average yearly amount of sugar, in kg consumption per person x years after 2000. In about what year will sugar consumption average about 18 kg per person?

Homework 11.2 pg 776 #18-32 even, 38-47, 52-56

Algebra I Chapter 11 Section 3 Exponential growth and decay

Warm-up

Simplify each expression.

1. (4 + 0.05)²

2. 25(1 + 0.02)³

3. 1 + 0.03/4

4. The first term in a geometric sequence is 3 and the common ratio is 2. What is the 5 th term of the sequence?

5. The function y=2(4)x models an insect population after x days. What is the insect population after 3 days?

Exponential growth occurs when a quantity increases by the same rate r in each time period t.

An exponential growth function has the form y=a(1+r )t where a > 0 y represents a represents r represents t represents

The original value of a painting is $1,400 and the value increases by 9% each year. Write an exponential growth function to model this situation. Then find the value of the painting in 25 years.

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. Then find the sculpture’s value in 2006.

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.

A common application of exponential growth is __________________________________.

Compound Interest

A=P(1+ rn )nt

A represents P represents r represents n represents t represents

Write a compound interest function to model each situation. Then find the balance after the given number of years.

$1000 invested at a rate of 3% compounded quarterly: 5 years

$1200 invested at a rate of 3.5% compounded quarterly: 4 years

$4000 invested at a rate of 3% compounded monthly: 8 years

$1200 invested at a rate of 2% compounded quarterly: 3 years

$15,000 invested at a rate of 4.8% compounded monthly: 2 years

Exponential decay occurs when a quantity decreases by the same rate r in each time period t.

An exponential decay function has the form y=a(1−r)t where a > 0

y represents a represents r represents t represents

The population of a town is decreasing at a rate of 1% per year. In 2000 there were 1300 people. Write an exponential decay function to model this situation. Then find the population in 2008

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.

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. Then find the population in 2012.

A common application of exponential decay is ______________.

A=P(0.5)t

A represents P represents t represents

Fluorine 20 has a half life of 11 seconds. Find the amount of fluorine 20 left from a 40 gram sample after 44 seconds.

Find the amount of fluorine 20 left from a 40 gram sample after 2 minutes.

Homework 11.3 pg 785 #10-20, 22-32 even, 45-47, 55-60

Algebra I Chapter 11 section 4 Linear, Quadratic and Exponential Models

1. Find the slope and the y intercept of the line that passes through (4,20) and (20, 24)

The population of a town is decreasing at a rate of 1.8% per year. In 1990, there were 4600 people.

2. Write an exponential decay function to model this situation.

3. Find the population in 2010

Look at the tables and graphs below. The data show three ways you have learned that variable quantities can be related.

Training heart rateAge Beats/min20 17030 161.540 15350 144.5

Volleyball heightTime height0.4 10.440.8 12.761 12

1.2 9.96

Volleyball tournamentRound Teams left

1 162 83 44 2

In the real world, people often gather data and then must decide what kind of relationship (if any) they think best describes their data.

Graph each data set. Which kind of model best describes the data?

Look for a pattern in each data set to determine which kind of model best describes the data

Height of golf ball

Oven temperature

Time 0 1 2 3Bacteria 10 20 40 80

Time 0 1 2 3Bacteria 24 96 384 1536

Boxes 1 5 20 50Reams 10 50 200 500

Time(s) 0 1 2 3 4Height (ft) 4 68 100 100 68

Time (min) 0 10 20 30Temperature

(f)375 325 275 225

Money in CD

General Forms of functions

LINEAR QUADRATIC EXPONENTIAL

Use the data in the table to describe how the ladybug population is changing. Then write a function that models the data. Use the function to predict the ladybug population after one year.

Ladybug Population

Use the data in the table to describe how the number of people changes. Then write a function that models the data. Use the function to predict the number of people who receive the e-mail after one week.

E-mail forwarding

Time (yr) 0 1 2 3Amount ($) 1000.00 1169.86 1368.57 1601.04

Time (mo) 0 1 2 3Ladybugs 10 30 90 270

Time (days) 0 1 2 3# of people who

receive the e-mail8 56 392 2744

Homework 11.4 pg 793 #8-14, 16-22, 27-29, 32-40

Algebra I Chapter 11 section 6 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 product of prime numbers

5. 36

6. 64

7. 196

8. 24

An expression that contains a radical sign ( ____) is a ______________________. The expression under a radical sign is the _____________

Simplest form of a square-root expressionAn expression containing square roots is in simplest form when

the radicand has ________________________ factors other than 1 the radicand has no _________________ there are no square roots in any _________________

Below are some simplifies square root expressions

Simplifying square-root expression

√ 272 √32+42 √ 2564

√40+9 √52+122 √ 250Product property of square roots

WORDS NUMBERS ALGEBRAFor any nonegtive real numbers a and b, the square root of ab is equal to the square root of a times the square root of b

Simplify. All variables represent nonnegative numbers

√128 √ x3 y2

√48ab2 √24

√72 √200a

Quotient property of square roots

WORDS NUMBERS ALGEBRAFor any nonegtive real numbers a and b, (a >0 and b> 0) the square root of a/b is equal to the square root of a divided the square root of b

√ 59 √ 1227

√ 36x4 √ y64

√ 54 √ m39m

√ 10825 √ 9x316

√ 4 x59 √ p6q10

A quadrangle on a college campus is a square with sides of 250 feet. If a student takes a shortcut by walking diagonally across the quadrangle, how far does he walk? Give the answer as a radical expression in simplest form.

Homework 11.6 pg 808 # 24-42 evens, 46-60 evens, 67-69, 74-78

Algebra I Chapter 11 section 7 Adding and subtracting radical expressions

Warm-up

Simplify each expression

1. 14x + 15y – 12y + x

2. 9xy + 2xy - 8xy

3. -3(a + b) + 5(2 + 2b)

Simplify.

4. √96

5. √ x9 y10

Square root expressions with the same radicand are examples of _______________________

Add or subtract

3√5+7√5 5√7−6 √7

8√3−5√3 4 √x+2√ x

√2x−√5 x+9√5x 9√3+4 √3

6√ x−7√ y 2√xy−√2 y+9√ xy

√45−√20 9√75+2√50

√75 y−2√27 y+√48 y

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

Homework 11.7 pg 813 #15-29, 32-46 even, 59-61

Algebra I Chapter 11 Section 8 multiplying and dividing radical expressions

Warm up

Simplify each expression

√72

√ x5

√ 249Multiply. Write each product in simplest form

√5√10 √2m√14m (3√7 )2

√6 (√8−3 ) √5 (√10+4√3 ) √3 (7−√8 )

√2 (√8+√18 )

(4+√3 ) (5+√3 ) (3−√8 ) (2+√8 )

(3+√3 ) (8−√3 )

(4+√3 )2 (3−√2 )2

A quotient with a square root in the denominator is not simplified. To simplify these expressions, multiply by a form of 1 to get a perfect square radicand in the denominator. This is called ___________________________ the denominator

Simplify each quotient

√7√2

√11√3

√13m√5

√7 a√12

Homework 11.8 pg 819 #27-52, 56-66 even, 75 – 77

Algebra I Chapter 11 Section 9 solving radical equations

Warm-up

Solve each equation

3x + 5 = 7 4x + 1 = 2x – 3 x/7 = 5

Power property of equality

WORDS NUMBERS ALGEBRAYou can square both sides of an equation and the resulting equation is still true

Solve each equation.

√ x=8 √ x=6 √27 x=9 √3 x=1

√ x+3=10 √ x−2=1 √ x+7=5

√2x−1+4=7 √3 x+7−1=3

2√x=22 2=√ x4

2√ x5

=4

Homework pg 825 #2-24 even, 42-52 even, 87-89,101-102