LESSON #59 - LINEAR FUNCTIONS AND MODELING COMMON CORE ALGEBRA II

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1 1 LESSON #59 - LINEAR FUNCTIONS AND MODELING COMMON CORE ALGEBRA II A linear function is any pattern where the function increases or decreases by the same numerical constant per unit. It is a function where the rate of change is always constant. The two most important qualities of a linear function are its starting point on the y-axis, known as the y- intercept, and its constant rate of change or slope. These two qualities tell us where the function starts and where it is going. The most basic form, slope-intercept, is the one we will be using predominantly in this course. The point-slope form is also included in the textbox below. TWO COMMON FORMS OF A LINE Slope-Intercept: where m is the slope (or average rate of change) of the line and Point-Slope: represents one point on the line. Exercise #1: Write a function to model each of the patterns in the tables below. x y x y Exercise #2: Graph each equation. Equation Graph Equation Graph y 2x 7 2 f ( x) x 3 5

2 2 Exercise #3: Write an equation for each graph. Equation Graph Equation Graph Exercise #4: Dia was driving away from New York City at a constant speed of 58 miles per hour. He started 45 miles away. (a) Write a linear function that gives Dia s distance, D, from New York City as a function of the number of hours, h, he has been driving. Let h = (b) If Dia s destination is 270 miles away from New York City, algebraically determine to the nearest tenth of an hour how long it will take Dia to reach his destination. Let D = Exercise #5: Two students have bank accounts. Student A starts with $600 in her bank account and takes out $20 each month. Student B starts with $900 in his bank account and takes out $50 each month. (a) Create linear functions for amount of money, in each account after x months. Let A(x) = (b) Algebraically determine exactly how many months it will take for Student A and Student B to have the same amount in their accounts. Let B(x) = Let x =

3 Exercise #3: A factory is currently printing sci-fi paperback novels. Each day it costs $1000 to run the factory and pay the workers. It also costs $3.50 per book to make the books. Write a function, C(b), to model the total cost of producing b books each day. 3 The factory sells the books to a distributor for $4.75 per book. Write a function, R(b), to model the revenue for the books each day. (a) Use your graphing calculator to sketch and label each of these linear functions for the interval 0 b Be sure to label both axes with a scale. Let b = Let C(b) = Let R(b) = (b) Use your calculator s INTERSECT command to determine the number of sci-fi books, b, that must be produced for the revenue to equal the cost. (c) If profit is defined as the revenue minus the cost, create an equation in terms of w for the profit, P.

4 4 LESSON #59 - LINEAR MODELING COMMON CORE ALGEBRA II HOMEWORK 1. Write a function to model each of the patterns in the tables below. x y x y Graph each equation. Equation Graph Equation Graph 1 g( x) 3x 2 y x Write an equation for each graph. Equation Graph Equation Graph

5 5 APPLICATIONS 4. Which of the following would model the distance, D, a driver is from Chicago if they are heading towards the city at 58 miles per hour and started 256 miles away? (1) D 256t 58 (3) D 58t 256 (2) D t (4) D t 5. The cost, C, of producing x-bikes is given by C 22x 132. The revenue gained from selling x-bikes is given by R 350x. If the profit, P, is defined as P R C, then which of the following is an equation for P in terms of x? (1) P 328x 132 (3) P 328x 132 (2) P 372x 132 (4) P 372x The average temperature of the planet is expected to rise at an average rate of 0.04 degrees Celsius per year due to global warming. The average temperature in the year 2000 was degrees Celsius. (a) Write a function to represent the average temperature of the planet, C(x), where x represents the number of years since (b) Algebraically determine the number of years, x, it will take for the temperature, C, to reach 20 degrees Celsius. Round to the nearest year. (c) Sketch a graph of the average yearly temperature below for the interval. Be sure to label your y-axis scale as well as two points on the line (the y- intercept and one additional point). (d) What does this model project to be the average global temperature in 2200?

6 6 LESSON #60 - EXPONENTIAL GROWTH AND DECAY COMMON CORE ALGEBRA II Last lesson, we looked at situations that can be modeled by linear functions because the values increased or decreased by equal differences over equal intervals. There are many things in the real world that grow faster as they grow larger or decrease slower as they get smaller. One specific example of this is situations where the increase or decrease by equal factors over equal intervals. These types of phenomena are known as exponential growth and decay, respectively. Exercise #1: Last lesson, you learned the most common way to write the equation of a line is in the form,. What information can be determined by an equation in this form? Exercise #2: Last unit, you were introduced to exponential functions in the form. What information can be determined by an equation in this form? The value of a is the. The value of b is the. Exercise #3: Write a function to model each of the patterns in the tables below. x y x y

7 Exercise #4: The number of people who have heard a rumor often grows exponentially. Consider a rumor that starts with 3 people and where the number of people who have heard it doubles each day that it spreads. 7 (a) Why does it make sense that the number of people who have heard a rumor would grow exponentially? (b) Fill in the table below for the number of people, N, who knew the rumor after it has spread a certain number of days, d. d N 3 6 (c)we d like to determine the number of people who know the rumor after 20 days, but to do that, we need to develop a formula to predict N (the number knowing the rumor) if we know d (the number of days it has been spreading). Write a function to model this situation. (d) Graph N over the interval. (e) How many people would know the rumor after 20 days? (f) Exponential growth can be very fast. Assuming our equation from (b) holds, how many days will it take for the number of people knowing the rumor to surpass the population of the United States, which is approximately 315 million people?

8 8 Let s now look at developing a fairly simple exponential decay problem. Exercise #5: Helmut (from Finland) is heading towards a lighthouse in a very peculiar way. He starts 160 feet from the lighthouse. On his first trip he walks half the distance to the light house. On his next trip he walks half of what is left. On each consecutive trip he walks half of the distance he has left. We are going to model the distance, D, that Helmut has remaining to the lighthouse after n-trips. 160 ft (a) Fill in the table below for the amount of distance that Helmut has left after n-trips. n D (ft) (b) Each entry in the table could be found by multiplying the previous by what number? This is important because we always want to think about exponential functions in terms of multiplying. (c) Exponential decay formula: (e) How far is Helmut from the windmill after 6 trips? Provide a calculation that justifies your answer and don t forget those units! (f) Helmut believes he will reach the windmill after 10 trips. Is he correct? (g) Explain why Helmut will never reach the windmill? Remember, when things get crazy next unit, those exponential functions still model situations where there the growth or decay shows a multiplying pattern. (equal factors over equal intervals). When the formulas get crazy, any formula* can be rewritten in the form to identify the start value or y-intercept, a, and growth factor, b. *(The only exception are exponential formulas with a vertical shift).

9 9 LESSON #60 - EXPONENTIAL GROWTH AND DECAY COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Write a function to model each of the patterns in the tables below. a) x y b) x y APPLICATIONS 2. A typical cell phone is 5 ounces (oz.). When a cell phone is thrown in the garbage and decomposes over time, half of it is absorbed into the ground every year. (a) Fill in the table below for the ounces of remaining cell phone, Z, remaining after a certain number of years, y. y Z (b) Using the table, determine an exponential decay formula for the number of ounces remaining, Z, after y years. (c) Using your formula from (b), how many ounces will be left after 15 years?

10 3. 5 students at our school have a stomach virus. The number of students infected with the virus triples each day. (a) Fill in the table below for the number of people, N, that have the virus after a certain number of days, d. d N (b) Using the table, determine an exponential growth formula for the number of students infected, N, after d days. (c) Graph N on the grid to the right over the interval 0 d 5. (d) Using your formula from (b), how many students will be infected with the stomach virus by day 8? (e) Is your answer in part (c) reasonable? Explain.

11 11 LESSON #61 - LINEAR VERSUS EXPONENTIAL COMMON CORE ALGEBRA II Linear and exponential functions share many characteristics. This is because they are based on two different, but similar, sets of principles. LINEAR VERSUS EXPONENTIAL Linear functions are based on repeatedly adding the same amount (the slope). Exponential functions are based on repeatedly multiplying by the same amount (the base). Exercise #1: The two tables below represent a linear function and an exponential function. Determine which function is linear and which function is exponential and write a function rule for each. TABLE 1 TABLE 2 x y x y Exercise #2: There were 50 squirrels at a park initially. It has been noted that each year, the number of squirrels grows by 20%. Fill out the table below. Round the number of squirrels to the nearest whole number. a) Year t Number of Squirrels S(t) b) Based on the table to the right, are functions that grow by a constant percent linear or exponential functions? c) A function, S(t) to model the number of squirrels after t years.

12 You will learn more about this next unit, but a situation where the data increases or decreases by a constant percent can be modeled by an exponential function. Exercise #3: Answer the following questions about linear vs. exponential functions. a) Which situation could be modeled with an exponential function? (1) The amount of money in a savings account where $150 is deducted every month (2) The amount of money in Suzy s piggy bank which she adds $10 to each week (3) The amount of money in a certificate of deposit that gets 4% interest each year (4) The amount of money in Jaclyn s wallet which increases and decreases by a different amount each week 12 b) Which statement below is true about a linear function? (1) Linear functions grow by equal factors over equal intervals. (2) Linear functions grow by equal differences over equal intervals. (3) Linear functions grow by equal differences over unequal intervals. (4) Linear functions grow by unequal factors over equal intervals. c) The selling prices for a group of cars were recorded when the cars were new and for an additional five years. The results are summarized in the tables below. Which car s price dropped at a constant percent rate each year? d) Joseph s taxi charges $10.00 for the initial service of any drive. The fee for each mile of the taxi ride is $0.75. Which type of function is represented by this situation? (1) Linear (2) Exponential (3) Quadratic (4) Absolute Value

13 We want to be very sure that we understand the various constants or parameters that come up in linear and exponential functions. Because these parameters always have a meaning in a physical situation. 4. emathinstruction is keeping track of the number of views on a newly released math lesson screencast. They record the total number of views as a function of the number of days since it launched, with the launch day being x 0. The data does not follow a perfect linear or exponential pattern, so they found both the best linear and exponential models for the data. y 68x 157 y x (a) How can you interpret the parameter 68 in the linear model in terms of the views of the website? 13 (b) How do you interpret the parameter 1.18 in the exponential model in terms of the views of the website? (c) Why is the interpretation of the 157 in the linear model unreasonable or nonviable? Exercise #3: A situation modeled by a linear function has a starting value of 1000 and increases by 100 each day. A situation modeled by an exponential function has a starting value of 10 and doubles each day. (a) Write a function, L(x), to model the linear function after x days. (b) Write a function, E(x), to model the exponential function after x days. (c) Use your graphing calculator to graph and label L(x) and E(x). Use an appropriate viewing window to find the intersection of the two functions to the nearest tenth. The graph above illustrates an important point about linear and exponential functions. Even if the linear function has a large starting value and a very steep slope, an increasing exponential function will eventually have a larger function value because its slope is always increasing.

14 14 FLUENCY LESSON #61 - LINEAR VERSUS EXPONENTIAL COMMON CORE ALGEBRA II HOMEWORK 1. For each of the following problems a table of values is given where x 1. For each, first determine if the table represents a linear function, of the form y mx b Then, write its equation. (a) x y , or an exponential function, of the form y a b x (b) x y ,000. Type: Type: Equation: Equation: 2. The data shown in the table below represents either a linear or an exponential function. Which of the equations below best models this data set? (1) 52 x y (3) y 2x 10 (2) 10 2 x y (4) y 10x 5 x y Answer the following questions about linear vs. exponential functions. a) Which situation could be modeled by a linear function? (1) The height of a ball that is thrown in the air (2) The price of a car that depreciates 20% per year (3) The amount of money Jonathan pays for a certain number of gallons of gas at $3.85 per gallon (4) A bacteria colony which doubles in number every 4 hours b) Joseph conducted a science experiment involving the growth of bacteria. He measured the number of bacteria hourly for 6 hours. The data is summarized in the accompanying table. What type of regression would best fit the data? (1) Linear (2) Exponential (3) Quadratic (4) Absolute Value Hours Number of Bacteria

15 15 c) The tables below show the amount of money in different bank customer s accounts on the first day of each month for five months. Which customer s account increased at a constant numerical rate each month? 4. Two scenarios are modeled using in (a) a linear function and in (b) an exponential function. In each case interpret the parameters that help define the functions. (a) Plant managers at a local tire factory model the cost, c, in dollars of producing n-tires in a day by the equation: c n n ,245 Interpret the parameter values of 6.50 and 1,245. Include units in your answer. (b) Biologists model the population, p, of lactic acid bacteria in yogurt as a function of the number of minutes, m, since they added the bacteria using the equation: m p m Interpret the parameter values of 135 and Include units in your answer. 5. In the lesson and in the question above, we saw exponential functions with a constant percent increase. The following function models a situation where there is a constant percent decrease. f( x) 5(.75) x. By what percent is this function decreasing?

16 16 LESSON #62 - WRITING EQUATIONS OF LINEAR AND EXPONENTIAL FUNCTIONS FROM TWO POINTS COMMON CORE ALGEBRA II For both linear and exponential functions only two points are necessary to determine the equation of the curve. Exercise #1: Consider the two points in y mx b 0,12 and form and an exponential equation in y a b x your calculator, graph both using a WINDOW of 2 x 2 and 5 y 15. 1, 3. Create a linear equation that passes through these points form that also passes through them. Then, using Linear: m = b = Exponential: a = b = y Equation: Equation: x The situation above was pretty simple because the y-intercept of the function was given as well as the point where x=1. The equation of any linear or exponential function can be found if there are two points, even if those points do not include the y-intercept or consecutive integer x-values. This is possible because these two points can create a system of two equations to solve for the two parameters in the function that are needed. For linear functions these parameters are m, the slope, and b, the y-intercept. For exponential functions, these parameters are a, the y-intercept, and b, the growth factor. Exercise #2: Find a linear equation that passes through the points Steps 1. Plug each point into the equation, y=mx+b to create a system of two equations. 2. Solve for b in one of the equations. 3. Solve for m using substitution. 4. Substitute the value of m into one of the two equation to solve for b. 5. Plug the values of m and b into y=mx+b. 2, 36 and 5,121.5.

17 17 Exercise #3: Find an exponential function that passes through the points Steps 1. Plug each point into the equation, y a() b x to create a system of two equations. 2. Divide one of the functions by the other to eliminate a. 3. Solve for b. 4. Substitute the value of b into one of the two equations to solve for a. 5. Plug the values of a and b into y a() b x. 2, 36 and 5, Write an exponential and a linear function that pass through the points (4,98) and (9,189). Linear: Exponential (round values to the nearest hundredth): In the above problem, what additional information would you need to help you determine if an exponential function or a linear function or something else would best model this situation? There are multiple answers to this question.

18 Exercise #4: Find the equation of the exponential function shown graphed below. Be careful in terms of your x exponent manipulation. State your final answer in the form y a b. y 18 x

19 19 LESSON #62 - WRITING EQUATIONS OF LINEAR AND EXPONENTIAL FUNCTIONS FROM TWO POINTS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Consider the points 0, 5 and 1,15. (a) Write the equation of the line that passes between these two points in y mx b form. (c) Using your calculator, sketch the two curves on the axes below. Label with their equations. 30 (b) Write the equation of the exponential that x passes between these two points in y a b form Find the equation of the linear function and the exponential function that passes through 2,192 and 5, Show the work that you use to arrive at your answer. Linear: Exponential:

20 Water Depth (ft) 3. Find the equation for an exponential function that passes through the points 2,14 and 7, 205 in y a b x form. When you find the value of b do not round your answer before you find a. Then, find both to the nearest hundredth and give the final equation Engineers are draining a water reservoir until its depth is only 10 feet. The depth decreases exponentially as shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours to be 28 feet. The engineers found the exponential function y 84.31(0.76) x to model the depth of the water after x hours. Graph the horizontal line y 10 and find its intersection to determine the time, to the nearest tenth of an hour, when the reservoir will reach a depth of 10 feet. Time (hrs)

21 21 LESSON #63 USING REGRESSION TO FIND EQUATIONS OF FUNCTIONS COMMON CORE ALGEBRA II In this lesson we will look at a number of different types of problems we have done throughout the year where we found equations of functions given sets of points. Each of the problems used a different algebraic method because a different type of function is being found.. We will be looking at linear, exponential, quadratic, and cubic problems. First, we will review how to find a regression equation which you learned in CC Algebra. Steps 1) Press STAT 2) Choose #1: Edit 3) Enter the two lists in L1 and L2. 4) Press STAT. Move over to Calc. 5) Choose one of the following: 4: LinReg (ax+b) 5: QuadReg 6: CubicReg 0:ExpReg 6) Press ENTER through the options in the menu. 7) The equation set-up will be given to you in the first line. Plug in the values of a,b,c, and d. (Some types of functions do not have all of these values.) Exercise #1: Using the table below, find the following regression equations. x y a) Find the linear regression equation that passes through the points in the table. Round coefficients to the nearest hundredth. b) Find the exponential regression equation for the points in the table. Round coefficients to the nearest hundredth. c) Find the quadratic regression for the points in the table. Round coefficients to the nearest hundredth. d) Find the cubic regression for the points in the table. Round coefficients to the nearest hundredth. Each of the regression equations in exercise #1 gives the function of BEST FIT (linear, exponential, quadratic or cubic depending on what is chosen). That means the function does not necessarily pass through all of the points in the table, but it comes as close as possible to the points for that type of function. In each of the following problems, we are interested in finding the EXACT equation of a certain type of function with the given qualities.

22 22 Exercise #2: Find the equation of a line passes through the points 5, 2 and 20, 4. a) Traditional Work: b) With a regression: Type: Equation: Exercise #3: Find the equation of the exponential function, in the form 2,192 and 5, a) Traditional Work: x y a b that pases through the points b) With a regression: Type: Equation:

23 23 Exercise #4: Create the equation of a quadratic polynomial, in standard form, that has zeroes of 5 and 2 and which passes through the point 3, 24. Sketch the graph of the quadratic below to verify your result. a) Traditional Work: b) With a regression: Type: Equation: Exercise #5: Create an equation for a cubic function, in standard form, that has x-intercepts given by the set 3,1, 7 2, 54. Sketch your result on the axes shown below. and which passes through the point a) Traditional Work: b) With a regression: Type: Equation:

24 24 LESSON #63 - USING REGRESSION TO FIND EQUATIONS OF FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK Directions: For problems 1-5, use regression to find the type of function that fits the given criteria. 1. Find the equation of the exponential function, in the form x ( 2, 45) and ( 4, 405). Type: Equation: 2. Find the equation of a line passes through the points ( 2, 45) and ( 4, 405). Type: Equation: y a b that passes through the points 3. Create an equation for a cubic function, in standard form, that has x-intercepts given by the set -8,-3,1 ( ). { } and which passes through the point -4, - 40 Type: Equation: 4. Create the equation of a quadratic polynomial, in standard form, that has zeroes of -1 and 3 and which passes through the point ( 6,126). Type: Equation:

25 5. In problems 1 and 2, you found the equation of an exponential function and a linear function that passes through the points ( 2, 45) and ( 4, 405). a) What happens when you try to find the equation of a quadratic function that passes through those two points? 25 b) Why do you think this happens? (Hint: Compare the information you were given in this problem to the other two quadratic problems on the previous page). 6. In the next lesson we will be looking for exponential and linear patterns in data. Determine if each of the following scatter plots shows an exponential pattern or a linear pattern.

26 26 LESSON #64 LINEAR AND EXPONENTIAL REGRESSION COMMON CORE ALGEBRA II Real life data often forms patterns that can be modeled with functions. For this course, we will be looking at data that can be modeled by linear and exponential functions as well as trigonometric functions next unit. Below is a summary of what to look for when deciding if data should be represented by a linear function or an exponential function. Type Connection Examples linear y = ax + b y=2x-7 y=-4x+3 exponential y = ab x y = 2 x y = 100(.7) x Qualities Does the plotted data resemble a straight line? The slope may be either positive or negative. Does the plotted data appear to grow (or decline) by percentage increases (decreases)? Remember the shape of the exponential function. The range must be: y > 0 Exercise #1: Which type of function (linear or exponential) would best model the data in each scatter plot?

27 Creating a Scatter Plot Go to STAT #1 (Edit). Enter the two lists in L1 and L2. Press Y= Move your cursor up to PLOT 1 and press ENTER to turn it on. It should be highlighted. Press ZOOM 9 (ZoomStat) Label the window Exercise #2: Create a scatter plot for the data. Based on the scatter plot, what type of function would best model this data? Hours since observation # of bacteria in the sample began In each of the following problems, bivariate (two variable) data was collected which can be modeled by either a linear or an exponential function. Exercise #3: A scientist in a laboratory collected data about the number of bacteria in a sample. The results are recorded in the table below. a) Based on a scatter plot, what type of regression would best model the data (see above)? Hours since observation began # of bacteria in the sample b) Find the appropriate regression equation with coefficients rounded to the nearest hundredth. c) Using your regression, how many bacteria will be in the sample 4.5 hours after the observation began? Round to the nearest whole number. d) Using a graph of your regression, after how many hours, to the nearest tenth of an hour, will there be 2000 bacteria in the sample? (You do not know how to solve this problem algebraically, yet). 27

28 28 The following problem has variables defined in a way that makes the equation less messy. The trade-off is that you will have to think about these definitions when you are making predictions. Exercise #4: The availability of leaded gasoline in New York State is decreasing, as shown in the accompanying table where x is defined as years after Year Gallons Available (in thousands) a) Based on a scatter plot, what type of regression would best model the data? b) Find the appropriate regression equation with coefficients rounded to the nearest tenth. c) Using your regression, how many gallons will be left in 2018 to the nearest gallon? d) If this relationship continues, during what year will leaded gasoline first become unavailable in New York State? e) Using your regression equation, when will 25,000 gallons of gasoline be left in New York State?

29 Exercise #5: The data at the below shows the cooling temperatures of a freshly brewed cup of coffee after it is poured. The brewing pot temperature is approximately 180º F. 29 Time (mins) Temp (º F) a) Based on a scatter plot, what type of regression would best model the data? b) Find the appropriate regression equation with coefficients rounded to the nearest hundredth. c) Using the regression, to the nearest minute, when is the coffee at a temperature of 106 degrees? d) Using the regression equation, what is the predicted temperature of the coffee after 1 hour to the nearest degree? e) In 1992, a woman sued McDonald's for serving coffee at a temperature of 180º that caused her to be severely burned when the coffee spilled. As a result of this famous case, many restaurants now serve coffee at a temperature around 155º. Using this regression, how many minutes should restaurants wait before serving coffee, to ensure that the coffee is not hotter than 155º? Round to the nearest minute.

30 30 LESSON #64 - LINEAR AND EXPONENTIAL REGRESSION COMMON CORE ALGEBRA II HOMEWORK 1. Which of the following scatter plots shows data best modeled by an exponential equation? y y (1) (2) (3) (4) y y x x x x 2. The total fat and the number of calories in various McDonalds sandwiches are recorded below. Sandwich Total Fat (g) Total Calories Hamburger Cheeseburger ¼ pounder (QP) QP with cheese Big Mac Fish Fillet a) Based on a scatter plot, what type of regression would best model the data? b) Find the appropriate regression equation with coefficients rounded to the nearest hundredth. c) Using your regression, how much fat was is in a sandwich that has 700 calories? Round to the nearest tenth of a gram. d) Use your regression to predict the total calories in a sandwich that has 25 grams of fat. Round to the nearest calorie.

31 31 3. The table below, created in 1996, shows a history of transit fares from 1975 to Note: x is measured in years after Year Fare ($) a) Based on a scatter plot, what type of regression would best model the data? b) Find the appropriate regression equation with coefficients rounded to the nearest thousandth. c) Using your regression, predict when the transit fare will reach $3.00 to the nearest year. d) Using your regression, what will the fare be in the year 2017 to the nearest cent? e) Using your regression, predict when the transit fare will reach $6.00 to the nearest year. f) It took a long time for the price of the fare to reach $3.00. Why was there much less time between when the fare reached $3.00 and when it reached $6.00?