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A Resource Aligned with New York State Algebra 1 Module 3

Table of Contents

Arithmetic Sequences ………………………………………………………………………………………………………………….…….5

Geometric Sequences ……………………………………………………………………………………………………………………...11

Why Do Banks Pay You to Provide Their Services? …………………………………………………………………………..18

Lake Algae Exploration ………………………………………………………………………………………………………………….…24

The Power of Exponential Growth ………………………………………………………………………………………………..…25

Exponential Growth …………………………………………………………………………………………………………………...……31

Review Lessons 1–6 ……………………………………………………………………………………………………………………...…39

Exponential Decay ……………………………………………………………………………………………………………………………42

Why Stay With Whole Numbers? ………………………………………………………………………………………………….…50

Relations and Functions ………………………………………………………………………………………………………………..…57

Interpreting Functions………………………………………………………………………………………………………….………..…63

Representing, Naming, and Evaluating Functions ………………………………………………………………….…………66

Exponential Functions ………………………………………………………………………………………………….…….……………70

Linear and Exponential Functions ……………………………………………………………………………………….…..………75

Multiple Representations of Linear and Exponential Functions …………………………………………….…………79

Mathematical Models and Regression Analysis …………………………………………………………………….…………82

Review of Linear and Exponential Functions ………………………………………………………………………………..…90

Absolute Value Functions ……………………………………………………………………………………………………………….94

Piecewise Functions ………………………………………………………………………………………………………………………100

Step Functions ………………………………………………………………………………………………………………………………103

Graphs Can Solve Equations Too …………………………………………………………………………………………..………105

Vertical Translations of Functions …………………………………………………………………………………………………109

Horizontal Translations of Functions ………………………………………………………………………………………………113

Transformations With Functions ……………………………………………………………………………………………………118

End of Module Review (Sample Regents Exam Questions) …………………………………………………….………123

Name ______________________ Arithmetic SequencesShow all your work! Opening Exercises

Describe each pattern in the sequences below and find the next five numbers.

1) 5, 10, 15, _______, _______, _______, _______, _______

2) 5, 7, 9, 11, _______, _______, _______, _______, _______

3) 1, –1, –3, –5, _______, _______, _______, _______,_______

4) 5.5, 9, 12. 5, _______, _______, _______, _______, _______

Arithmetic Sequences

Learning Target: I can write arithmetic sequences recursively and with an explicit formula.

A sequence is an ordered list of numbers. Each number of a sequence is called a term (or element) of the sequence. An arithmetic sequence follows a pattern of adding a fixed amount from one term to the next.

An explicit formula describes the nth term of a sequence using the number n.

1. A sequence has an explicit formula an=3n−2. What are the first 5 terms of the sequence? What is the common difference between terms?

A recursive formula states the first term of the sequence and relates subsequent terms to the one before.

2. State the first 4 terms of the sequence: a1=12

an=an−1+5 What is the common difference?

3. Is the sequence 1, 4, 9, 16, 25 and arithmetic sequence? Justify your reasoning.

4. What is the 100th term of the arithmetic sequence 6, 11, 16, … ?

5. What is the explicit formula for the nth term of the sequence 1, 3, 5, 7, … ?

6. Write a recursive formula for the sequence 1, 3, 5, 7, …

7. What is the missing term of the arithmetic sequence … , 15, _____ , 59, … ?

Name ______________________ Arithmetic SequencesShow all your work! Problem Set

1. What is a formula for the nth term of sequence B shown below?

1)2)3)4)

2. What is the common difference of the arithmetic sequence 5, 8, 11, 14?

3. Which arithmetic sequence has a common difference of 4?

1) 4, 16, 64, 2562) 8, 4, 0, –43) 6, 10, 14, 184) 4, 8, 14, 18

4. Find the first four terms of the recursive sequence defined below.

5. Find the third term in the recursive sequence an=an−1−1, where a1=3.

6. What are the first five terms of the sequence an=4 n−7. What is the common difference?

7. State the first four terms of the sequence: a1=23

an=an−1−9 What is the common difference?


9. What is the explicit formula for the nth term of the sequence 10, 12.5, 15, 17.5 … ?

10. Write a recursive formula for the sequence 10, 8, 6, 4, …

11. What is the missing term of the arithmetic sequence … , 18, _____ , 92, … ?

Name ______________________ Arithmetic SequencesShow all your work! Exit Ticket

1. Consider the sequence given by a “plus 8” pattern: 2, 10, 18, 26, …. Shae says that the formula for the sequence is f (n )=8n+2. Marcus tells Shae that she is wrong because the formula for the sequence is f (n )=8n−6.

a. Which formula generates the sequence by starting at 𝑛=1?

b. Find the 100th term in the sequence.

Name ______________________ Geometric SequencesShow all your work! Opening Exercises

1. What are the first 4 terms of the sequence an=n2? Is this an arithmetic sequence? Explain.

2. What are the first 4 terms of the sequencea1=−4

an=an−1+3 ? Is this an arithmetic sequence? Explain.


4. What is the missing term in the arithmetic sequence –37, _______, 23?

5. Write an explicit formula and a recursive formula for the sequence 5, 10, 15, 20, ….

Geometric Sequences

Learning Target: I can write geometric sequences recursively and with an explicit formula.

A geometric sequence follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term. The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term yields this common multiple. To find the common ratio, divide the second term by the first term.

Recall: an explicit formula describes the nth term of a sequence using the number n.

1. What are the first 5 terms of a sequence with an explicit formula an=2 ∙3n−1 ?

What is the initial value? What is the common ratio between terms?

Recall: a recursive formula states the first term of the sequence and relates subsequent terms to the one before.

2. State the first 4 terms of the sequence: a1=5

an=an−1 ∙(−2) What is the initial value? What is the common ratio?

3. Is the sequence 4 ,83, 169

, 3227

, 6481 a geometric sequence? Justify your reasoning.

4. What is the tenth term of the geometric sequence 4, 12, 36, … ?

5. What is the explicit formula for the nth term of the sequence 3, 6, 12, 24, 48, … ?

6. Write a recursive formula for the sequence 3, 6, 12, 24, 48, … ?

7. What is the missing terms of the geometric sequence 2, _____ , _____, –54, … ?

8. The third term of a geometric sequence is 3 and the sixth term is 19 . Find the first term.

9. A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

Name ______________________ Geometric SequencesShow all your work! Problem Set

1. A sequence has the following terms: a1=4, a2=10, a3=25, a4=62.5. Which formula represents the nth term in the sequence?

1)2)3)4)

2. What is the explicit formula for the nth term of the sequence 6, 18, 54, … ?

3. What is the common ratio of the geometric sequence 2, 8, 32 ?

4. What is the tenth term of the sequence 5, 10, 20, 40, 80, … ?

5. What is the next term of the geometric sequence 1, –5, 25, –125, _______ ?

6. What are the first 4 terms of a sequence with an explicit formula an=6 ∙(2)n−1 ?

What is the initial value? What is the common ratio between terms?

7. State the first 5 terms of the sequence: a1=

14

an=an−1 ∙( 32 ) What is the initial value? What is the common ratio?

8. Is the sequence 4 ,8 ,12 ,16 a geometric sequence? Justify your reasoning.

9. Write a recursive formula for the sequence 1, 3, 9, 27 … ?

10. What are the missing terms of the geometric sequence 5, _____ , _____, 320, … ?

11. The fourth term of a geometric sequence is 4 and the seventh term is 116 . Find the first term.

Name ______________________ Geometric SequencesShow all your work! Exit Ticket

Identify each sequence as arithmetic or geometric. Explain your answer, and write an explicit formula for the sequence.

Name ______________________ Why do banks pay you to provide their services?Show all your work! Opening Exercises

1. Why banks pay you interest to provide a service to you? Have you ever thought it odd that banks pay YOU interest for the honor of looking after your money for you?

2. How do banks make money?

Why do banks pay you to provide their services?

Learning Target: I can compare the rate of change for simple and compound interest and recognize situations in which a quantity grows by a constant percent rate per unit interval.

Example 1

Kyra has been babysitting since 6th grade. She has saved $1000 and wants to open an account at the bank so that she will earn interest on her savings. Simple Bank pays simple interest at a rate of 10%. How much money will Kyra have after 1 year?

After 2 years, if she does not add money to her account?

Is there a formula we can create to find this amount? After 5 years?

Raoul needs $200 to start a snow cone stand for this hot summer. He borrows the money from a bank that charges 4% simple interest a year.

a. How much will he owe if he waits 1 year to pay back the loan?

2 years?

3 years?

4 years?

5 years?

b. Write a formula for the amount he will owe after tyears.

Example 2

Jack has $500 to invest. The bank offers an interest rate of 6% compounded annually. How much money will Jack have after 1 year? 2 years? 5 years?

Year Amount of money Interest Total amount

Example 3

If you have $200 to invest for 10 years, would you rather invest your money in a bank that pays 7% simple interest or 5% interest compounded annually? Is there anything you could change in the problem that would make you change your answer? (HINT: Use the 2 formulas we created in the previous problems)

Lesson Summary

Simple Interest – Interest is calculated once per year on the original amount borrowed or invested. The interest does not become part of the amount borrowed or owed (the principal). I=PrtCompound Interest – Interest is calculated once per period on the current amount borrowed or invested. Each period, the interest becomes a part of the principal. A=P(1+r )t

Name ______________________ Why do banks pay you to provide their services?Show all your work! Problem Set

1. $250 is invested at a bank that pays 7% simple interest. Calculate the amount of money in the account after 1 year; 3 years; 7 years; 20 years.Work space:

1 year: ______________________

3 years: ______________________

7 years: ______________________

20 years: _____________________

2. $325 is borrowed from a bank that charges 4% interest compounded annually. How much is owed after 1 year; 3 years; 7 years; 20 years?Work space:

1 year: _______________________

3 years: _______________________

7 years: _______________________

20 years: _____________________

3. Joseph has $10,000 to invest. He can go to Yankee Bank that pays 5% simple interest or Met Bank that pays 4% interest compounded annually. After how many years will Met Bank be the better choice?

Name ______________________ Why do banks pay you to provide their services?Show all your work! Exit Ticket

A youth group has a yard sale to raise money for a charity. The group earns $800 but decides to put its money in the bank for a while. Calculate the amount of money the group will have if:

a. Cool Bank pays simple interest at a rate of 4%, and the youth group leaves the money in for 3 years.

b. Hot Bank pays an interest rate of 3% compounded annually, and the youth group leaves the money in for 5 years.

c. If the youth group needs the money quickly, which is the better choice? Why?

Lake Algae Exploration

On June 1, a fast growing species of algae is accidentally introduced into a lake in a city park. It starts to grow and cover the surface of the lake in such a way that the area covered by the algae doubles every day. If it continues to grow unabated, the lake will be totally covered and the fish in the lake will suffocate. At the rate it is growing, this will happen on June 30.

(a) When will the lake be covered half-way?

(b) On June 26, a pedestrian who walks by the lake every day warns that the lake will be completely covered soon. Her friend just laughs. Why might her friend be skeptical of the warning?

(c) On June 29, a clean-up crew arrives at the lake and removes almost all of the algae. When they are done, only 1% of the surface is covered with algae. How well does this solve the problem of the algae in the lake?

Name ______________________ The Power of Exponential GrowthShow all your work! Opening Exercise

Two equipment rental companies have different penalty policies for returning a piece of equipment late:Company 1: On day 1, the penalty is $5. On day 2, the penalty is $10. On day 3, the penalty is $15. On day 4, the penalty is $20 and so on, increasing by $5 each day the equipment is late.

Company 2: On day 1, the penalty is $0.01. On day 2, the penalty is$0.02. On day 3, the penalty is $0.04. On day 4, the penalty is $0.08 and so on, doubling in amount each additional day late.

Jim rented a digger from Company 2 because he thought it had the better late return policy. The job he was doing with the digger took longer than he expected, but it did not concern him because the late penalty seemed so reasonable. When he returned the digger 15 days late, he was shocked by the penalty fee. What did he pay, and what would he have paid if he had used Company 1 instead?

Company 1 Company 2Day Penalty Day Penalty1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10

11 11

12 12

13 13

14 14

15 15

The Power of Exponential Growth

Learning Target: I can solve problems involving exponential formulas.

Use the tables from the opening exercise to answer the following questions.

1. Which company has a greater 15 day late charge?

2. Describe how the amount of the late charge changes from any given day to the next successive day in both companies 1 and 2.

3. How much would the late charge have been after 20 days under Company 2?

Example

Let us understand the difference between f (n)=2n and f (n)=2n.

a. Complete the tables below, and then graph the points (n , f (n)) on a coordinate plane for each of the formulas.

n f (n)=2n

−2

−1

0

1

2

3

b. Describe the change in each sequence when n increases by 1 unit for each sequence.

n f (n)=2n−2

−1

0

1

2

3

Exercise

A typical thickness of toilet paper is 0.001 inches. Seems pretty thin, right? Let’s see what happens when we start folding toilet paper.

a. How thick is the stack of toilet paper after 1 fold? After 2 folds? After 5 folds?

b. Write an explicit formula for the sequence that models the thickness of the folded toilet paper after n folds.

c. After many folds will the stack of folded toilet paper pass the 1 foot mark?

d. The moon is about 240,000 miles from Earth. Compare the thickness of the toilet paper folded 50 times to the distance from Earth.

Watch the following video “How folding paper can get you to the moon” (http://www.youtube.com/watch?v=AmFMJC45f1Q )

http://www.youtube.com/watch?v=AmFMJC45f1Q

http://ed.ted.com/lessons/how-folding-paper-can-get-you-to-the-moon

Name ______________________ The Power of Exponential GrowthShow all your work! Problem Set

1. A bucket is put under a leaking ceiling. The amount of water in the bucket doubles every minute. After 8 minutes, the bucket is full. After how many minutes is the container half full?

2. A three-bedroom house in Burbville was purchased for $190,000. If housing prices are expected to increase 1.8% annually in that town, write an explicit formula that models the price of the house in t years. Find the price of the house in 5 years.

3. In 2013, a research company found that smartphone shipments (units sold) were up 32.7% worldwide from 2012, with an expectation for the trend to continue. If 959 million units were sold in 2013, how many smartphones can be expected to be sold in 2018 at the same growth rate? (Include the explicit formula for the sequence that models this growth.) Can this trend continue?

Name ______________________ The Power of Exponential GrowthShow all your work! Exit Ticket

Chain emails are emails with a message suggesting you will have good luck if you forward the email on to others. Suppose a student started a chain email by sending the message to 3 friends and asking those friends to each send the same email to 3 more friends exactly 1 day after they received it.

a. Write an explicit formula for the sequence that models the number of people who will receive the email on the 𝑛𝑡ℎ day. (Let the first day be the day the original email was sent.) Assume everyone who receives the email follows the directions.

b. Which day will be the first day that the number of people receiving the email exceeds 100?

Name ______________________ Exponential GrowthShow all your work! Opening Exercise

Callie and Joe are examining the population data in the graphs below for a history report. Their comments are as follows:

Callie: It looks like the U.S. population grew the same amount as the world population, but that can’t be right, can it?

Joe: Well, I don’t think they grew by the same amount, but they sure grew at about the same rate. Look at the slopes.

a. Is Callie’s observation correct? Why or why not?

b. Is Joe’s observation correct? Why or why not?

c. Use the World Population graph to estimate the percent increase in world population from 1950 to 2000.

d. Now use the U.S. Population graph to compute the percent increase in the U.S. population for the same time period.

Exponential Growth

Learning Target: I can compare linear and exponential models of population growth.

Example

a. How is this graph similar to the World Population graph in Example 1? How is it different?

b. Does the behavior of the graph from 1950–2000 match that shown on the graph in Example 1?

c. Why is the graph from Example 1 somewhat misleading?

d. An exponential formula that can be used to model the world population growth from 1950 through 2000 is as follows: f (t)=2,519(1.0177t)

where 2,519 represents the world population in the year 1950, and t represents the number of years after 1950. Use this equation to calculate the world population in 1950, 1980, and 2000. How do your calculations compare with the world populations shown on the graph?

1700 1750 1800 1850 1900 1950 20000

1000

2000

3000

4000

5000

6000

7000

World Population

Year

Popu

latio

n (in

mill

ions

)

e. Following is a table showing the world population numbers used to create the graphs above.

YearWorld

Population(in millions)

1700 6401750 824

1800 978

1850 1,244

1900 1,650

1950 2,5191960 2,9821970 3,6921980 4,435

1990 5,263

2000 6,070

How do the numbers in the table compare with those you calculated in part (d) above?

f. How is the formula in part (d) above different from the formula in Example 1 part (g)? What causes the difference? Which formula more closely represents the population?

Exercises

4. The table below represents the population of the U.S. (in millions) for the specified years.

Year U.S. Population(in millions)

1800 51900 762000 282

a. If we use the data from 1800–2000 to create an exponential equation representing the population, we generate the following formula for the sequence, where f (t) represents the U.S. population and t represents the number of years after 1800.

f (t)=5 (1.0204)t

Use this formula to determine the population of the U.S. in the year 2010.

b. If we use the data from 1900–2000 to create an exponential formula that models the population, we generate the following, where f (t) represents the U.S. population and t represents the number of years after 1900.

f (t)=76 (1.013)t

Use this formula to determine the population of the U.S. in the year 2010.

c. The actual U.S. population in the year 2010 was 309 million. Which of the above formulas better models the U.S. population for the entire span of 1800–2010? Why?

d. Complete the table below to show projected population figures for the years indicated. Use the formula from part (b) to determine the numbers.

Year World Population(in millions)

202020502080

e. Are the population figures you computed reasonable? What other factors need to be considered when projecting population?

5. The population of the country of Oz was 600,000 in the year 2010. The population is expected to grow by a factor of 5% annually. The annual food supply of Oz is currently sufficient for a population of 700,000 people and is increasing at a rate which will supply food for an additional 10,000 people per year.a. Write a formula to model the population of Oz. Is your formula linear or exponential?

b. Write a formula to model the food supply. Is the formula linear or exponential?

c. At what point does the population exceed the food supply? Justify your response.

d. If Oz doubled its current food supply (to 1.4 million), would shortages still take place? Explain.

e. If Oz doubles both its beginning food supply and doubles the rate at which the food supply increases, would food shortages still take place? Explain.

Name ______________________ Exponential GrowthShow all your work! Problem Set

1. The table below represents the population of the state of New York for the years 1800–2000. Use this information to answer the questions.

Year Population1800 300,0001900 7,300,0002000 19,000,000

a. Using the year 1800 as the base year, an explicit formula for the sequence that models the population of New York is P(t)=300000(1.021)t, where t is the number of years after 1800. Using this formula, calculate the projected population of New York in 2010.

b. Using the year 1900 as the base year, an explicit formula for the sequence that models the population of New York is P(t)=7300000(1.0096)t, where t is the number of years after 1900. Using this equation, calculate the projected population of New York in 2010.

c. Using the internet (or some other source), find the population of the state of New York according to the 2010 census. Which formula yielded a more accurate prediction of the 2010 population?

Name ______________________ Exponential GrowthShow all your work! Exit Ticket

Do the examples below require a linear or exponential growth model? State whether each example is linear or exponential, and write an explicit formula for the sequence that models the growth for each case. Include a description of the variables you use.

1. A savings account accumulates no interest but receives a deposit of $825 per month.

2. The value of a house increases by 1.5% per year.

3. Every year, the alligator population is 97 of the previous year’s population.

4. The temperature increases by 2° every 30 minutes from 8:00 a.m. to 3:30 p.m. each day for the month of July.

5. Every 240 minutes, 13 of the rodent population dies.

Name ____________________________ Review of Lessons 1 – 6 Show all your work!

1. Write an explicit formula for the sequence below. Is the sequence arithmetic or geometric? Justify your reasoning.

12, 16, 20, 24, …

2. Write an explicit formula for the sequence below. Is the sequence arithmetic or geometric? Justify your reasoning.

2, 8, 32, 128, …

3. Janice deposits $400 in an account that pays 3% simple interest annually. If there are no additional deposits or withdrawals, how much money is in the account after 5 years?

4. Mike deposits $350 in an account that pays 2% interest compounded annually. If there are no additional deposits or withdrawals, how much money is in the account after 6 years?

5. Denise runs three miles each day for ten consecutive days. William runs 0.1 mile on the first day, 0.2 mile on the second day, 0.4 mile on the third day and continues to double the distance he runs each day until day 10.

A. Complete the table below to show the distance that Denise and William run each day.

Denise WilliamDay Distance (miles) Day Distance (miles)1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10

B. Write an explicit formula that models the distance Denise runs on day n.

C. Write an explicit formula that models the distance William runs on day n.

D. Who ran more total miles over the ten day period? Justify your reasoning.

6. The U.S. population was 281 million people in 2000. The population increases at an annual rate of 1.24%. Write an explicit formula that models the U.S. population growth, in millions, after 2000. Use this formula to predict the population in 2020, rounded to the nearest million.

Name ______________________ Exponential DecayShow all your work! Opening Exercise

Malik bought a new car for $15,000. As he drove it off the lot, his best friend, Will, told him that the car’s value just dropped by 15% and that it would continue to depreciate 15% of its current value each year. If the car’s value is now $12,750 (according to Will), what will its value be after 5 years?

Complete the table below to determine the car’s value after each of the next five years.

Number of years, t , passed since

driving the car off the lot

Car value after t years

15% depreciation of current car value

Car value minus the 15% depreciation

0 $12,750.00 $1,912.50 $10,837.50

1 10,837.50

2

3

4

5

Exponential Decay

Learning Target: I can describe and analyze exponential decay models

a. Using the table from the opening exercise, write an explicit formula for the sequence that models the value of Malik’s car t years after driving it off the lot.

b. Use the formula from part (b) to determine the value of Malik’s car five years after its purchase. Round your answer to the nearest cent. Compare the value with the value in the table. Are they the same?

c. Use the formula from part (b) to determine the value of Malik’s car 7 years after its purchase. Round your answer to the nearest cent.

Exercises

1. Identify the initial value in each formula below, and state whether the formula models exponential growth or exponential decay. Justify your responses.

a. f (t)=2(25 )t

b. f (t)=2(53 )t

c. f (t)=23

¿

d. f (t)=23 ( 13 )

t

e. f (t)=32 ( 23 )

t

2. If a person takes a given dosage (d) of a particular medication, then the formula f (t)=d¿ represents the concentration of the medication in the bloodstream t hours later. If Charlotte takes 200 mg of the medication at 6 :00 a.m., how much remains in her bloodstream at 10 :00 a.m.? How long does it take for the concentration to drop below 1mg?

3. Ryan bought a new computer for $2,100. The value of the computer decreases by 50% each year. When will the value drop below $300?

4. According to the International Basketball Association (FIBA), a basketball must be inflated to a pressure such that, when it is dropped from a height of 1,800 mm, it will rebound to a height of 1,300 mm. Maddie decides to test the rebound-ability of her new basketball. She assumes that

the ratio of each rebound height to the previous rebound height remains the same at 1,3001,800 . Let

f (n) be the height of the basketball after n bounces. Complete the chart below to reflect the heights Maddie expects to measure.

n f (n)

0 1,800

1

2

3

4

a. Write the explicit formula for the sequence that models the height of Maddie’s basketball after any number of bounces.

b. Plot the points from the table. Connect the points with a smooth curve, and then use the curve to estimate the bounce number at which the rebound height will drop below 200 mm.

Name ______________________ Exponential DecayShow all your work! Problem Set

1. A construction company purchased some equipment costing $300,000. The value of the equipment depreciates (decreases) at a rate of 14% per year.a. Write a formula that models the value of the equipment.

b. What is the value of the equipment after 9 years?

c. Graph the points (t , v ( t ) ) for integer values of 0≤ t ≤15.

d. Estimate when the equipment will have a value of $50,000.

2. The number of newly reported cases of HIV (in thousands) in the United States from 2000 to 2010 can be modeled by the following formula:

f ( t )=41¿, where t is the number of years after 2000.a. Identify the growth factor.

b. Calculate the estimated number of new HIV cases reported in 2004.

c. Graph the points (t , f ( t ) ) for integer values of 0≤ t ≤10.

d. During what year did the number of newly reported HIV cases drop below 36,000?

3. Sam bought a used car for $8,000. He boasted that he got a great deal since the value of the car two years ago (when it was new) was $15,000. His friend, Derek, was skeptical, stating that the value of a car typically depreciates about 25% per year, so Sam got a bad deal. a. Use Derek’s logic to write a formula for the value of Sam’s car. Use t for the total age of the

car in years.

b. Who is right, Sam or Derek?

Name ______________________ Exponential DecayShow all your work! Exit Ticket

A huge Ping-Pong tournament is held in Beijing, with 65,536 participants at the start of the tournament. Each round of the tournament eliminates half the participants.

a. If f (r ) represents the number of participants remaining after 𝑟 rounds of play, write a formula to model the number of participants remaining.

b. Use your model to determine how many participants remain after 10 rounds of play.

c. How many rounds of play will it take to determine the champion Ping-Pong player?

Name ______________________ Why Stay With Whole numbers?Show all your work! Opening Exercise

The sequence of perfect squares {1,4,9,16,25 ,… } earned its name because the ancient Greeks realized these quantities could be arranged to form square shapes.

1st 2nd 3rd 4th 5th

If S(n) denotes the nth square number, what is a formula for S(n)?

Why Stay With Whole numbers?

Learning Target: I can use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Exercises

1. Prove whether or not 169 is a perfect square.

2. Prove whether or not 200 is a perfect square.

3. If S(n)=225, then what is n?

Instead of arranging dots into squares, suppose we extend our thinking to consider squares of side length x cm.

4. Create formula for the area A(x ) cm2 of a square of side length x cm. A (x )=¿ ___________

5. Use the formula to determine the area of squares with side lengths of 3 cm, 10.5 cm, and π cm.

6. What does A(0) mean?

7. What does A(−10) and A(√2) mean?

The triangular numbers are the numbers that arise from arranging dots into triangular figures as shown:

8. What is the100th triangular number?

9. Find a formula forT (n), the nth triangular number (starting with n=1).

10. How can you be sure your formula works?

11. Create a graph of the sequence of triangular numbers, T (n )=n(n+1)2

, where n is a positive

integer.

12. Create a graph of the triangle area formula T (x)= x (x+1)2

, where x is any positive real number.

Name ______________________ Why Stay With Whole numbers?Show all your work! Problem Set

1. A tile pattern is shown below.

Figure 1 Figure 2 Figure 3 Figure 4

a. How is this pattern growing?

b. Create an explicit formula that could be used to determine the number of squares in the nth

figure.

c. Evaluate your formula for n = 0 and n = 2.5. Draw Figure 0 and Figure 2.5, and explain how you decided to create your drawings.

Name ______________________ Why Stay With Whole numbers?Show all your work! Exit Ticket

Recall that an odd number is a number that is one more than or one less than twice an integer. Consider the sequence formed by the odd numbers {1,3,5,7,…}.

1. Find a formula for f(𝑛), the 𝑛th odd number starting with 𝑛=1.

2. Write a convincing argument that 121 is an odd number.

3. What is the meaning of f (17)?

Relations and Functions

Learning Target: I understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range and can apply this definition in the context of a situation.

A relation is a set of pairs of input and output values. You can represent a relation in 4 different ways as shown below.

A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched to one and only one element of 𝑌. The set 𝑋 is called the domain of the function.

1. The function f has a domain of {9 ,11 ,13 ,15} and a range of {8 ,10 ,12}. Could f be represented by { (9 ,8 ) , (11 ,10 ) , (13 ,12 ) }? Justify your reasoning.

2. Determine if the relation is a function. Justify your reasoning.

3. Determine if the relation is a function. Justify your reasoning.

You can use a vertical line test to determine whether a relation is not a function over a specified domain. The vertical line test states that if a vertical line passes through more than one point on a graph of a relation, then the relation is not a function.

4. Determine if the graph below could represent a function. Justify your reasoning.

5. Determine if the graph below could represent a function over the real numbers. Justify your reasoning.

It is important to note that the vertical line test is not sufficient to show that a relation is a function without considering a specified domain.

Consider the following graph in which the domain is the set of integer years from 1988 to 1997.

6. The graph below shows the circulation of newspapers in a town for integer years. According to the graph, is the circulation of newspapers a function of the years shown? Justify your reasoning.

7. Which statement is true about the relation shown on the graph below?

1) It is a function because there exists one x-coordinate for each y-coordinate.

2) It is a function because there exists one y-coordinate for each x-coordinate.

3) It is not a function because there are multiple y-values for a given x-value.

4) It is not a function because there are multiple x-values for a given y-value.

8. If f ( x )=−2x+5, determine the value of each of the following:

f (−3 )=¿

f (0 )=¿

f ( 14 )=¿

Name ______________________ Relations and FunctionsShow all your work! Problem Set

1. The table below shows the cost of parking in a 24-hour garage for a given number of hours 0<t<24. Does this correspondence represent a function? Justify your reasoning.

2. Is the relation y=g ( x )=√2−x a function from the Real numbers to the Real numbers? Justify your reasoning.

3. Is the relation below a function from the Real numbers to the Real numbers? Justify your reasoning.

y= p ( x )={x+2 x ≤2x x≥0

4. Is y a function of x? Justify your reasoning.

f ( x )={−x x<0x x ≥0

5. Does each element of the domain correspond to exactly one element in the range in the following sequence? Justify your reasoning.

f (1 )=2f (n )=f (n−1 )+2n for all natural numbers n>1

6. If f ( x )=13x+9, which statement is always true?

(1) f ( x )<0 (3) If x<0, then f ( x )<0.

(2) f ( x )>0 (4) If x>0, then f ( x )>0.

FUNCTION

Definition

Nonexamples

Examples

Facts/Characteristics

Interpreting Functions

1. Use the graph (for example, by marking specific points) to illustrate the statements in (a)–(d). Ifpossible, label the coordinates of any points you draw.

a. f (0 )=2

b. f (−3 )=f (3 )= f (9 )=0

c. f (2 )=g(2)

d. g ( x )> f (x ) for x>2

2. Let f (t ) be the number of people, in millions, who own cell phones years after 1990. Explain themeaning of the following statements.

a. f (10 )=100.3

b. f ( a )=20

c. f (20 )=b

d. n=f (t)

3. The figure shows the graph of T , the temperature (in degrees Fahrenheit) over one particular20-hour period in Santa Elena as a function of time t .

a. Estimate T(14).

b. If t=0 corresponds to midnight, interpret what we mean by T (14) in words.

c. Estimate the highest temperature during this period from the graph.

d. When was the temperature decreasing?

e. If Anya wants to go for a two-hour hike and return before the temperature gets over 80 degrees, when should she leave?

Name ___________________ Representing, Naming, and Evaluating FunctionsShow all your work! Opening Exercise

Study the 4 representations of a function below. How are these representations alike? How are they different?

TABLE:

Input 0 1 2 3 4 5Output 1 2 4 8 16 32

FUNCTION: DIAGRAM:

Let f : {0,1,2,3,4,5 }→ {1,2,4,8,16,32} such that x↦ 2x.

SEQUENCE:

Let an+1=2an , a0=1 for 0≤n≤5 where n is an integer.

f

1

2

4

8

16

32

0

1

2

3

4

5

Representing, Naming, and Evaluating Functions

Learning Target: I can use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context

Exercise 1

Let X={0,1,2,3,4,5 }. The function f follows the rule:

f : X→Y

Assigneach x∈X ¿the expression 2x .

What are f (0), f (1), f (2), f (3), f (4 ), and f (5)?

What is the range of f ?

Exercise 2

The squaring function is defined as follows:

Let f : X→Y be the function such that x↦ x2, where X is the set of all real numbers.

What are f (0), f (3), f (−2), f (√3), f (−2.5), f ( 23 ), f (a), and f (3+a)?

What is the range of f ?

Name ___________________ Representing, Naming, and Evaluating FunctionsShow all your work! Problem Set

1. Let f ( x )=6x−3 , and let g ( x )=0.5 (4 )x. Find the value of each function for the given input.

f (0 )=¿ g (−1 )=¿

f (−10 )=¿ g (0 )=¿

f (2 )=¿ g (2 )+g(1)=¿

2. Since a variable is a placeholder, we can substitute letters that stand for numbers in for x. Let f ( x )=6x−3 , and let g ( x )=0.5 (4 )x, and suppose a and b are real numbers. Find the value of each function for the given input.

f (a )=¿

g (b )=¿

3. Provide a suitable domain and range to complete the definition of each function.

a. Let f ( x )=2x+3.

b. Let f ( x )=2x.

c. Let C ( x )=9x+130, where C (x) is the number of calories in a sandwich containing x grams of fat.

d. Let B (x )=100 (2 )x, where B(x) is the number of bacteria at time x hours over the course of one day.

Name ___________________ Representing, Naming, and Evaluating FunctionsShow all your work! Exit Ticket

Given 𝑓 as described below. 𝑓:{𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}→{𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠} Assign each whole number to its largest place value digit.

For example, f (4 )=4, f (14 )=4, and f (194 )=9.

a. What is the domain and range of 𝑓?

b. What is f (257)?

c. What is f (0)?

d. What is f (999)?

e. Find a value of 𝑥 that makes the equation f ( x )=7 a true statement.

Name ___________________ Exponential FunctionsShow all your work! Opening Exercise

Let f ( x )=3 (4 )x. Complete the table below.

Exponential Functions

Learning Target: I can graph exponential functions and identify key features of the graph including end-behavior.

1. a. Make a table of values and a graph for the equation y=2x.

b. Is y=2x a function? Justify your reasoning.

c. What is the domain of y=2x? Justify your reasoning.

d. What is the range of y=2x? Justify your reasoning.

e. What is the y-intercept of y=2x? Justify your reasoning.

f. On the graph of y=2x, as the value of x increases, the value of y ______________.

2. Make a table of values and a graph for the equation y=(12 )x

.

b. Is ¿( 12 )x

a function? Justify your reasoning.

c. What is the domain of ¿( 12 )x

? Justify your reasoning.

d. What is the range of ¿( 12 )x


e. What is the y-intercept of ¿( 12 )x


f. On the graph of ¿( 12 )x

, as the value of x increases, the value of y ______________.

3. You invested $1000 in a savings account at the end of 6th grade. The account pays 5% annual interest. How much money will be in the account after six years?

4. Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no additional deposits or withdrawals, how many years will it take for the account to grow to at least $1500?

5. Matt bought a new car at a cost of $25,000. The car depreciates approximately 15% of its value each year. What is the decay factor? What will the car be worth in 10 years?

Linear and Exponential Functions

Learning Target: I can distinguish between situations that can be modeled with linear functions and with exponential functions

Linear Functions

A. Sketch points A (0,4 ) and B (3,10 ). Draw the line that passes through points A and B.

B. Write the equation of the line that passes through the points A (0,4 ) and B (3,10 ).

C. Is the relation represented by the graph in part A and the equation in part B a function over the real numbers? Justify your reasoning.

D. Sketch points C (0,4 ) and D (0 ,−2 ). Draw the line that passes through points C and D.

E. Write the equation of the line that passes through the points C (0,4 ) and D (0 ,−2 ).

F. Is the relation represented by the graph in part D and the equation in part E a function over the real numbers? Justify your reasoning.

Exponential Functions

A. A lab researcher records the growth of the population of a yeast colony and finds that the population doubles every hour. Complete the researcher’s table of data:

B. What is the exponential function that models the growth of the colony’s population?

Graphs (c) and (d) are both graphs of an exponential function of the form g ( x )=abx. Rewrite the function g(x ) using the values of a and b required for the graph shown to be a graph of g.

e. g(x )=¿

f. g(x )=¿

(−2,0.5 )

(0,2 )

(−1,2 )

(2 , 274 )

Hours into study 0 1 2 3 4

Yeast colony population (thousands)

5

Name ___________________ Linear and Exponential FunctionsShow all your work! Exit Ticket

A. Write an equation of the function represented in the table below.

x g(x)–3 –7–2 –3–1 10 51 92 13

B. Write an equation of the function represented in the table below.

x h(x)–1 1.250 51 202 803 3204 1280

C. How does the y-intercept of the function g(x) compare to the y-intercept of h(x)?

D. How does the rate of change of the function g(x) compare to the rate of change of h(x)?

Multiple Representations of Linear and Exponential Functions

Learning Target: I can write an equation for a linear or exponential function represented on a table or graph.

1. Write an equation of the function represented in the table below.

x –1 0 1 2 3 4f(x) 1 3 5 7 9 11


x –1 0 1 2 3 4

f(x)23 2 6 18 54 162


x –3 –2 –1 0 1 2

f(x) 5.5 5 4.5 4 3.5 3


x –3 –2 –1 0 1 2

f(x) 32 16 8 4 2 1

5. Write an equation for each of the functions represented in the graphs below.

6. Graph the function ¿3 ∙2x .

7. Graph the function y=3.

Mathematical Models and Regression Analysis

Learning Target: I can fit a linear or exponential function to a set of data.

Before attempting a regression analysis of data, it is often helpful to examine a scatter plot of the data to see which regression model is most likely going to be a good fit. Keep in mind that when working with real world data, it is unlikely that any regression model is going to be a "perfect" fit. The goal is to find the model that fits as many of the data points as possible and will be the best indicator of trends in the data.

A scatter plot graphically displays two related sets of data. Such a visual representation can indicate patterns, trends and relationships.

1. Which type of function (linear or exponential) would best model the data in each of the scatter plots shown below? Explain your reasoning.

2. A rapidly growing bacteria has been discovered. Its growth rate is shown in the chart.

Hours Since Observation Began Number of Bacteria in Sample0 201 402 753 1504 2975 510

A. Prepare a scatter plot of the data with hours as the independent variable and the number of bacteria as the dependent variable.

B. Determine which regression model will best approximate your data. Justify your reasoning.

C. Write the regression equation for your model, rounding values to three decimal places.

D. Using your regression equation, predict how many bacteria, to the nearest integer, will be present in 12 hours. Does your prediction seem reasonable? Explain

E. Using your regression equation, determine how many bacteria, to the nearest integer, will be present in 3.5 hours. Is your answer reasonable? Explain.

3. A California Population Projection Engineer in 1920 was tasked with finding a model that predicts the state’s population growth. He modeled the population growth as a function of time, t years since 1900. Census data shows that the population in 1900, in thousands, was 1,490. In 1920, the population of the state of California was 3,554 thousand. He decided to explore both a linear and an exponential model.

a. Use the data provided to determine the equation of the linear function that models the population growth from 1900 to 1920.

b. Use the data provided and your calculator to determine the equation of the exponential function that models the population growth.

c. Use the two functions to predict the population for the following years:

Projected Population based on Linear Function,

f (t)(thousands)

Projected Population based on Exponential

Function, g(t )(thousands)

Census Population Data and Intercensal Estimates for

California(thousands)

1935

1960

2010

d. Which function is a better model for the population growth of California in 1935 and in 1960?

e. Does either model closely predict the population for 2010? What phenomenon explains the real population value?

1880 1900 1920 1940 1960 1980 2000 2020 -

5,000,000 10,000,000 15,000,000 20,000,000 25,000,000 30,000,000 35,000,000 40,000,000

California Population Growth: 1900-2010

Year

Popu

latio

n in

Tho

usan

ds

Name _________________________ Mathematical Models and Regression AnalysisShow all your work! Problem Set

1. Australia experienced a major pest problem in the early 20th century. The pest? Rabbits. In 1859, 24 rabbits were released by Thomas Austin at Barwon Park. In 1926, there were an estimated 10 billion rabbits in Australia. Based on this information, write an exponential function that would model Australia’s rabbit population growth.

2. The population of a town in 2007 is 15,000 people. The town has gotten its fresh water supply from a nearby lake and river system with the capacity to provide water for up to 30,000 people. Due to its proximity to a big city and a freeway, the town’s population has begun to grow more quickly than in the past. The table below shows the population counts for each year from 2007 to 2012.

Year Years past 2007Population of the

town

2007 0 15,000

2008 1 15,600

2009 2 16,224

2010 3 16,873

2011 4 17,548

2012 5 18,250

A. Write a linear function that models this data. Use your function to predict the population in 2015.

B. Write an exponential function that models this data. Use your function to predict the population in 2015.

C. Which of the two functions is a better model for this data? Justify your reasoning.

3. A real estate agent plans to compare the price of a cottage, y, in a town on the seashore to the number of blocks, x, the cottage is from the beach. The accompanying table shows a random sample of sales and location data. Write a linear regression equation that relates the price of a cottage to its distance from the beach. Use the equation to predict the price of a cottage, to the nearest dollar, located three blocks from the beach.

4. A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 10-minute period.

Time in Minutes (x) 0 2 4 6 8 10Temperature in oF (y) 180.2 165.8 146.3 135.4 127.7 110.5

A. Determine which regression model will best approximate your data. Justify your reasoning.

B. Write the regression equation for your model, rounding values to the nearest thousandth.

C. Use your equation to predict the time in minutes at which the temperature of the soup will be 100℉.

Name __________________________ Mathematical Models and Regression AnalysisShow all your work! Additional Practice

1. The accompanying table shows the enrollment of a preschool from 1980 through 2000. Write a linear regression equation to model the data in the table. Use x=0 to represent the year 1980. Use your equation to predict the enrollment in 2015.

2. The data table below shows water temperatures at various depths in an ocean. Write the linear regression equation for this set of data, rounding all values to the nearest thousandth. Using this equation, predict the temperature, to the nearest integer, at a water depth of 255 meters.

3. A population of single-celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table below. Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 18th hour.

4. The table below shows the results of an experiment involving the growth of bacteria.

A. Prepare a scatter plot of the data.

B. Determine which regression model will best approximate your data. Justify your reasoning.

C. Write the regression equation for your model, rounding values to three decimal places.

D. Using this equation, predict the bacteria’s growth, to the nearest integer, after 15 minutes.

Name ____________________________ Linear and Exponential Functions ReviewShow all your work!

1. Let f ( x )=8x−2. Find the value of f (4 ).

2. Let g ( x )=3(6)x. Find the value of g(4).

3. Make a table of values for the function g ( x )=75( 14 )x

.

A. Make a table of values for x-values of –1, 0, 1, 2, 3.

x h(x)

–1

0

1

2

3

B. How many times greater is h(2) than h(3)?

4. Michael’s baseball card collection is valued at $500. He predicts that the value of his collection will increase by 15% each year.

A. Write a function that represents the value of Michael’s card collection after x years.

B. Use your function to determine the value of Michael’s collection in 10 years.

5. Consider the following functions: f ( x )=3 x+5 and g ( x )=5 (3x)

A. On the same set of axes graph and label the functions f (x) and g(x ).

B. Use the equations and graphs in Part A to explain the difference between linear and exponential functions.


x –2 –1 0 1 2 3f(x) 0.375 1.5 6 24 96 384


x –2 –1 0 1 2 3

f(x) 14 10 6 2 –2 –6

8. Write the equation of the line that passes through the points (–5, 4) and (7, 40).

9. Write an equation for the function represented in the graph below.

10. The number of reported crime cases in a county in New York State is shown in the accompanying table. Write the linear regression equation that represents this set of data. (Let represent 1999.) Using this equation, find the projected number of new cases for 2015, rounded to the nearest whole number.

Name __________________________ Absolute Value FunctionsShow all your work! Opening Exercises

Solve each one variable equation.

1. |x|=6

2. |x−5|=4

3. 2|x+3|=−10

Absolute Value Functions

Learning Target: I can graph an absolute value function and identify its key features.

1. a. Make a table of values and a graph for the equation y=|x|.

b. Isy=|x| a function? Justify your reasoning.

c. What is the domain ofy=|x|? Justify your reasoning.

d. What is the range ofy=|x|? Justify your reasoning.

e. What are the x and y-intercepts of y=|x|?

2. a. Make a table of values and a graph for the equation y=|x−5|.

b. Isy=|x−5| a function? Justify your reasoning.

c. What is the domain ofy=|x−5|? Justify your reasoning.

d. What is the range ofy=|x−5|? Justify your reasoning.

e. What are the x and y-intercepts of y=|x−5|?

3. a. Make a table of values and a graph for the equationx=|y|.

b. Isx=|y| a function? Justify your reasoning.

c. How are the graphs of y=|x|, y=|x−5|,∧x=|y| similar?

d. How are the graphs of y=|x|, y=|x−5|,∧x=|y| different?

Name __________________________ Absolute Value FunctionsShow all your work! Problem Set

1. Graph f ( x )=¿x+3∨¿ for the domain–5≤ x≤3

2. Graph f ( x )=¿2 x∨¿ for the domain –3≤ x≤3

3. Graph f ( x )=|2x−5| for 0≤ x≤5

Piecewise Functions

Learning Target: I can graph a piecewise function and identify its key features.

The absolute value function f is defined by setting f (x)=¿x∨¿ for all real numbers. Another way to write f is as a piecewise linear function:

f ( x )={−x x<0x x ≥0

Example

Let g(x )=¿ x−5∨¿. The graph of g is the same as the graph of the equation y=¿x−5∨¿ you drew in Exercise 3. Use the redrawn graph below to re-write the function g as a piecewise function.

1. The Heaviside function is defined using the formula below. Graph this function and state its domain and range.

H ( x )={−1, x<00 , x=01, x>0

2. Graph the following piecewise function for the specified domain: f ( x )={ x if x ≤0x+1 if x>0

3. Graph the following piecewise function for the specified domain: f ( x )={2 x+3 if x←13−x if x ≥−1

4. Write a piecewise function for each graph below. a.

b.

c.

1

1

Step Functions

Learning Target: I can graph a step function and identify its key features.

1. The following piecewise function is an example of a step function.

S ( x )={3 −5≤ x←21 −2≤x<32 3≤ x≤5

a. Graph this function and state the domain and range.

b. Why is this type of function is called a step function?

2. Let f ( x )=|x|x

wherex can be any real number except 0.

a. Why is the number 0 excluded from the domain of f ?

b. What is the range of f?

c. Make a table of values and a graph of f .

d. Express f as a piecewise function.

Name ___________________ Graphs Can Solve Equations TooShow all your work! Opening Exercise

1. Solve for x in the following equation algebraically: |x+2|−3=0.5 x+1

Graphs Can Solve Equations Too

Learning Target: I can solve an equation of the form f ( x )=g (x) visually by looking for the intersection points of the graphs of 𝑦=(𝑥) and 𝑦=𝑔(𝑥).

1. Let f (x)=¿x+2∨−3 and g(x )=0.5 x+1. When does f ( x )=g (x)? To answer this, first graph y=f (x ) and y=g (x) on the same set of axes.

2. When does f ( x )=g (x)? What is the visual significance of the points where f ( x )=g (x)?

3. Use your calculator to solve |0.5 x|−5=−|x−3|+4 .

4. Use your calculator to solve −|x−3.5|+4=−0.25 x−1.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

5. Use your calculator to approximate the values of the solution set of: 2 (1.5 )x=2+1.5 x

6. Use your calculator to find all three solutions to the equation x2=2x.

7. In each exercise, the graphs of the functions f and g are shown on the same Cartesian plane. Estimate the solution set to the equation f (x)=g (x). Assume that the graphs of the two functions only intersect at the points shown on the graph.

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y

Vertical Translations of Functions

Learning Target: I can identify the effect on a graph by replacing f (x) by f ( x )+k .

Let f ( x )=|x|, g ( x )=f (x)−3, h ( x )= f (x )+2 for any real number x.

1. Write an explicit formula for g(x ) in terms of |x| (i.e., without using f (x) notation):

2. Write an explicit formula for h(x ) in terms of |x| (i.e., without using f (x) notation):

3. Complete the table of values for these functions.

x f ( x )=|x| g ( x )=f (x)−3 h ( x )=f (x)+2

−3

−2

−1

0

1

2

3

4. Graph all three equations: y=f (x ), y=f ( x )−3, and y=f ( x )+2.

5. What is the relationship between the graph of y=f (x ) and the graph of y=f ( x )+k?

6. How do the values of g and h relate to the values of f ?

Let f ( x )=¿x∨¿ for every real number x. The graph of y=f (x ) is shown below. Describe how the graph for each function below is a transformation of the graph of y=f (x ). Then use this same set of axes to graph each function.

a ( x )=|x|+ 32

e (x )=|x|−3

Name __________________________ Vertical Translations of FunctionsShow all your work! Exit Ticket

Given the graph of

A. Graph and label f ( x )−3 B. Graph and label f ( x )+2

Horizontal Translations of Functions

Learning Target: I can identify the effect on a graph by replacing f (x) by f ( x+k ).

Let f ( x )=|x|

g ( x )=f (x−3)

h ( x )= f (x+2)

1. Write the formula for g(x ) in terms of |x| (i.e., without using f (x) notation):

2. Write the formula for h(x ) in terms of |x| (i.e., without using f (x) notation):

3. Complete the table of values for these functions.

x f ( x )=¿x∨¿ g ( x )=¿ h ( x )=¿

−3

−2

−1

0

1

2

3

4. Graph all three equations: y=f (x ), y=f (x−3), and y=f (x+2).

5. How does the graph of y= f (x ) relate to the graph of y=f (x−3)?

6. How does the graph of y= f (x ) relate to the graph of y=f (x+2)?

7. How does the graph of y=|x|−3 and the graph of y=|x−3| relate differently to the graph of y=¿x∨¿?

8. How do the values of g and h relate to the values of f ?

Exercises

1. Karla and Isamar are disagreeing over which way the graph of the function g ( x )=|x+3| is translated relative to the graph of f ( x )=|x|. Karla believes the graph of g is “to the right “of the graph of f , Isamar believes the graph is “to the left.” Who is correct? Use the coordinates of the vertex of f and g and to support your explanation.

2. Let f ( x )=|x| where x can be any real number. Write a formula for the function whose graph is the transformation of the graph of f given by the instructions below.a. A translation right 5 units.

b. A translation down 3 units.

c. A translation up 5 units.

d. A translation left 4 units.

e. A translation 1 unit up and 2 units to the right.

3. Write the formula for the function depicted by the graph.

4. Write the formula for the function depicted by the graph.

5. The vertex of the quadratic function f ( x )=x2 is at (0,0), which is the minimum for the graph of f . Based on your work in this lesson, to where do you predict the vertex will be translated for the graphs of g ( x )=(x−2)2 and h ( x )=(x+3)2?

6. Complete the table of values.

x f ( x )=x2 g ( x )=(x−2)2 h ( x )=(x+3)2

−3−2−1

0123

Name __________________________ Translations of FunctionsShow all your work! Exit Ticket

Given the graph of

A. Graph and label f ( x )−3 B. Graph and label f ( x )+2

A. Graph and label f ( x−3 ) B. Graph and label f ( x+2 )

Transformations with Functions

There are sets of functions, called families, in which each function is a transformation of a special function called the parent. A parent function is the simplest form in a set of functions that form a family. Each function in the family is a transformation of the parent function.

A translation is a transformation that shifts the graph of a function horizontally or vertically.

Vertical Translation

1. On the same set of axes graph and label f ( x )=x2, f ( x )+2, and f ( x )−3.

The graph of f ( x )+2 is a _____________shift. It translates the graph of f ( x ) _____ units _______.

The graph of f ( x )−3 is a _____________shift. It translates the graph of f ( x ) _____ units _______.

In general, f ( x )+a translates f (x) _______________. If a > 0, the graph slides ______ and if a < 0, the graph slides ______.

Horizontal Translation

2. On the same set of axes graph and label f ( x )=|x|, f ( x+4 ), and f ( x−5 ).

The graph of f ( x+4 ) is a _____________shift. It translates the graph of f ( x ) _____ units _______.

The graph of f ( x−5 ) is a _____________shift. It translates the graph of f ( x ) _____ units _______.

In general, f (x+a) translates f (x) _______________. If a > 0, the graph slides ______ and if a < 0, the graph slides ______.

Vertical Scaling

A vertical stretch multiplies all y-values of a function by the same factor greater than 1. A vertical stretch will stretch a graph away from the x-axis.

A vertical compression reduces all y-values of a function by the same factor between 0 and 1. A vertical compression will squeeze a graph towards the x-axis.

3. On the same set of axes graph and label f ( x )=|x| and 2 f ( x ) and 12f (x).

Compared to the graph of f (x), the graph of af ( x) is a _________________ when a > 0 and a ___________________ when 0 < x < 1.

Horizontal Scaling

A horizontal compression multiplies all x-values of a function by the same factor greater than 1. A horizontal compression will squeeze a graph towards the y-axis.

A horizontal stretch reduces all x-values of a function by the same factor between 0 and 1. A horizontal stretch will stretch a graph away from the y-axis.

4. On the same set of axes graph and label f ( x )=x2 and f (2 x ) and f ( 12 x).

Compared to the graph of f (x), the graph of f (kx ) is a _________________ when k > 0 and a ___________________ when 0 < k < 1.

Name __________________________ Transformations with FunctionsShow all your work! Exit Ticket

Given the graph of

A. Graph and label B. Graph and label

C. Graph and label 2( f ( x )) D. Graph and label f (2x )

End of Module Review (Sample Regents Exam Questions)

1. Emma recently purchased a new car. She decided to keep track of how many gallons of gas she used on five of her business trips. The results are shown in the table below.

Write the linear regression equation for these data where miles driven is the independent variable. (Round all values to the nearest hundredth.)

2. On the set of axes below, graph the function y=|x+1|.

State the range of the function.

State the domain over which the function is increasing.

3. The table below lists the total cost for parking for a period of time on a street in Albany, N.Y. Thetotal cost is for any length of time up to and including the hours parked. For example, parking forup to and including 1 hour would cost $1.25; parking for 3.5 hours would cost $5.75.

Graph the step function that represents the cost for the number of hours parked.

Explain how the cost per hour to park changes over the six-hour period.

4. About a year ago, Joey watched an online video of a band and noticed that it had been viewedonly 843 times. One month later, Joey noticed that the band’s video had 1708 views. Joey madethe table below to keep track of the cumulative number of views the video was getting online.

a) Write a regression equation that best models these data. Round all values to the nearesthundredth. Justify your choice of regression equation.

b) As shown in the table, Joey forgot to record the number of views after the second month.Use the equation from part a to estimate the number of full views of the online video thatJoey forgot to record.

5. Given the functions g(x), f(x), and h(x) shown below:

The correct list of functions ordered from greatest to least by average rate of change over the interval 0≤ x≤3 is (1) f(x), g(x), h(x) (2) h(x), g(x), f(x) (3) g(x), f(x), h(x) (4) h(x), f(x), g(x)

6. The value of the x-intercept for the graph of 4 x−5 y=40 is

(1) 10 (3) −45

(2) 45 (4) −8

7. Draw the graph of y=√x−1 on the set of axes below.

8. A population that initially has 20 birds approximately doubles every 10 years. Which graph represents this population growth?

9. Which situation could be modeled by using a linear function?

(1) a bank account balance that grows at a rate of 5% per year, compounded annually(2) a population of bacteria that doubles every 4.5 hours(3) the cost of cell phone service that charges a base amount plus20 cents per minute(4) the concentration of medicine in a person’s body that decays by a factor of one-third every hour

10. The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a1, which is an equation for the nth term of this sequence?

(1) an=8n+10 (3) an=16n+10(2) an=8n−14 (4) an=16n−38

11. The table below represents the function F.

The equation that represents this function is

(1) F ( x )=3x (3) F ( x )=2x+1(2) F ( x )=3 x (4) F ( x )=2 x+3

12. A company that manufactures radios first pays a start-up cost, and then spends a certain amount of money to manufacture each radio. If the cost of manufacturing r radios is given by the function c (r )=5.25 r+125, then the value 5.25 best represents

(1) the start-up cost(2) the profit earned from the sale of one radio(3) the amount spent to manufacture each radio(4) the average number of radios manufactured

13. The breakdown of a sample of a chemical compound is represented by the functionp (t )=300(0.5)t, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

14. If f (1 )=3 and f ( n )=−2 f (n−1 )+1, then f (5 )=¿

(1) –5 (3) 21

(2) 11 (4) 43

15. A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Which function(s) shown below can be used to determine the height, f (n), of the sunflower in n weeks?

I. f (n )=2n+3

II. f (n )=2n+3(n−1)

III. f (n )=f (n−1 )+2 where f (0 )=3

(1) I and II (3) III, only

(2) II, only (4) I and III

16. Officials in a town use a function, C, to analyze traffic patterns. C (n) represents the rate of traffic through an intersection where n is the number of observed vehicles in a specified time interval. What would be the most appropriate domain for the function?

(1) {…−2 ,−1 ,0 ,1 ,2 ,3 ,…} (3) {0 ,12,1 ,1 1

2,2,2 1

2}

(2) {−2 ,−1 ,0 ,1 ,2 ,3 } (4) {0, 1, 2, 3, …}

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