14.3 Constructing Exponential Functions
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1 Name Class Date 1.3 Constructing Eponential Functions Essential Question: What are discrete eponential functions and how do ou represent them? Resource Locker Eplore Understanding Discrete Eponential Functions Recall that a discrete function has a graph consisting of isolated points. A The table represents the cost of tickets to an annual event as a function of the number t of tickets purchased. Complete the table b adding 1 to each successive cost. Plot each ordered pair from the table. Tickets t Cost ($) (t, f (t) ) 1 1 (1, 1) (, ) 3 (3, ) (, ) 5 (5, ) Houghton Mifflin Harcourt Publishing Compan Cost ($) Tickets Module 1 3 Lesson 3
2 B The number of people attending an event doubles each ear. The table represents the total attendance at each annual event as a function of the event number n. Complete the table b multipling each successive attendance b. Plot each ordered pair from the table. Event Number n Attendance (n, g (n) ) 1 (1, ) (, ) 3 (3, ) (, ) 5 (5, ) 3 3 Attendance 1 C Event Number Complete the table. Function Linear? Discrete? f (t) Yes g (n) Houghton Mifflin Harcourt Publishing Compan Reflect 1. Communicate Mathematical Ideas What are the limitations on the domains of these functions? Wh? Module 1 Lesson 3
3 Eplain 1 Representing Discrete Eponential Functions An eponential function is a function whose successive output values are related b a constant ratio. An eponential function can be represented b an equation of the form ƒ () = ab, where a, b, and are real numbers, a, b >, and b 1. The constant ratio is the base b. When evaluating eponential functions, ou will need to use the properties of eponents, including zero and negative eponents. Recall that, for an nonzero number c: Eample 1 c = 1, c c -n = c n, c. Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table. A ƒ () = 3 (_ 1 ) with a domain of -1,, 1,, 3, ƒ (-1) = 3 ( -1 ) = 3 = 3 = ( ) f () (, f () ) Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3
4 B ƒ () = 3 ( _ 3 ) ; domain = -, -1,, 1,, 3 ƒ (-) = 3 ( _ 3 ) - = 3 = 3 _ = 3 _ = _ = 1 _ ( _ ) f () (, f () ) - (-, ) -1 (-1, ) (, ) 1 (1, ) (, ) ( 3, 7 9) - - Reflect. What If What would happen to the function ƒ () = a b if a were? What if b were 1? Houghton Mifflin Harcourt Publishing Compan 3. Discussion Wh is a geometric sequence a discrete eponential function? Module 1 Lesson 3
5 Your Turn Make a table for the function using the given domain. Then graph the function using the ordered pairs from the table.. ƒ () = 3_ ( ; domain = ) -3, -, -1,, 1, f () (, f () ) - - Eplain Constructing Eponential Functions from Verbal Descriptions Houghton Mifflin Harcourt Publishing Compan Image Credits: Digital Vision/Gett Images You can write an equation for an eponential function ƒ () = ab b finding or calculating the values of a and b. The value of a is the value of the function when =. The value of b is the common ratio of successive function ƒ ( + 1) values, b =. For discrete functions with integer or whole number domains, these will be successive values of ƒ () the function. Eample A Write an equation for the function. When a piece of paper is folded in half, the total thickness doubles. Suppose an unfolded piece of paper is.1 millimeter thick. The total thickness t (n) of the paper is an eponential function of the number of folds n. The value of a is the original thickness of the paper before an folds are made, or.1 millimeter. Because the thickness doubles with each fold, the value of b (the constant ratio) is. n The equation for the function is t (n) =.1 (). Module 1 7 Lesson 3
6 B A savings account with an initial balance of $1 earns 1% interest per month. That means that the account balance grows b a factor of 1.1 each month if no deposits or withdrawals are made. The account balance in dollars B (t) is an eponential function of the time t in months after the initial deposit. Let B represent the balance in dollars as a function of time t in months. The value of a is the original balance,. The value of b is the factor b which the balance changes ever month,. The equation for the function is B(t) =. Reflect 5. Wh is the eponential function in the paper-folding eample discrete? Your Turn. A piece of paper that is. millimeters thick is folded. Write an equation for the thickness t of the paper in millimeters as a function of the number n of folds. Eplain 3 Constructing Eponential Functions from Input-Output Pairs You can use given two successive values of a discrete eponential function to write an equation for the function. Eample 3 Write an equation for the function that includes the points. A (3, 1) and (, ) Find b b dividing the function value of the second pair b the function value of the first: b = Evaluate the function for = 3 and solve for a. Write the general form. Substitute the value for b. ƒ () = ab ƒ () = a Substitute a pair of input-output values. 1 = a 3 Simplif. 1 = a Solve for a. a = 3_ Use a and b to write an equation for ƒ () = 3_ the function. _ 1 =. Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 3
7 B (1, 3) and (, 9_ ) Find b b dividing the function value of the second pair b the first: b = 9_ 3 =. Write the general form. ƒ () = Substitute the value for b. ƒ () = a Substitute a pair of input-output values. Simplif. = a ( 3 _ ) 3 = a Solve for a. a = Use a and b to write an equation for ƒ () = the function. Your Turn Write an equation for the function that includes the points. 7. ( -, _ 5) and (-1, ) Elaborate. Eplain wh the following statement is true: For < b < 1 and a >, the function ƒ () = ab decreases as increases. Houghton Mifflin Harcourt Publishing Compan 9. Eplain wh the following statement is true: For b > 1 and a >, the function ƒ () = ab increases as increases. 1. Essential Question Check-In What propert do all pairs of adjacent points of a discrete eponential function share? Module 1 9 Lesson 3
8 Evaluate: Homework and Practice Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table. Online Homework Hints and Help Etra Practice 1. ƒ () = ; domain = -, -1,, 1,. f () (, f () ) - -. ƒ () = 9 ( ; domain = 3), 1,, 3,, 5. f () (, f () ) 3. ƒ () = _ ( ; domain = 3) -1,, 1,, 3,. f () (, f () ) Houghton Mifflin Harcourt Publishing Compan Module 1 7 Lesson 3
9 . ƒ () = ( _ 3 ) ; domain = -3, -, -1,, 1 f () (, f () ) Write an equation for each function. 5. Business A recent trend in advertising is viral marketing. The goal is to convince viewers to share an amusing advertisement b or social networking. Imagine that the video is sent to 1 people on da 1. Each person agrees to send the video to 5 people the net da, and to request that each of those people send it to 5 people the da after the receive it. The number of viewers v (n) is an eponential function of the number n of das since the video was first shown. Houghton Mifflin Harcourt Publishing Compan Image Credits: David J. Green - technolog/alam. A pharmaceutical compan is testing a new antibiotic. The number of bacteria present in a sample when the antibiotic is applied is 1,. Each hour, the number of bacteria present decreases b half. The number of bacteria remaining r (n) is an eponential function of the number n of hours since the antibiotic was applied. 7. Optics A laser beam with an output of 5 milliwatts is directed into a series of mirrors. The laser beam loses 1% of its power ever time it reflects off of a mirror. The power p (n) is a function of the number n of reflections. Module 1 71 Lesson 3
10 . The NCAA basketball tournament begins with teams, and after each round, half the teams are eliminated. The number of remaining teams t (n) is an eponential function of the number n of rounds alread plaed. Write an equation for the function that includes the points. 9. (, 1) and (3, 1) 1. (-, ) and (-1, ) 11. ( 1, _ 5 ) and (, _ 3) 1. ( -3, 1) and ( -, 3_ ) Use two points to write an equation for the function. 13. f () 1 _ 7 3 _ 9 _ 33 Houghton Mifflin Harcourt Publishing Compan Module 1 7 Lesson 3
11 1. f () (1, ) (, 3) (1, ) (, ) Houghton Mifflin Harcourt Publishing Compan 17. The height h (n) of a bouncing ball is an eponential function of the number n of bounces. One ball is dropped and on the first bounce reaches a height of feet. On the second bounce it reaches a height of feet. Module 1 73 Lesson 3
12 1. A child starts a plaground swing from standing and doesn t use her legs to keep swinging. On the first swing she swings forward b 1 degrees, and on the second swing she onl comes 13.5 degrees forward. The measure in degrees of the angle m (n) is an eponential function of the number n of swings. 19. Make a Prediction A town s population has been declining in recent ears. The table shows the population since 19. Is this data consistent with an eponential function? Eplain. If so, predict the population for 1 assuming the trend holds. Year Population A piece of paper has a thickness of.15 millimeters. Write an equation to describe the thickness t(n) of the paper when it is repeatedl folded in thirds, where n is the number of foldings. Houghton Mifflin Harcourt Publishing Compan Image Credits: Steve Hi/ Somos Images/Corbis Module 1 7 Lesson 3
13 1. Probabilit The probabilit of getting heads on a single coin flip is _. The probabilit of getting nothing but heads on a series of coin flips decreases b _ for each additional coin flip. Write an eponential function for the probabilit p (n) of getting all heads in a series of n coin flips.. Multipart Classification Determine whether each of the functions is eponential or not. a. ƒ () = Eponential Not eponential b. ƒ () = 3 Eponential Not eponential c. ƒ () = 3 Eponential Not eponential d. ƒ () = 1. 1 Eponential Not eponential e. ƒ () = 3 Eponential Not eponential f. ƒ () = 1 5 Eponential Not eponential H.O.T. Focus on Higher Order Thinking Houghton Mifflin Harcourt Publishing Compan Image Credits: mj7/ Shutterstock 3. Eplain the Error Biff observes that in ever math test he has taken this ear, he has scored points higher than the previous test. His score on the first test was 5. He models his test scores with the eponential function s (n) = n where s (n) is the score on his nth test. Is this a reasonable model Eplain.. Find the Error Kalee needed to write the equation of an eponential function from points on the graph of the function. To determine the value of b, Kalee chose the ordered pairs (1, ) and (3, 5) and divided 5 b. She determined that the value of b was 9. What error did Kalee make? Module 1 75 Lesson 3
14 Lesson Performance Task In ecolog, an invasive species is a plant or animal species newl introduced to an ecosstem, often b human activit. Because invasive species often lack predators in their new habitat, their populations tpicall eperience eponential growth. A small initial population grows to a large population that drasticall alters an ecosstem. Feral rabbits that populate Australia and zebra mussels in the Great Lakes are two eamples of problematic invasive species that grew eponentiall from a small initial population. An ecologist monitoring a local stream has been collecting samples of an unfamiliar fish species over the past four ears and has summarized the data in the table. Here are the results so far: Year Average Population Per Mile a. Look at the data in the table and confirm that the growth pattern is eponential. b. Write the equation that represents the average population per square meter as a function of ears since 9. c. Predict the average populations epected for 13 and 1. d. Graph the population versus time since 9 and include the predicted values. Average population per mile P t Houghton Mifflin Harcourt Publishing Compan Time (ears) Module 1 7 Lesson 3
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