Linear and Exponential Function 2.2 Date: 09/24/18 1. Grab your Binder 2. Copy down the Essential Question (EQ). 3. Work on the Warm-up. Essential Question When is the Domain discrete and when is the domain continuous? Warm Up: 2.2 SET #5 and #6

Identify whether the following items best fit with a discrete or a continuous model. Then determine whether it is a linear (arithmetic) or exponential (geometric) relationship that is being described. 5. The freeway construction crew pours 300 ft of concrete in a day. 6. For every hour that passes, the amount of area infected by the bacteria doubles.

Purpose The purpose of this task is for students to explicitly consider when a discrete or continuous model is appropriate for a given context.

1. The Library of Congress in Washington D. C 1. The Library of Congress in Washington D.C. is considered the largest library in the world. They often receive boxes of books to be added to their collection. Since books can be quite heavy, they aren’t shipped in big boxes. If, on average, each box contains about 8 books, how many books are received by the library in 6 boxes, 10 boxes, or n boxes? Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function.

𝑫𝒐𝒎𝒂𝒊𝒏 𝑹𝒂𝒏𝒈𝒆 Equation 𝑦=8𝑥+0 Domain is Boxes Boxes are discrete

2. Many of the books at the Library of Congress are electronic 2. Many of the books at the Library of Congress are electronic. If about 13 e-books can be downloaded onto the computer each hour, how many e-books can be added to the library in 3 hours, 5 hours, or n hours (assuming that the computer memory is not limited)? Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function.

Equation 𝑦=13𝑥+0 Domain is hours hours(time) is continuous 𝑫𝒐𝒎𝒂𝒊𝒏 𝑹𝒂𝒏𝒈𝒆 Equation 𝑦=13𝑥+0 Domain is hours hours(time) is continuous

3. The librarians work to keep the library orderly and put books back into their proper places after they have been used. If a librarian can sort and shelve 3 books in a minute, how many books does that librarian take care of in 3 hours, 5 hours, or n hours? Use a table, a graph, and an equation to model this situation.

𝑫𝒐𝒎𝒂𝒊𝒏 𝑹𝒂𝒏𝒈𝒆 Equation 𝑦=8𝑥+0 Domain is hours hours(time) is continuous

4. Would it make sense in any of these situations for there to be a time when 32.5 books had been shipped, downloaded into the computer or placed on the shelf?

5. Which of these situations (in problems 1-3) represent a discrete function and which represent a continuous function? Justify your answer. Problem 1 is is discrete because box are finite object Problem 2 and 3 are continuous because time is not finite, you can have 1.5 time

6. A giant piece of paper is cut into three equal pieces and then each of those is cut into three equal pieces and so forth. How many papers will there be after a round of 10 cuts? 20 cuts? n cuts?

Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function. Domain is the cuts because the cuts determine the amount of paper. c. Would it make sense to look for the number of pieces of paper at 5.2 cuts? Why? d. Would it make sense to look for the number of cuts it takes to make 53.6 papers? Why?

𝑫𝒐𝒎𝒂𝒊𝒏 𝑹𝒂𝒏𝒈𝒆 Equation 𝑓 𝑛 =1∙ 3 𝑥 Equation𝑓 𝑛 =3∙ 3 𝑥−1

Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function. c. Would it make sense to look for the number of pieces of paper at 5.2 cuts? Why? NO because the domain is discrete and not continuous d. Would it make sense to look for the number of cuts it takes to make 53.6 papers? Why? NO because the range is also discrete and not continuous

7. Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of 60 milligrams of anti-parasite medicine is given to a dog and the medicine breaks down such that 20% of the medicine becomes ineffective every hour. How much of the 60 milligram dose is still active in the dog’s bloodstream after 3 hours, after 4.25 hours, after n hours?

7. Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of 60 milligrams of anti-parasite medicine is given to a dog and the medicine breaks down such that 20% of the medicine becomes ineffective every hour. How much of the 60 milligram dose is still active in the dog’s bloodstream after 3 hours, after 4.25 hours, after n hours? Question: This is arithmetic or geometric ? Answer: Geometric because of the 20% Question: What is the decay rate? Answer: .80 because 1-.30 = .80

Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function. c. Would it make sense to look for an amount of active medicine at 3.8 hours? Why? d. Would it make sense to look for when there is 35 milligrams of medicine? Why?

Equation 𝑓 𝑛 =60∙ (.80) 𝑥 𝑫𝒐𝒎𝒂𝒊𝒏 𝑹𝒂𝒏𝒈𝒆 Equation𝑓 𝑛 =48∙ (.80) 𝑥−1

Use a table, a graph, and an equation to model this situation. b. Identify the domain of the function. c. Would it make sense to look for an amount of active medicine at 3.8 hours? Why? Yes because the domain is continuous. Time is fluid and can have infinite points. d. Would it make sense to look for when there is 35 milligrams of medicine? Why?

8. Which of the functions modeled in #6 and #7 are discrete and which are continuous? Why?

9. What needs to be considered when looking at a situation or context and deciding if it fits best with a discrete or continuous model? Look at the domain and apply the 1.5 Rule. “ Can I have 1.5 of *blank*?”

10. Describe the differences in each representation (table, graph, and equation) for discrete and continuous functions.

11. Which of the functions modeled above are linear 11. Which of the functions modeled above are linear? Which are exponential? Why? Exponential have a common ratio

Linear, Exponential or Neither? 2.3 Date: 09/25/18 1. Grab your Binder 2. Copy down the Essential Question (EQ). 3. Work on the Warm-up. Essential Question How is being linear different than being exponential? Warm Up: 2.3 READY #1 & # 2

Identify whether situation “a” or situation “b” has a greater rate of change.

Purpose The purpose of this task is to develop fluency in determining if a function is linear or exponential using various representations.

For each representation of a function, decide if the function is linear, exponential, or neither. Give at least 2 reasons for your answer.

Linear Exponential Neither Why?

Linear Exponential Neither Why? Straight line Has a decrease constant slope Cross the x-axis only once Cross the y-axis only once

Linear Exponential Neither Why?

Linear Exponential Neither Why? There is a common ratio of 1 2 Decrease extreme fast

Linear Exponential Neither Why?

Linear Exponential Neither Why? looks like its in 𝑦=𝑚𝑥+𝑏 there is a slope of 4 The exponent on the variables is only 1

Linear Exponential Neither Why?

Linear Exponential Neither Why? there is a constant rate (something) ?

Linear Exponential Neither Why?

Linear Exponential Neither Why?

Linear Exponential Neither Why?

Linear Exponential Neither Why? No constant slope of growth. No common ratio of growth. The growth rate is sometime zero

Linear Exponential Neither Why?

Linear Exponential Neither Why? Can be re-arrange into 𝑦=𝑚𝑥+𝑏 There is a constant slope of −3 4 The exponent on the variables is only 1

Linear Exponential Neither Why?

Linear Exponential Neither Why? There is a constant slope Cross the x-axis once Cross the y-axis once Straight line

Linear Exponential Neither Why?

Linear Exponential Neither Why? No constant slope No common ratio Is not straight like the linear function Does not curve like the exponential Doesn’t cross any of the axis's.

Linear Exponential Neither Why?

Linear Exponential Neither Why? Increase by 75% shows a common ratio Growth rate value of r=1.75

Linear Exponential Neither Why?

Linear Exponential Neither Why? Start growing slow then the growth become extreme large The slope is changing The curve bend upwards cross the y-axis once Never cross the x-axis

Linear Exponential Neither Why?

Linear Exponential Neither Why? No constant slope There is both slow down and speed up

Linear Exponential Neither Why?

Linear Exponential Neither Why? There is both a decrease and increase The exponent is to the power of two

Linear Exponential Neither Why?

Linear Exponential Neither Why? Decay rate of 1 3 The slope is changing base on a common ratio The graph curves down The graph never touches the x-axis

Linear Exponential Neither Why?

Linear Exponential Neither Why? It curves but there is no common ratio Its grow but doesn’t grow fast enough like the exponential This is the Fibonacci sequence which is neither linear nor exponential

Linear Exponential Neither Why?

Linear Exponential Neither Why? the equation is the recursive formula for a geometric sequences Common ratio of 2/3