Lesson 7: Exponential Decaymrpaccione.weebly.com/uploads/9/3/6/4/93644172/paccione...models...

Name ___________________________________________________ Date____________________

Lesson 7: Exponential Decay Exit Ticket

1. A huge Ping-Pong tournament is held in Beijing with 65,536 participants at the start of thetournament. Each round of the tournament eliminates half the participants.a. If (𝑟) represents the number of participants remaining after 𝑟 rounds of play, write a formula to

model the number of participants remaining. Identify and label all key elements.

b. Refer to the formula you created in question 1. What is the value of the growth factor? Explainwhy that value makes sense given the context of the problem.

c. How many rounds of play will it take to determine the champion Ping-Pong player? Justify yourresponse. (Show your work!)

2. Analyze the following model: $1,000 ∙ (9

10)𝑡, where t represents years. Describe what you believe

will occur each year according to the model. Justify your response.

3. What aspects of Exponential Decay would you like to understand better?

Lesson 1 - Assessments

NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 7 ALGEBRA I

Problem Set

1. From 2000 to 2013, the value of the U.S. dollar has been shrinking. The value of the U.S. dollar over time ( ( )) canbe modeled by the following formula:( ) = 1.36(0.9758) , where is the number of years since 2000

a. How much was a dollar worth in the year 2005?b. Graph the points , ( ) for integer values of 0 14.c. Estimate the year in which the value of the dollar fell below $1.00.

2. 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 each year.b. What is the value of the equipment after 9 years?c. Graph the points , ( ) for integer values of 0 15.d. Estimate when the equipment will have a value of $50,000.

3. The number of newly reported cases of HIV (in thousands) in the United States from 2000 to 2010 can be modeledby the following formula: ( ) = 41(0.9842) , where 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 , ( ) for integer values of 0 10.d. During what year did the number of newly reported HIV cases drop below 36,000?

4. Doug drank a soda with 130 mg of caffeine. Each hour, the caffeine in the body diminishes by about 12%.

a. Write a formula to model the amount of caffeine remaining in Doug’s system each hour.b. How much caffeine remains in Doug’s system after 2 hours?c. How long will it take for the level of caffeine in Doug’s system to drop below 50 mg?

Lesson Summary

The explicit formula ( ) = models exponential decay, where represents the initial value of the sequence, < 1 represents the growth factor (or decay factor) per unit of time, and represents units of time.

Lesson 7: Exponential Decay S.41

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5. 64 teams participate in a softball tournament in which half the teams are eliminated after each round of play.

a. Write a formula to model the number of teams remaining after any given round of play.b. How many teams remain in play after 3 rounds?c. How many rounds of play will it take to determine which team wins the tournament?

6. 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 depreciatesabout 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 for the total age of the car in years.b. Who is right, Sam or Derek?

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Name:_______________________ Lesson 2 – Pre-assessment 3/23/17 per:____

Multiple Choice: In questions 1-5 select the best answer and write its matching letter to the left of the question

1. What are the three essential elements of a function:

a) start value, rate of change, vertical line test

b) domain, range, relationship

c) domain, start value, vertical line test

d) input, domain, start value

2. Which of the following is the set for input values?

a) range b) rate of change c) domain d) real numbers

3. Unless modified by a restriction the domain of a function should be . . .

a) all real numbers b) all integers c) positive real numbers d) non-negative integers

4. What is the relationship of a function?

a) each x has only 1 y b) the input of the function

c) the input of the function d) equation mapping x to y

5. How can the vertical line test be explained?

a) all relationships are increasing b) output values increase vertically

c) every x value has only 1 y value d) non negatives under the square root symbol

6. Which of the following is not a discrete function?

a) y = 6x2 for all Real Numbers b) y = 0.5x2 for all Integers

c) y = 4x + 3 for x > 1 d) {1, 4, 9, 16, 25}



Problem Set

1. The first four terms of two different sequences are shown below. Sequence is given in the table, and sequence is graphed as a set of ordered pairs.

a. Create an explicit formula for each sequence.b. Which sequence will be the first to exceed 500? How do you know?

2. 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 th figure.c. Evaluate your formula for = 0, and = 2.5. Draw Figure 0 and Figure 2.5, and explain how you decided to

create your drawings.

( ) 1 15 2 31 3 47 4 63

Lesson 8: Why Stay With Whole Numbers? S.47

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3. The first four terms of a geometric sequence are graphed as a set of ordered pairs.

a. What is an explicit formula for this sequence?b. Explain the meaning of the ordered pair (3, 18).c. As of July 2013, Justin Bieber had over 42,000,000 Twitter followers. Suppose the sequence represents the

number of people that follow your new Twitter account each week since you started tweeting. If yourfollowers keep growing in the same manner, when will you exceed 1,000,000 followers?

Lesson 8: Why Stay With Whole Numbers? S.48

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© 2015 Great Minds. eureka-math.orgALG I-M3-SE-1.3.0-07.2015



Problem Set

1. Which of the following are examples of a function? Justify your answers.

a. The assignment of the members of a football team to jersey numbers.

b. The assignment of U.S. citizens to Social Security numbers.

c. The assignment of students to locker numbers.

d. The assignment of the residents of a house to the street addresses.

e. The assignment of zip codes to residences.

f. The assignment of residences to zip codes.

g. The assignment of teachers to students enrolled in each of their classes.

h. The assignment of all real numbers to the next integer equal to or greater than the number.

i. The assignment of each rational number to the product of its numerator and denominator.

2. Sequences are functions. The domain is the set of all term numbers (which is usually the positive integers), and therange is the set of terms of the sequence. For example, the sequence 1, 4, 9, 16, 25, 36, … of perfect squares is thefunction: : {positive integers} {perfect squares}Assign each term number to the square of that number.

a. What is (3)? What does it mean?

b. What is the solution to the equation ( ) = 49? What is the meaning of this solution?

c. According to this definition, is 3 in the domain of ? Explain why or why not.

d. According to this definition, is 50 in the range of ? Explain why or why not.

3. Write each sequence as a function.

a. {1, 3, 6, 10, 15, 21, 28}b. {1, 3, 5, 7, 9, … }c. = 3 , = 1, where is a positive integer greater than or equal to 1.

Lesson 9: Representing, Naming, and Evaluating Functions S.53


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Name ___________________________________________________ Date____________________

Lesson 4: Representing, Naming, and Evaluating Functions Exit Ticket

1. Jenna knits scarves and then sells them on Etsy, an online marketplace. Let𝐶(𝑥)=4𝑥+20 represent the cost 𝐶 in dollars to produce 1 to 6 scarves.

a. Create a table to show the relationship between the number of scarves 𝑥and the cost 𝐶.

b. What are the domain and range of 𝐶?

c. What is the meaning of 𝐶(3)?

d. What is the meaning of the solution to the equation 𝐶(𝑥)=40?

2. Reflect on your understanding of Representing, Naming and Evaluating Functions:

a. What aspects of Representing, Naming and Evaluating Functions do youunderstand well? Explain your understanding.

b. What aspects of Representing, Naming and Evaluating Functions still leave youconfused, if any?



Problem Set

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

a. (0) j. (0)b. ( 10) k. ( 1)c. (2) l. (2)d. (0.01) m. ( 3)e. (11.25) n. (4)f. 2 o. 2g. 53 p. 12h. (1) + (2) q. (2) + (1)i. (6) (2) r. (6) (2)

Lesson Summary

ALGEBRAIC FUNCTION: Given an algebraic expression in one variable, an algebraic function is a function : such that for each real number in the domain , ( ) is the value found by substituting the number into all instances of the variable symbol in the algebraic expression and evaluating.

The following notation will be used to define functions going forward. If a domain is not specified, it is assumed to be the set of all real numbers.

For the squaring function, we say Let ( ) = .

For the exponential function with base 2, we say Let ( ) = 2 .

When the domain is limited by the expression or the situation to be a subset of the real numbers, it must be specified when the function is defined.

For the square root function, we say Let ( ) = for 0.

To define the first 5 triangular numbers, we say Let ( ) = ( +1)2 for 1 5 where is an integer.

Depending on the context, one either views the statement “ ( ) = ” as part of defining the function or as an equation that is true for all in the domain of or as a formula.



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2. Since a variable is a placeholder, we can substitute letters that stand for numbers in for . Let ( ) = 6 3, andlet ( ) = 0.5(4) , and suppose , , , and are real numbers. Find the value of each function for the giveninput.

a. ( ) h. ( )b. (2 ) i. ( + 3)c. ( + ) j. (3 )d. (2 + ) k. ( 3)e. ( + ) l. ( + )f. ( + 1) ( ) m. ( + 1) ( )g. ( + ) ( )

3. What is the range of each function given below?

a. Let ( ) = 9 1.

b. Let ( ) = 3 .

c. Let ( ) = 4.

d. Let ( ) = + 2.

e. Let ( ) = + 2 such that is a positive integer.

f. Let ( ) = 5 for 0 4.

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

a. Let ( ) = 2 + 3.

b. Let ( ) = 2 .

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

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

5. Let : , where and are the set of all real numbers, and and are real numbers.a. Find a function such that the equation ( + ) = ( ) + ( ) is not true for all values of and .

Justify your reasoning.b. Find a function such that equation ( + ) = ( ) + ( ) is true for all values of and .

Justify your reasoning.c. Let ( ) = 2 . Find a value for and a value for that makes ( + ) = ( ) + ( ) a true number

sentence.



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6. Given the function whose domain is the set of real numbers, let ( ) = 1 if is a rational number, and let( ) = 0 if is an irrational number.

a. Explain why is a function.

b. What is the range of ?

c. Evaluate for each domain value shown below.

23 0 5 2 ( )

d. List three possible solutions to the equation ( ) = 0.



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Lesson 5 - Stations Activity - Observations Sheets

PERIOD: TEAM 1 TEAM 2 TEAM 3

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QUESTION 1

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