PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS

Topic 21: Problem solving with eponential functions 323 PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS Lesson 21.1 Finding function rules from graphs 21.1 OPENER 1. Plot the points from the table onto the graph below. 2. Draw slope triangles between the points ou plotted. - 2-4 - 1-2 0 0 1 2 2 4 3 6 4 8

324 Unit 7 Eponential relationships 21.1 CORE ACTIVITY 1. Look back at our slope triangles from the Opener. a. What do ou notice about the heights of the slope triangles as increases? b. Write an algebraic rule for the relationship between and in question 1 of the Opener. 2. Given the following table and graph - 2 0.25-1 0.5 0 1 1 2 2 4 3 8 4 16 a. Plot the points from the table onto the curve and draw slope triangles between the points ou plotted. b. How does the height of each slope triangle compare with the one before it (moving from left to right)? B what factor does it change for each new triangle? c. What does this factor, or constant multiplier, tell us about the algebraic rule for this relationship? d. What is the - intercept (the value of when = 0)? e. What does the - intercept tell us about the algebraic rule for this relationship? f. Write an algebraic rule for the relationship between and.

Topic 21: Problem solving with eponential functions 325 3. Given the following table and graph - 1 120 0 60 1 30 2 15 3 7.5 a. Plot the points from the table onto the curve and draw slope triangles between the points ou plotted. b. How does the height of each slope triangle compare with the one before it (moving from left to right)? B what factor does it change for each new triangle? c. What does this factor, or constant multiplier, tell us about the algebraic rule for this relationship? d. What is the - intercept (the value of when = 0)? e. What does the - intercept tell us about the algebraic rule for this relationship? f. Write an algebraic rule for the relationship between and.

326 Unit 7 Eponential relationships 21.1 CONSOLIDATION ACTIVITY In this activit, ou will work with our partner to match different representations of functions to descriptions of how the functions grow. Objective: Create sets of matching cards. Matching is defined as representing the same relationship. Each set will have a table card, an equation card, a graph card, and a growth card that describes the growth of the relationship. On the growth card, ou ma be asked to write some additional information about the relationship to complete the set. Materials: Your teacher will give ou and our partner pages with cards on them to cut out. There are si graph cards (labeled A- F), si table cards (labeled G- L), and si growth cards (labeled M- R). Instructions: Work with our partner to find a set of matching cards. When ou both agree on a set of matching cards, tape the cards that form that set together. Justif the growth card ou selected b filling in the information that is asked for on the card. So that ou can more easil check our answers, tape each set with the graph card on the left, the table card in the middle, and the growth card on the right, as shown here. Graph card Table card Growth card

Topic 21: Problem solving with eponential functions 327 HOMEWORK 21.1 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Plot the points from the table onto the curve and draw slope triangles between the points ou plotted. - 1 3.33 0 10 1 30 2 90 3 270 2. How does the height of each slope triangle compare with the one before it? B what factor does it increase for each unit increase of? 3. What is the value of when = 0? 4. Write an algebraic rule for the relationship between and.

328 Unit 7 Eponential relationships STAYING SHARP 21.1 Practicing algebra skills & concepts 1. What is the product of 4 and 5? 2. The product of two eponential epressions is 5 2 z. One of the epressions is 2. What is other epression? 3. Describe the pattern of the - values in this table. 4. Graph the values in Question 3. Is the graph linear? Eplain. Preparing for upcoming lessons 0 1 1 2 3!! 1 2!! 1 4 1 8!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 5. Write each number in scientific notation. a. 1,320,000 6. Write each number in standard notation. e. 8.00 10 3 Focus skill: Reasoning with quantities b. 0.0006 c. 75,000,000,000,000,000,000 d. 0.00000000204310 f. 3.206 E 12 g. 9.9 10 5 h. 3.206 E 12

Topic 21: Problem solving with eponential functions 329 Lesson 21.2 Geometric sequences and eponential functions 21.2 OPENER Martina and Karina each wrote sequence puzzles on a strip of paper for each other to figure out. Martina s Sequence 2, 6, 10, 14,,,... Karina s Sequence 2, 6, 18, 54,,,... 1. Find the net two terms for Martina s sequence. Describe the pattern and eplain how ou found it. 2. Find the net two terms for Karina s sequence. Describe the pattern and eplain how ou found it. 21.2 CORE ACTIVITY Compare the process ou used to find the net two terms for Martina s pattern and Karina s pattern in the Opener. 1. How did each term compare to the one before it for Martina s sequence? 2. How did each term compare to the one before it for Karina s sequence? 3. For which sequence were the terms related b constant differences? What was the constant difference? 4. For which sequence were the terms related b constant ratios? What was the constant ratio?

330 Unit 7 Eponential relationships 5. You alread saw that each term in Martina s sequence is 4 units more than the previous term. In other words, the common difference is 4. Fill in the table to epress this relationship using function notation. Term number, n Process Term, f(n) 1 2 2 f(2) = 2 + 4 = f(1) + 4 6 3 f(3) = 6 + 4 = f(2) + 4 10 4 f(4) = 10 + 4 = f(3) + 4 14 5 f(5) = 14 + 4 = f(4) + 4 18 6 f(6) = 18 + 4 = + 4 22 7 f(7) = + 4 = + 4 8 f(8) = + 4 = + 4 9 f(9) = + 4 = + 4 6. Write a rule to represent the sequence. 7. Complete the table to write both tpes of rules for the sequence in smbolic and verbal form. Recursive rule Eplicit rule Verbal rule Function rule

Topic 21: Problem solving with eponential functions 331 Toda is Amanda s fifteenth birthda. On the da Amanda was born, her grandmother invested $100 for her in a special account. Toda she wants to know how much it is worth. Suppose the account earns 6% interest on Amanda s birthda each ear. Since the interest is alwas added to the amount in the account, each ear s interest is based on a larger amount than the ear before. 8. Fill in the table to figure out how much mone is in the account on Amanda s fifteenth birthda. 9. Once ou found the amount in the account after Amanda s eighth birthda (n = 8), how did ou figure out the amount on her ninth birthda (n = 9)? Number of Amount, A ($) ears, n 0 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10. Describe how ou used the amount in the previous ear to calculate the amount in Amanda s account for each ear. 11. Write two different rules for finding the amount in Amanda s account. Your recursive rule should tell how to find the amount based on the amount from the previous ear. Your eplicit rule should tell how to find the amount based on the number of ears, n. Recursive rule Eplicit rule Verbal rule Function rule

332 Unit 7 Eponential relationships A Round- A- Bound is a to ball that bounces unusuall high. The diagram shows the path of a Round- A- Bound ball that is dropped from a height of 50 feet. On its first bounce, it reaches a height of 40 feet. On its second bounce, it reaches a height of 32 feet. Each successive bounce height decreases b a constant ratio. Drop height: 50 ft bounce 1 bounce 2 bounce 3 bounce 4 bounce 5 bounce 6 12. Find a pattern to complete the rest of the table for the path of the Round- A- Bound ball. Bounce, b Height, h (ft) 0 50 1 40 2 32 3 4 5 6 13. What is the constant ratio for this function? Eplain how ou found it.

Topic 21: Problem solving with eponential functions 333 14. Write two different rules for finding the bounce height of the Round- A- Bound. Your recursive rule should tell how to find the bounce height using the height from the height of the previous bounce. Your eplicit rule should tell how to find the bounce height based on the number of bounces, b. Recursive rule Eplicit rule 21.2 CONSOLIDATION ACTIVITY In this activit, ou will work with our partner to find different representation of the same function. Each of the functions can be represented as a recursive function, as an eplicit function, and as an input- output table. One representation of the function is given in each row of the table. Complete the table b filling in the missing two representations for each function. Recursive function Eplicit function Input/output table 0 3 1 7 2 11 3 15 4 19 The first term is 80. To get each net term, divide the previous term b 2 (or multipl it b ½). 0 1 2 3 4 0 80 1 79.5 2 79 3 78.5 4 78

334 Unit 7 Eponential relationships Recursive function Eplicit function Input/output table The first term is 10. Then double each term to get the net term. 0 1 2 3 4 0 1 1 5 2 25 3 125 4 625 = ( 1.1) 0 1 2 3 4 Now create a geometric sequence (i.e., eponential function) of our own, and represent it using onl one of the boes below. Then have our partner fill in the other two representations. Check each other s work when ou are both done. Recursive function Eplicit function Input/output table 0 1 2 3 4

Topic 21: Problem solving with eponential functions 335 HOMEWORK 21.2 Notes or additional instructions based on whole- class discussion of homework assignment: Marcos invested $1,000 in a savings account. His mone will grow at a rate of 15% a ear. 1. Create a table and graph showing the amount of mone in his account for the first 5 ears of his investment. (Number of ears) (Amount of mone in $) 0 1,000 1 2 3 4 5 2. Write a recursive function rule to model the situation. (How can ou find each value of from the value that comes before it?) 3. Write an eplicit function rule to model the situation. (How can ou find each value of from each value of?) What is the constant multiplier in this relationship? How does each number in this function rule relate to the data in the table and the graph? 4. Use the table, graph, or function rule ou wrote to answer the following questions. Eplain how ou found the answer. a. How much mone will be in the account after 5 ears? How I figured out the answer: b. After about how man ears will the amount of mone in the account be $1,500? How I figured out the answer: c. How much mone will be in the account in 9 ears? How I figured out the answer:

336 Unit 7 Eponential relationships STAYING SHARP 21.2 Practicing algebra skills & concepts 1. Solve the following equation: 2( + 1) = 3 1. Justif each step ou take to solve the problem. 2. Use first differences to determine whether or not the table represents a linear relationship: - 2 8-1 2 0 0 1 2 2 8 3. Find second differences of the - values in the table from problem 2. What do ou notice? 4. Graph the points from problem 2. What do ou notice? Preparing for upcoming lessons 5. Justif each step in the multiplication of (8.7 10 4 ) (6.1 10 9 ). 6. Write the following numbers in order from least to greatest. Focus skill: Reasoning with quantities Step 8.7 (10 4 6.1) 10 9 8.7 (6.1 10 4 ) 10 9 (8.7 6.1) (10 4 10 9 ) Justification 9.999999 10 3 0.00000007668 100,000,000 126 million 3.9 E - 9 5,900,000,000 2.75 E 9 1.1 X 10 1 8.0006 10 10 1.0001 10 18

Topic 21: Problem solving with eponential functions 337 Lesson 21.3 Transforming eponential functions 21.3 OPENER 1. Determine whether each table represents a linear relationship, an eponential relationship, or neither. Then, eplain how ou know. a. 1-2 2 3 3 8 4 13 5 18 6 23 Circle one: Linear Eponential Neither Eplain how ou know: b. - 3 1-2 2-1 4 0 8 1 16 2 32 Circle one: Linear Eponential Neither Eplain how ou know: c. Circle one: d. Circle one: 1 8 Linear Eponential Neither - 3 10 Linear Eponential Neither 2 12 3 18 Eplain how ou know: - 2 5-1 2 Eplain how ou know: 4 27 0 1 5 40.5 1 2 6 60.75 2 5 2. What did ou calculate to determine whether the tables above were linear or eponential? 21.3 CORE ACTIVITY 1. Recall Barr and Red s eperiment with insects from the topic Comparing Linear and Eponential Growth. Red s data from raising fire ants is shown in the table. Use the process column to show how the number of fire ants,, can be calculated mathematicall from the number of weeks,, in each row of the table. 2. Write an algebraic rule for the number of fire ants,, in terms the number of weeks,. Weeks Process Fire ants 0 20 1 40 2 80 3 160 4 320

338 Unit 7 Eponential relationships 3. Red and Barr are discussing Red s data. Work with our partner to evaluate each of the statements the make in their discussion. Tell whether ou agree or disagree with the statement. Give reasons wh ou agree or disagree. Statement Barr: The growth is linear with a slope of 2. Agree or disagree (with eplanation of reasoning)? Red: No, the growth is eponential with a constant ratio of 2. Barr: If there is a constant ratio of 2, that means the base of the eponential epression in the algebraic rule is 2. Red: Since I started with 20 ants, the base must be 20. Mabe the function rule should be: = 20 Barr: Red: Starting with 20 ants means ou have to multipl b a constant of 20. That s where the 20 comes from in = 20 2. I see. That means the - intercept of the graph should be at = 20. Barr: It also means that ever value of our function is 20 times larger than it would be for = 2. That s because ou started out with 20 ants instead of just one. 4. Sketch the graph of = 20 2 on the same aes as the function = 2. Label the graph of the new function. = 2 5. How does the graph of = 20 2 compare to the graph of = 2?

Topic 21: Problem solving with eponential functions 339 6. Suppose Red tries a second eperiment. This time he starts with 15 fire ants instead of 20. The population of fire ants grows the same wa as it did in the first eperiment. a. Fill in the data table to show how Red s data would change. Show how ou calculated the number of fire ants for each week in the process column. Weeks Process Fire ants 0 15 1 2 3 4 b. What is the new function rule for this second eperiment? c. Sketch and label the graph of the new function on the same aes as our graph from question 4. d. How does the graph of this new function compare to the graph of = 20 2 and the graph of = 2? e. How does this function compare to the eperiment that started with 20 ants? How are the two functions similar? How are the different? f. For the new eperiment, predict how man weeks it will take for the number of fire ants to reach 1000.

340 Unit 7 Eponential relationships 21.3 CONSOLIDATION ACTIVITY 1. Make a table comparing the - values of Red s new function rule, = 40 2 1, to those of the function rule ou found earlier. Weeks 0 Number of fire ants = 20 2 Number of fire ants = 40 2 1 1 2 3 4 2. Verif the rules = 20 2 and = 40 2 1 are equivalent using graphs. 3. Think about how ou can appl the laws of eponents ou learned in a previous topic to rewrite the epression 40 2 1. Use these laws to verif the rules = 40 2 1 and = 20 2 are equivalent b algebraic manipulation. 40 2 1 = 40 2 2 1 = 40 2 1 2 = 40 1 2 2 = 20 2

Topic 21: Problem solving with eponential functions 341 HOMEWORK 21.3 Notes or additional instructions based on whole- class discussion of homework assignment: A team of biologists is researching the population of white- tailed deer that live in a certain area of the countr. The have found that the deer population in the area is growing at a rate of about 25% per ear. There are currentl 32 deer living in the area. 1. Create a table and graph showing a prediction of number of deer in the area for each of the net 10 ears. (Number of ears) (Number of deer) 0 32 1 2 3 4 5 6 7 8 9 10 2. Find the following for this situation: a. The constant multiplier: b. The multiplication constant, or stretch factor : c. A function rule that fits this population model: 3. Use the table and graph ou created to predict the following. Eplain how ou made each prediction. a. The amount of time it will take for the deer population to reach 400 deer b. The number of deer that will be in the area in 15 ears 4. Suppose there were currentl 100 deer living in the area instead of 32. a. What new function rule would fit this situation? b. Make a sketch of the graph of this function on the same aes as the function ou graphed in question 1. How does the graph compare to the graph from question 1?

342 Unit 7 Eponential relationships STAYING SHARP 21.3 Practicing algebra skills & concepts 1. A 9 th grade math class has 27 students. There are twice as man girls in this class as there are bos. Write a sstem of equations that could be used to model this situation. 2. Solve the sstem of equations from problem 1 using an method. How man girls are in the class? Preparing for upcoming lessons 3. Describe the patterns ou see in this sequence of ordered pairs: - 4-3 - 2-14 - 7-2 - 1 1 0 2 1 1 2-2 3-7 4-14 4. Graph the ordered pairs from Problem 3. (Choose our scale carefull.) What do ou notice? Focus skill: Reasoning with quantities 5. A large tank is 120 meters long, 65 meters wide, and 48 meters high. Epress the volume of the tank in cubic meters using scientific notation. 6. One liter is equal to 0.001 cubic meters. Find the volume of the tank from question 5 in liters. Epress our answer using scientific notation.

Topic 21: Problem solving with eponential functions 343 Lesson 21.4 Eploring parameters 21.4 OPENER 1. Given the function = 80 5 a. What is the value of when = 0? 2. Given the function = 80 1 2 a. What is the value of when = 0? b. What is the common multiplier? b. What is the common multiplier? c. Complete this table of values. c. Complete this table of values. - 2-2 - 1-1 0 0 1 1 2 2 3 3 3. Describe two different methods ou could use to find the values in the table in question 2. 4. How are the function rules in questions 1 and 2 different? How are the the same?

344 Unit 7 Eponential relationships 21.4 CORE ACTIVITY In this activit, ou will use our graphing calculator to investigate the effect of changing parameters of eponential functions. Man eponential functions can be written in the form: = a b Two of the parameters of an eponential function are the values of a, the multiplication factor, and b, the base (or constant multiplier). As ou have alread seen, changing the values of a function s parameters changes the behavior of the function. You will predict how each parameter affects the function and then test our predictions using our graphing calculator. 1. Investigate the effect of the base, b, b following the steps below. a. Enter the function = 2 into Y1. (This is the original function that ou will compare our transformations with.) b. Select a new value for the base, b. Pick values that are greater than 2, between 1 and 2, between 0 and 1. c. Record our new function. Enter it into our calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on our calculator. Sketch it with a solid line. How does our prediction compare to the actual graph? e. Describe how the graph of the new function compares to the original function = 2. (a) Original function (in Y1) = 2 Multipli- cation factor, a 1 (b) New base, b Pick a value greater than 2 and less than 10, 2 < b < 10 (c) New function = b (in Y2) (d) Graph Prediction: Dotted line Actual: Solid line Compare (e) Describe how the graph of the new function compares to the original function b = 1 Pick a value greater than 1 and less than 2, 1 < b < 2 b = 1 Pick a value greater than 0 and less than 1, 0 < b < 1 b = Eplain how the value of the base affects the graph of an eponential function.

Topic 21: Problem solving with eponential functions 345 2. Investigate the effect of the multiplication factor, a, b following the steps below. a. Enter the original function = 2 into Y1. b. Select a new value for the multiplication factor, a. Pick values that are between 1 and 10 and between 0 and 1. c. Record our new function. Enter it into our calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on our calculator. Sketch it with a solid line. How does our prediction compare to the actual graph? e. Describe how the graph of the new function compares to the original function = 2. (a) Original function (in Y1) = 2 (b) New multipli- cation factor, a Base, b (c) New function = a 2 (in Y2) (d) Graph Prediction: Dotted line Actual: Solid line Compare (e) Describe how the graph of the new function compares to the original function Pick a value greater than 1 and less than 10, 1 < a < 10 2 a = Pick a value greater than 0 and less than 1, 0 < a < 1 2 a = Eplain how the value of the multiplication factor affects the graph of an eponential function.

346 Unit 7 Eponential relationships One additional parameter, c, can also be included in an eponential function. This parameter is called a constant term. It can be added to the function as shown below. The result is called the general form of an eponential function. = a b + c 3. Investigate the effect of adding a constant term b following the steps below. a. Enter the original function = 2 into Y1. b. Select a new value for the constant term, c. Use a variet of values: Ones that are between 0 and 10, between - 2 and 0. c. Record our new function. Enter it into our calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on our calculator. Sketch it with a solid line. How does our prediction compare to the actual graph? e. Describe how the graph of the new function compares to the original function = 2. (a) Original function (in Y1) Multipli- cation factor, a Base, b (b) New constant term, c (c) New function = 2 + c (in Y2) (d) Graph Prediction: Dotted line Actual: Solid line Compare (e) Describe how the graph of the new function compares to the original function = 2 1 2 Pick a value greater than 1 and less than 10, 1 < c < 10 1 2 c = Pick a value greater than 0 and less than 1, 0 < c < 1 c = Eplain how the constant term affects the graph of an eponential function. 21.4 ONLINE ASSESSMENT

Topic 21: Problem solving with eponential functions 347 HOMEWORK 21.4 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Without graphing the functions on a graphing calculator, describe the similarities and differences between the graphs of the functions = 100 0.2 and = 100 0.95. Sketch a graph of what ou think these two functions will look like. 2. Without graphing the functions on a graphing calculator, describe the similarities and differences between the graphs of the functions = 2 3 and = 10 3. Sketch a graph of what ou think these two functions will look like. 3. Match the curves on the graph with the function rule that best represents the curve. Graph A Graph B Graph C Graph D Graph E Graph F = 1 5 = 1 2 = 1 0.2 = 1 0.5 = 1 2.7 = 1 0.35 4. Describe how the parameters a and b affected the graph of = ab. Don t forget to mention specific values that these parameters cannot have. How the parameter a affects the graph of = ab How the parameter b affects the graph of = ab

348 Unit 7 Eponential relationships STAYING SHARP 21.4 Practicing algebra skills & concepts 1. While taking a road trip, Jose decides to keep track of his mileage. After 2 hours he has traveled a total of 120 miles and after 3 hours he has traveled a total of 180 miles. What is the rate of change between the two points? 2. What does the rate of change from Problem 1 represent? Preparing for upcoming lessons 3. Plot the following points on the coordinate plane provided: (- 1,1), (0,3), (1,1), (2,- 5). 4. Does the graph from problem 3 represent a linear function, an eponential function, or neither? Eplain how ou know. Focus skill: Reasoning with quantities 5. What was the estimated world population in 1940? Epress our answer in scientific notation. 6. Use the graph in question 5 to find the ear when the population of the world was approimatel 5.25 10 9.

Topic 21: Problem solving with eponential functions 349 Lesson 21.5 Behavior of eponential functions 21.5 OPENER Without calculating eact values, predict whether each of the following values will be ver small or ver large. Eplain the reasoning for each of our predictions. Then check our predictions using a calculator. Epression Prediction (circle one) Reasoning Calculated value (1/3) 10 ver small ver large 3.05 10 ver small ver large 5000 (0.005) 10 ver small ver large 5000 10 ver small ver large (1.005) 10 ver small ver large 21.5 CORE ACTIVITY The behavior of a function has to do with how its value changes at different locations. Work with a partner to answer questions 1 and 2 b analzing the behavior of the functions shown in the graph below. = (1.3) = (1.3) = 0.8 + 10 = 10 (0.8) P Q

350 Unit 7 Eponential relationships 1. Compare the four functions in the graph at the locations described in the table below. Then write the function from the graph that best fits into each bo in the table below. Location along -ais Ver far to the left (beond what ou can see on the graph) At = 0 A little farther to the right on the graph where = P A little farther to the right on the graph where = Q Ver far to the right (beond what ou can see on the graph) a. Which function has the greatest value? b. Which function has the least value? c. Which function shows the fastest growth? d. Which function shows the fastest deca? 2. Recall that the domain of a function is the set of possible - values. The range of a function is the set of possible - values. Find the domain and range of each of the functions in the graph. a. = (1.3) b. = (1.3) Domain: Range: Domain: Range: c. = 0.8 + 10 d. = 10 (0.8) Domain: Range: Domain: Range:

Topic 21: Problem solving with eponential functions 351 An insurance compan estimates that the value of a particular car depreciates b 15% each ear. The compan uses an eponential function to predict the value of a car, v, as a function of time, t, in ears. The function rule and graph are shown here. v = 22,000 (0.85) t 3. Write a paragraph to describe the behavior of this function model. In our description, discuss the following: Does the function represent eponential growth or deca? Eplain how ou know in as man was as ou can. What does the 22,000 represent in the function equation? What does the 0.85 represent in the equation? How is it related to the 15% depreciation? What is the domain and range of the function? Eplain our reasoning. When is the value of the car greatest? The least? Wh? Value of the car, v ($) Car Depreciation Time, t (ears) 4. Write an eponential function to predict the value of a car, v, as a function of time, t, in ears, for a car that is worth $28,000 when new and depreciates b 20% ever ear.

352 Unit 7 Eponential relationships 21.5 REVIEW ONLINE ASSESSMENT You will work with our class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Attributions for our successes: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Attributions for our difficulties: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that our class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:

Topic 21: Problem solving with eponential functions 353 HOMEWORK 21.5 Notes or additional instructions based on whole- class discussion of homework assignment: Net class period, ou will take an end- of- unit assessment. One good stud skill to prepare for tests is to review the important skills and ideas ou have learned. Use this list to help ou review these skills and concepts b reviewing related course materials. Important skills and ideas ou have learned in the unit Eponential and quadratic functions: 1. Rewriting epressions using the laws of eponents 2. Converting numbers between scientific and standard notation and computing in scientific notation 3. Connecting common differences and common multipliers to linear and eponential functions 4. Eamining the effects of a and b on the behavior of eponential functions 5. Eamining the effects of a and c on the behavior of quadratic functions 6. Comparing linear, eponential, and quadratic functions Homework Assignment Part I: Part II: Stud for the end- of- unit assessment b reviewing the ke ideas in the topic as listed above. Take the More practice from the topic Problem solving with eponential functions through the online services. Note the skills and ideas for which ou need more review, and refer back to related activities and animations from this topic to help ou stud. Part III: Complete Staing Sharp 21.5

354 Unit 7 Eponential relationships STAYING SHARP 21.5 Practicing algebra skills & concepts 1. Write an equation or inequalit that could be used to mathematicall represent the following statement: Five more than a number is less than twice the number minus one. 2. Solve the equation or inequalit from Question 1. 3. The following table relates the area of a square given a certain side length. Complete the table and sketch a graph of the data. 4. What function rule could be used to represent the data from Question 3? Preparing for upcoming lessons Side length Area 1 1 2 3 4 What is an appropriate domain for this function? Eplain. 5. Write the following distances in order from least to greatest. 6. Without using a calculator, rewrite the following product using scientific notation. Focus skill: Reasoning with quantities 25,000 cm; 3.2 10 4 km; 6.08 E 3 km; 5 million meters (7 10 3 ) (2 10 4 ) (3 10 8 )

Topic 21: Problem solving with eponential functions 355 Lesson 21.6 Checking for understanding 21.6 OPENER Three situations are described below. One is represented with a graph, one with a verbal description, and one with a table. For each situation, write a function rule to model the relationship. Then eplain how ou found our function rule. Relationship Function rule Eplanation a. The number of trees growing in an orchard each ear is plotted on a graph. b. A sand hill is 50 feet high. The wind and rain cause its height to decrease b 20% each ear. c. A runner keeps track of how man miles she runs each week. Weeks Number of miles run 0 10.00 1 11.00 2 12.10 3 13.31 4 14.64 21.6 END-OF-UNIT ASSESSMENT Toda ou will take the end of unit assessment.

356 Unit 7 Eponential relationships 21.6 CONSOLIDATION ACTIVITY 1. There are some important similarities between linear functions and eponential functions. Eplore these similarities b completing the table below for Function A and Function B. Function A Function B = 3 + 2 Tpe of function: (linear or eponential): = 3 2 Tpe of function: (linear or eponential): Table: Table: - 2-2 - 1-1 0 0 1 1 2 2 3 3 Sketch of graph: Sketch of graph: The constant difference between terms is: The constant multiplier between terms is: The coefficient multiplied to the variable,, is: The base of the eponent,, is: The - intercept is: The - intercept is: To find the net term for this function, I would To find the net term for this function, I would

Topic 21: Problem solving with eponential functions 357 2. Answer the following questions to reflect on our performance and effort this unit. a. Summarize our thoughts on our performance and effort in math class over the course of this unit of stud. Which areas were strong? Which areas need improvement? What are the reasons that ou did well or did not do as well as ou would have liked? b. Set a new goal for the net unit of instruction. Make our goal SMART. Description of goal: Description of enabling goals that will help ou achieve our goal:

358 Unit 7 Eponential relationships HOMEWORK 21.6 Notes or additional instructions based on whole- class discussion of homework assignment: 1. Evaluate the following epressions. (Don t forget to use the correct order of operations!) a. 4 3 + 2 2 + - 6 3 b. (2 + 3 4) 4 + (- 2 3 +1) 2 2. Complete the following table. The first row has been completed for ou. 3 2 2 2 3 2 + 2 2 5 2-4 3 (- 4) 2 = 48 2 (- 4) 2 = 32 3 (- 4) 2 + 2 (- 4) 2 = 80 5 (- 4) 2 = 80-2 0 3 5 3. What relationships do ou notice between the epressions 3 2 + 2 2 and 5 2? Eplain. 4. Complete the following table. The first row has been completed for ou. 2 + 2 9 ( 2 + ) + (2 9) 2 + 3 9-4 (- 4) 2 + - 4 = 12 2(- 4) 9 = - 17-2 0 3 5 ((- 4) 2 + - 4) + (2(- 4) 9) = 12 + - 17 = - 5 (- 4) 2 + 3(- 4) 9 = - 5 5. What relationships do ou notice: a. between the values in the table in Question 4 for the ( 2 + ) + (2 9) and 2 + 3 9 columns? b. between the epressions ( 2 + ) + (2 9) and 2 + 3 9? Eplain.

Topic 21: Problem solving with eponential functions 359 STAYING SHARP 21.6 Practicing algebra skills & concepts 1. At Pizzamania, the cost of a large pizza is $12 plus $1.75 for each topping. What function rule could ou use to find the cost c of a pizza with toppings? 2. Using the function rule from Problem 1, determine the number of toppings a large pizza has if it costs $20.75. Preparing for upcoming lessons 3. The following diagram models the epression ( 2 + ) + (2 2 + 3 + 1) Write a simpler epression for this sum.! " #!#! "!! "!! "!!!!!!! $#! 4. Find the perimeter of a rectangle with length 2 and width 8. Provide a sketch to support our work. 5. Calculate the value of E in the equation below. Epress our answer in scientific notation. 6. Complete the table b writing the amounts in scientific notation. Focus skill: Reasoning with quantities E = (7.6 10 30 ) (3.0 10 8 ) 2 Year National debt ($) 1791 75 million 1916 1 billion 1946 280 billion 2010 13.5 trillion National debt in scientific notation ($)