Algebra II Foundations Non Linear Functions Student Journal
Problems of the Da First Semester Page 35 Problem Set 35 CHALLENGE Tr the following problem, and eplain how ou determined our answer. If it takes 1 minute to la each brick, and 3 minutes to la cement between each row of bricks, how long will it take to build the wall shown to the left? Answer: WHAT IS THE MISCONCEPTION? The following problem is solved incorrectl. Find the correct solution. Allen scored the following grades on his last 4 tests. 84, 86, 90, 92 What score does he need on the net test to have a 90% average? 84 + 86 + 90 + 92 = 88 Correct Answer: 4 88 + 2 = 90 = 92
Non-Linear Functions Lesson 6: Eponential Functions Page 51 Lesson 6: Eponential Functions Activit 1 Bacterial Growth Respirator Sstem Model Respirator sicknesses (infections), such as bronchitis and pneumonia, are caused b bacteria. Once bacteria gets in our lungs, the can duplicate (reproduce) at a certain rate. The following eperiment will model the amount of bacteria present over time. In this eperimental model, we will use small construction paper squares of one color to represent the bacteria. Eperiment Step 1: Cut out 64 red construction paper squares. Each square should be the same size and shape. The best size is 1 inch b 1 inch or 1 centimeter b 1 centimeter. Use a ruler to draw the squares before cutting. Eperiment Step 2: Cut out the lungs template at the end of the activit. Eperiment Step 3: Place one red square on the lung template (an where inside the lung area.) This represents the initial amount of bacteria, a single cell. Note: Bacteria are actuall ver small in size. A single cell of bacteria is about 1/10,000 th of a centimeter. Eperiment Step 4: Ever minute, add enough red squares to double the amount ou had previousl. This represents the bacteria duplicating (reproducing itself) ever hour. While ou are waiting for each minute to end, count out the necessar squares that ou will be adding for the net minute. Also, record the time and amount of bacteria present in the lungs in the table provided below. Eperiment Step 5: Repeat Step 4 until all 64 squares have been placed "in" our lungs. Eperiment Step 6: You should realize that our table matches the table from the Setting the Stage transparenc. Table 1: Bacterial Growth Eperiment Hour Bacteria Count 0 1 1 2 3 4 5 6
Page 52 1. Create a scatter plot of the hours compared to the number of bacteria in the lungs. What tpe of pattern occurred in the scatter plot graph? 2. What is the rate that the bacteria are growing? 3. Graph a scatter plot of our data on a graphing calculator. Set the window range to an minimum of 2, maimum of 7, scale of 1, minimum of 2, maimum of, and scale of 10. Is the scatter plot linear? If not describe the shape of the graph. Bacteria Count 24 22 20 18 16 14 12 10 Bacteria Growth Eperiment 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Hour 4. How man bacteria do ou epect to be in the lungs after a 24 hour period? How might ou calculate this value? 5. Approimatel how man hours will it take until there are 1 trillion (1,000,000,000,000 or 1 10 12 ) bacteria in the lungs? NOTE: The graphing calculator ma displa 1 trillion as 1.0 E12.
Non-Linear Functions Lesson 6: Eponential Functions Page 53 Antibiotic Deca in the Blood Stream Eperimental Model To help cure illnesses antibiotics and/or medicines taken into the bod are circulated throughout the bod b the bloodstream. The kidnes take the drugs out of the blood. We saw, from the first part of the activit, how bacteria can duplicate and create enormous amounts of themselves in a relative short period of time. Bacteria left unchecked can cause major health problems. Sometimes the onl wa to become health again is b the use of antibiotics. The following eperiment will model the amount of antibiotics left in the bloodstream over time. In this eperimental model, we will use small construction paper squares of one color to represent the blood and small construction paper squares of another color to represent the antibiotics. Eperiment Step 1: Eperiment Step 2: Eperiment Step 3: Cut out 40 red construction paper squares and 20 blue construction paper squares. Each square should be the same size and shape. The best size is 1 inch b 1 inch. Use a ruler to draw the squares before cutting. Place 20 red squares and 20 blue squares in a container (bag or bo). This represents a bloodstream that is half blood and half antibiotics. Although in real life the blood stream would not consist of 50% antibiotics, this will produce a model quickl that represents the wa drugs leave the bloodstream. Shake the container and randoml remove 10 squares. Replace them with 10 red squares. Determine how man blood squares and antibiotic squares are now in the container. Place this information in Table 1 below. This step models the kidnes randoml cleaning one quarter of the blood each hour. Eperiment Step 4: Repeat Step 3 ten times. Place the information for each cleaning ccle in Table 2, Antibiotics Deca Eperiment, below. Table 2: Antibiotics Deca Eperiment Hour Blood Count Antibiotic Count 0 20 20 1 2 3 4 5 6 7 8 9 10
Page 54 6. Create a scatter plot of the hours compared to the number of antibiotics left in the bloodstream. What tpe of pattern occurred in the scatter plot graph? 7. Create a transparenc cop of our graph. Place all the transparencies from each group on the overhead at one time and line up the aes. What do ou notice about the graph? Antibiotic Count Antibiotic Deca Eperiment 24 22 20 18 16 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Hour 8. If no new antibiotics are added, what would the graph do if we continued with the eperiment? 9. Graph a scatter plot of our data on a graphing calculator. Set the window range to an minimum of 2, a maimum of 12, a minimum of 2, and a maimum of 24. Is the scatter plot linear? If not describe the shape of the graph. 10. Graph = 20(0.75) on the same graph as the scatter plot. Describe how the graph of = 20(0.75) fits the data from the scatter plot.
Non-Linear Functions Lesson 6: Eponential Functions Page 55 Lungs Template Cut Here
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Non-Linear Functions Lesson 6: Eponential Functions Page 57 Activit 2 In this activit, ou will determine the intercept, determine the tpe of graph, and draw a rough sketch of eponential functions. For Eercises 1 through 4: a. Determine the coordinates of the intercept b. Tpe of graph: growth or deca c. Draw a rough sketch of the eponential function on the grid provided. Note: set grid scale appropriatel. 1. = 12 2. 1 = 8 3. 2 1 = 9 5 4. = 7(4 )
Page 58 For Eercises 5 through 7, state the intercept and the tpe of graph. 5. 6. 7.
Non-Linear Functions Lesson 6: Eponential Functions Page 59 Activit 3 In this activit, ou will solve real world eponential problems. 1. Your grandparents put $10,000 in an investment account, which collects interest four times a ear, when ou were born for our college education. The future value of our college education fund can be determined b the 4 function S = 0(1.0375) t, where t represents the number of ears for the investment. How much mone will ou have available when ou start college? Assume ou will be 18 ears old when ou start college. Draw a rough sketch of the investment; set ais scales accordingl. 2. Viruses can produce man more offspring than bacteria per infection. Some viruses produce at an t eponential rate related to the function v= C() h, where v represents the number of viruses, C represents initial population of viruses, t represents amount of time in hours, and h is the number of hours to produce a new generation. How man viruses will be present after 24 hours if there initiall were 5 viruses and the viruses produce a new generation ever 4 hours? 3. It has been determined that a certain cit has been growing eponentiall over the last 20 ears according to the function P= P0 (1 + r)t, where P represents the town's population, P 0 is the initial population, r is the rate at which the town's population is increasing, and t is the amount of time in ears that the town has been increasing. If the town initiall had 450 people 20 ears ago and the now have 1,443 people, what was the rate of increase in population over the last 20 ears? Round our answer to the nearest whole percent. 4. A local retail store has determined that its sales could grow eponentiall based on the amount the spend on advertising each week b the function s= C(1.15) w, where s represents the number of sales per week, C represents their initial sales before advertising began, w represents the number of consecutive weeks the advertised. If the store averaged 125 sales per week before advertising began, how man sales can the epect to have, each week, after advertising for 4 consecutive weeks? Round our answer down to the nearest whole sale.
Page 60 5. The radio active deca of a material is given b the function ( 0.693 t/t A = A ) 0e, where A 0 is the initial amount of the material, t is the amount of time in ears, and T is the half life of the radio active material. Plutonium 240 has a half life of 6540 ears. If a nuclear power plant started with pounds of Plutonium 240, how much would be left after 20 ears? How man ounces of plutonium decaed during the 20 ears? Round our answers to the nearest hundredth pound and ounce.
Non-Linear Functions Lesson 6: Eponential Functions Page 61 Activit 4 In this activit, ou will use eponential regression to obtain an eponential function from real world data. 1. The following data table represents the dail costs of commuting (driving to work) versus the amount of commuters (people who drive to work) for a large metropolitan area. Cost (in $) 10 15 20 25 30 35 40 45 50 Commuters 225,000 145,000 110,000 68,000 35,000 13,000 8,000 5, 2, a. What tpe of graph does the data model? b. What is the eponential regression function? Round values to three decimal places. c. How man commuters would ou epect if the had to pa $75 each da in commuting epenses? Round our answer to the nearest commuter. 2. The following data table represents the population of the United States from the ears 1790 through, where ear 0 = 1790, 1 = 1820, etc. Year 0 (1790) 1 (1820) 2 (1850) 3 (1880) 4 (1910) 5 (1940) 6 (1970) 7 () Population (in millions) 3.93 9.64 23.19 50.16 91.97 131.67 204.05 281.42 a. What tpe of graph does the data model? b. What is the eponential regression function? Round values to four decimal places. c. Using this eponential equation, what might ou predict will be the size of the U. S. population in the ear 2060? Round our answer to the nearest ten thousandths. Note: Remember our current units for population is in millions.
Page 62 3. The following table represents the earl production of crude petroleum in the United States. Year 0 (1859) 10 (1869) 20 (1879) 30 (1889) 40 (1899) Oil Production (in barrels) 2,000 4,215,000 19,914,146 35,163,513 57,084,428 a. What tpe of graph does the data model? b. What is the eponential regression function? Round values to three decimal places. c. The U. S. oil production peaked in 1970. What could ou predict was our countr's peak output of oil in 1970? Round our answer to the nearest whole barrel. d. The actual U. S. oil production in 1970 was approimatel 3,,000,000 barrels. What can ou sa about our predicted value of production compared to the actual value of production? e. What suggestion would ou make on limiting the use of our eponential regression function?
Non-Linear Functions Lesson 6: Eponential Functions Page 63 Practice Eercises For Eercises 1 and 3: a. Determine the coordinates of the intercept. b. Tpe of graph: growth or deca. c. Draw a rough sketch of the eponential function on the grid provided. Note: set grid scale appropriatel. 1. = 53 ( ). 2. 1 = 9. 6 2 3. = 3( 4 ).
Page 64 4. On Januar 15 th, 9, the world's population was 6.75 billion people. It is predicted that it will take just 44 ears for the world's population to double. What is the rate, per ear, at which the world population is increasing? Round our answer to the nearest tenth of a percent. Note: Use P= P0 (1 + r)t. Refer back to Activit 3, Eercise 3. 5. A biologist is conducting an eperiment testing a new antibiotic on a certain strain of bacteria cells. According to the biologist's calculation, the cells are ding (decaing) at a rate given b the 0.223t function L= ae, where L represents the amount of cells left after time t (in minutes) and a represents the initial amount of bacteria cells present before the antibiotic is applied. How man bacteria cells are present ten minutes after the antibiotic was applied if there initiall 10 million bacteria cells? Round our answer to the nearest whole cell. Use 2.178 for the value of e. 6. A person invests $15,000 into an interest bearing account. After 10 ears the person's investment is now worth approimatel $25,966. Determine the annual interest rate if the future value of an investment can be determined with the function S = P(1+ r/12) 12t, where S is the value of the investment after t ears, P is the amount invested, and r is the annual interest rate. Round our answer to the nearest tenth of a percent. 7. The intensit of earthquakes is measured b using the Richter scale. We can determine how much more powerful one earthquake is compared to another earthquake, b the ratios of their intensities. The ratios of the intensities of two earthquakes can be determined b the function I = 10 d, where I is the ratio of intensities and d is the absolute value of the difference of the intensities of the earthquakes as measured b the Richter scale. It is estimated that the 4 Indian Ocean earthquake measured 9.2 on the Richter scale. In comparison, the earthquake that caused Mt. St. Helen's volcano to erupt on Ma 18 th 1980, measured 5.1 on the Richter scale. a. How much more powerful was the 4 Indian Ocean earthquake compared to the 1980 Mt. St. Helen's earthquake? Round our answer to the nearest whole number. b. What can ou conclude about the difference in the intensities of two earthquakes?
Non-Linear Functions Lesson 6: Eponential Functions Page 65 8. For the following graph: a. State the intercept. b. State the tpe of graph.
Page 66 Outcome Sentences Eponential growth is I know an eponential problem when Eponential deca is When graphing eponential functions The part about eponential functions I don't understand is