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(-4)2
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 / a₁
2nd term / 3(2) = 6
3rd term / 3(2²) = 12
4th term / 3(2)3 = 24
Nth 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 10th term of the sequence?
The first term of a geometric sequence is 500 and the common ratio is 0.2 . What is the 7th 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 / 300
2 / 150
3 / 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 / 10,000
2 / 8,000
3 / 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(3)3
4. 2/3(3)4
5. -5(2)5
6. -1/2(4)3
7. 100(0.5)2
8. 3000(0.95)8
An exponential function has the form of ______, where a ≠ 0, b ≠ 0, and b > 0The function fx= 2(3)x models an insect population after x days. What will the population be on the 5th day?
The function fx= 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 fx= 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 fx= 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 ______.
X / fx= 2(3)x1 / 6
2 / 18
3 / 54
4 / 102
Tell whether each set of ordered pairs satisfies an exponential function.
{(-1, 1.5), (0,3), (1, 6), (2, 12)}
X / y{(-1, 1), (0,0), (1,1), (2, 4)}
X / y{(-2,4), (-1,2), (0,1), (1,0.5)}
X / y{(0,4), (1,12), (2,36), (3,108)}
X / y{(-1,-64), (0,0), (1,64), (2,128)}
X / yGraph y=3(4)x
Graph y=0.5(2)x
Graph y=-5(2)x
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 5th 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=P1+rnnt· 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 $48,000. Write an exponential decay function to model this situation. Then find the population after 7 years.
The population of a town is decreasing at a rate of 3% per year. In 2000, there were 1700 people. Write an exponential decay function to model this situation. Then find the population in 2012.
A common application of exponential decay is ______.
A=P(0.5)t· A represents
· P represents
· t represents
Fluorine 20 has a half life of 11 seconds. Find the amount of fluorine 20 left from a 40 gram sample after 44 seconds.
Find the amount of fluorine 20 left from a 40 gram sample after 2 minutes.
Homework 11.3 pg 785 #10-20, 22-32 even, 45-47, 55-60
Algebra I Chapter 11 section 4 Linear, Quadratic and Exponential Models
1. Find the slope and the y intercept of the line that passes through (4,20) and (20, 24)
The population of a town is decreasing at a rate of 1.8% per year. In 1990, there were 4600 people.
2. Write an exponential decay function to model this situation.
3. Find the population in 2010
Look at the tables and graphs below. The data show three ways you have learned that variable quantities can be related.
Training heart rateAge / Beats/min
20 / 170
30 / 161.5
40 / 153
50 / 144.5
Time / height
0.4 / 10.44
0.8 / 12.76
1 / 12
1.2 / 9.96
Round / Teams left
1 / 16
2 / 8
3 / 4
4 / 2
In the real world, people often gather data and then must decide what kind of relationship (if any) they think best describes their data.
Graph each data set. Which kind of model best describes the data?
Time / 0 / 1 / 2 / 3Bacteria / 10 / 20 / 40 / 80
Bacteria / 24 / 96 / 384 / 1536
Boxes / 1 / 5 / 20 / 50
Reams / 10 / 50 / 200 / 500
Look for a pattern in each data set to determine which kind of model best describes the data
Height of golf ball
Time(s) / 0 / 1 / 2 / 3 / 4Height (ft) / 4 / 68 / 100 / 100 / 68
Oven temperature
Time (min) / 0 / 10 / 20 / 30Temperature (f) / 375 / 325 / 275 / 225
Money in CD
Time (yr) / 0 / 1 / 2 / 3Amount ($) / 1000.00 / 1169.86 / 1368.57 / 1601.04
General Forms of functions
LINEAR / QUADRATIC / EXPONENTIALUse 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
Time (mo) / 0 / 1 / 2 / 3Ladybugs / 10 / 30 / 90 / 270
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 (days) / 0 / 1 / 2 / 3# of people who receive the e-mail / 8 / 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+122 250
Product 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 x3y2
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
4x59 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. x9y10
Square root expressions with the same radicand are examples of ______
Add or subtract
35+75 57-67
83-53 4x+2x
2x-5x+95x 93+43
6x-7y 2xy-2y+9xy
45-20 975+250
75y-227y+48y
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
249
Multiply. Write each product in simplest form
510 2m14m 372
68-3 510+43 37-8
28+18
4+35+3 3-82+8
3+38-3
4+32 3-22
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
72 113
13m5
7a12
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 27x=9 3x=1
x+3=10 x-2=1 x+7=5
2x-1+4=7 3x+7-1=3
2x=22 2=x4
2x5=4
Homework pg 825 #2-24 even, 42-52 even, 87-89,101-102