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1 Functions Contents WS07.0 Applications of Sequences and Series... Task Investigating Compound Interest... Task Reducing Balance... WS07.0 Eponential Functions... 4 Section A Activity : The Eponential Function, f ( ) Section A Activity : The Eponential Function, g ( ) Section A Activity 3: Compare the graph of f ( ) with the graph of g ( ) Section A Activity 4: Understand the characteristics of f( ) a, a Section B Activity : The Eponential Function, Section B Activity : The Eponential Function, f ( ) g ( ) Section B Activity 3: Compare the graph of f ( ) with the graph of g ( ) Section B Activity 4: Understand the characteristics of f( ) a, 0 a Section C Activity : Compare the graph of f( ) with the graph of f( ) Section C Activity : Compare the graph of g( ) 3 with the graph of g( ) Section C Activity 3: Now I see Section C Activity 4: Which of the following equations represent eponential functions?... 6 Problem Solving Questions on Eponential Functions... 7 WS07.03 Looking at Eponential Functions in Another Way... 9 Activity : Making the most of a Euro... 0 Activity : Further Eploration of Eponential Functions... WS07.04 Transformations of the Quadratic... 3 Activity : A to G... 3 Activity : Review of Graphs of Functions Activity 3: Different forms of the Quadratic... 36

2 WS07.0 Task Applications of Sequences and Series Investigating Compound Interest. If each block represents 0, shade in 00.. Then, using another colour, add 0% to the original shaded area. 3. Finally, using a third colour, add 0% of the entire shaded area. 4. What is the value of the second shaded area? 5. What is the value of the third shaded area? 6. Why do they not have the same amount? 7. Complete the following table and investigate the patterns which appear. Days (Time Elapsed) Amount Increase by % Total decimal Pattern/Total Amount of money received per day % Can you find a way of getting the value for day 0 without having to do the table to day 0? 9. Use your whiteboard to graph amount against time. Is the relationship linear? Task Reducing Balance David and Michael are going on the school tour this year. They are each taking out a loan of 600, which they hope to pay off over the net year. Their bank is charging a monthly interest rate of.5% on loans. David says that with his part-time work at present he will be able to pay 00 for the first 4 months but will only be able to pay off 60 a month after that. Michael says that he can only afford to pay 60 for the first 4 months and then 00 after that. Michael reckons that they are both paying the same amount for the loan. Why? Note: This problem is posed based on the following criteria: (a) A loan is taken out (b) after month interest is added on (c) the person then makes his/her monthly repayment. This process is then repeated until the loan is fully paid off.

3 Time 0 Monthly Total David Interest Payment Monthly Total Michael Interest Payment What do David and Michael have in common at the beginning of the loan period?. Calculate the first 3 months transactions for each. (How much, in total, had they each paid back after 3 months?) David Michael. 3. What is the total interest paid by each? David Michael. 4. Based on your answers to the first 3 questions, when would you recommend making the higher payments and why? 5. Is Michael s assumption that they will eventually pay back the same amount valid? 6. Using your whiteboard, plot the amount of interest added each month to both David s and Michael s account. 7. Looking at the graph, who will pay the most interest overall? 3

4 WS07.0 Eponential Functions Aim: To study the properties of eponential functions and learn the features of their graphs Section A Activity : The Eponential Function, f ( ).. For f ( ) : (i) The base of f ( ) is (ii) The eponent of f ( ) is (iii) What is varying in the function f ( )? (iv) What is constant in the function f ( )?. For f ( ) : What are the possible inputs i.e. values for (the domain)? Natural numbers Irrational numbers Integers Rational numbers 3. Set up a table of values and draw the graph of f ( ) X y f( ) Describe the graph of f ( ) : 6 Real numbers on your whiteboard: (i) Is it a straight line? (ii) Is y f( ) increasing or decreasing as increases? (iii) From the table above, find the average rate of change over different intervals. For eample from to and to 3. What do you notice? (iv) Describe how the curvature/rate of change is changing. 4

5 5. For f ( ) : (i) What are the possible outputs (range) for f ( ). (ii) Is it possible to have negative outputs? Eplain why? (iii) What happens to the output as decreases? (iv) Is an output of 0 possible? Why do you think this is? (v) What are the implications of this for the -intercept of the graph? (vi) What is the y-intercept of the graph of f ( )? 5

6 Section A Activity : The Eponential Function, g ( ) 3.. For g ( ) 3 : (i) The base of g ( ) 3 is (ii) The eponent of g ( ) 3 is (iii) What is varying in the function g ( ) 3? (iv) What is constant in the function g ( ) 3?. For g ( ) 3 : What are the possible inputs i.e. values for (the domain)? Natural numbers Irrational numbers Integers Rational numbers 3. Set up a table of values and draw the graph of g ( ) 3 X 3 y g( ) In relation to the graph of g ( ) 3 : Real numbers on your whiteboard: (i) Is it a straight line? (ii) Is y g( ) increasing or decreasing as increases? (iii) From the table above, find the average rate of change over different intervals. For eample from to and to 3. What do you notice? (iv) Describe how the curvature/rate of change is changing. 6

7 5. For g ( ) 3 : (i) What are the possible outputs (range) for g ( ) 3. (ii) Is it possible to have negative outputs? Eplain why? (iii) What happens to the output as decreases? (iv) Is an output of 0 possible? Why do you think this is? (v) What are the implications of this for the -intercept of the graph? (vi) What is the y-intercept of the graph of g ( ) 3? 7

8 Section A Activity 3: Compare the graph of f ( ) with the graph of g ( ) 3.. How are they similar and how do they differ?. Consider the relations (, y), y, y and (, ),, 3 Are they functions? Eplain. y y y. 3. What name do you think is given to this type of function and why do you think it is given this name? Section A Activity 4: Understand the characteristics of f( ) a, a.. What is the domain of f( ) a, a?. In relation to the graph of f( ) a, a. (i) Is it a straight line? (ii) Is y f( ) increasing or decreasing as increases? (iii) (iv) (v) Does it have a maimum value? Does it have a minimum value? Describe how its curvature/rate of change is changing. 3. What is the range of f( ) a, a? 4. What is the intercept of the graph f( ) a, a? 5. What is the y intercept of the graph f( ) a, a? 8

9 Section B Activity : The Eponential Function, f ( ).. For (i) (ii) (iii) (iv) f ( ) : The base of The eponent of f ( ) is f ( ) What is varying in the function f ( )? What is constant in the function f ( )?. For f ( ) : What are the possible inputs i.e. values for (the domain)? Natural numbers Irrational numbers Integers Rational numbers is Real numbers 3. Set up a table of values and draw the graph of f ( ) on your whiteboard: X y f( )

10 4. In relation to the graph of f ( ) : (i) Is it a straight line? (ii) Is y f( ) increasing or decreasing as increases? (iii) From the table above, find the average rate of change over different intervals. For eample from to and to 3. What do you notice? (iv) Describe how the curvature/rate of change is changing. 5. For (i) f ( ) : What are the possible outputs (range) for f ( ). (ii) Is it possible to have negative outputs? Eplain why? (iii) What happens to the output as decreases? (iv) Is an output of 0 possible? Why do you think this is? (v) What are the implications of this for the -intercept of the graph? (vi) What is the y-intercept of the graph of f ( )? 0

11 Section B Activity : The Eponential Function, g ( ). 3. For (i) (ii) (iii) (iv) g ( ) 3 : The base of The eponent of g ( ) 3 is g ( ) 3 What is varying in the function g ( ) 3? What is constant in the function g ( ) 3?. For g ( ) 3 : What are the possible inputs i.e. values for (the domain)? Natural numbers Irrational numbers Integers Rational numbers is Real numbers 3. Set up a table of values and draw the graph of g ( ) 3 below on your whiteboard: X y g( )

12 4. In relation to the graph of g ( ) : 3 (i) (ii) (iii) Is it a straight line? Is y increasing or decreasing as increases? From the table above, find the average rate of change over different intervals. For eample from to and to 3. What do you notice? (iv) Describe how the curvature/rate of change is changing. 5. For (i) g ( ) : 3 What are the possible outputs (range) for g ( ). 3 (ii) Is it possible to have negative outputs? Eplain why? (iii) What happens to the output as decreases? (iv) Is an output of 0 possible? Why do you think this is? (v) What are the implications of this for the -intercept of the graph? (vi) What is the y-intercept of the graph of g ( ) 3?

13 Section B Activity 3: Compare the graph of f ( ) with the graph of g ( ). 3. How are they the same and how do they differ?. Consider the relations (, y), y, y and (, y), y, y 3. Are they functions? Eplain. 3. What name do you think we give to this type of function and why do you think it is given this name? Section B Activity 4: Understand the characteristics of f( ) a, 0 a.. What is the domain of f( ) a, 0 a? In relation to the graph of. f( ) a, 0 a. (i) Is it a straight line? (ii) Is y f( ) increasing or decreasing as increases? (iii) (iv) (iv) Does it have a maimum value? Does it have a minimum value? Describe how its curvature/rate of change is changing. 3. What is the range of f( ) a, 0 a? 4. What is the intercept of the graph f( ) a, 0 a? 5. What is the y intercept of the graph f( ) a, 0 a? 3

14 Note: For all the following, you should assume that the domain is. Section C Activity : Compare the graph of f( ) with the graph of f( ).. How are the graphs similar?. How are the graphs different? 3. Rewrite k f( ) in the form f( ). 4. What transformation maps the graph of f( ) onto the graph of f( )? Section C Activity : Compare the graph of g( ) 3 with the graph of g( ). 3. How are the graphs similar?. How do the graphs differ? k 3. Rewrite g( ) in the form g( ) What transformation maps the graph of g( ) 3 onto the graph of g( )? 3 4

15 Section C Activity 3: Now I see.... If f( ) a, a, a, then the properties of the eponential function are:. If f( ) a, a, a, then the features of the eponential graph are: 3. If f( ) a, a, 0 a, then the properties of the eponential function are: 4. If f( ) a, a, 0 a, then the features of the eponential graph are: 5

16 Section C Activity 4: Which of the following equations represent eponential functions? Function f ( ) Is it an eponential Function? Yes/No Reason f( ) f ( ) ( ) f ( ) (3) f ( ) f( ) 3( ) f ( ) (0.9) 6

17 Problem Solving Questions on Eponential Functions Note: Etension Activities are required to strengthen students abilities in the following areas from the syllabus: Level Syllabus Page JCHL ( ) f a and f( ) a3, where a,. Page 3 LCFL ( ) f a and f( ) a3, where a,. Page 3 LCOL f( ) ab, where a, b,. Page 3 LCHL f( ) ab, where a, b,. Page 3. A cell divides itself into two every day. The number of cells C after D days is obtained from the function: C D (a) Draw a graph of the function for 0 D 6. (b) Find the number of cells after 5 days.. The value of a mobile phone M (in cents) after T years can be obtained from the following function: T M k,where k is a constant. (a) Draw a graph of the function for 0 T 6. (b) Find the value of k given that the value of the mobile phone after 3 years is 00. (c) Find the value of the phone after 7 years. 3. The number of bacteria B in a sample after starting an eperiment for m minutes is given by: B 50(3) 0.04m (a) Find the number of bacteria in the sample at the start of the eperiment. (b) Find the number of bacteria in the sample after starting the eperiment for 3 hours. 4. The graph of f( ) ka is shown: (a) Find the value of k and a. (b) Hence find the value of f ( ) when 8. 7

18 Q5. Olive finds that the number of bacteria in a sample doubles every 5 hours. Originally there are 8 bacteria in the sample. Complete the table below: Number of hours (hrs.) Number of bacteria (b) (a) Epress b in terms of h. (b) Find the number of bacteria in the sample after 3 hours. (c) How many hours later will the number of bacteria be more than 00. Q6. When a microwave oven is turned on for minutes the relationship between the temperature C inside the oven is given by C ( ) (0.9) where 0. (a) Find the value of C (0). (b) Eplain the meaning of C (0). (c) Can the temperature inside the microwave oven reach 550C? Answers h 5 Q (b) 3,768 cells, Q (b) 800 (c) 6.5, Q3 (a) 50 (b) 36,0 bacteria, Q4 (a) k, a 3 (b) 3,, Q5 (b) b 8() (c) 48 bacteria (d) 8. hrs., Q6 (a) 0 o C (c) No 8

19 WS07.03 Looking at Eponential Functions in Another Way ph Compound interest Absorption of drugs in the body Google Ranking of pages? Sound levels Astronomical calculations Simplifying graphical analysis Comparing earthquake sizes 9

20 Activity Making the Most of a Euro Invest for year at 00% compound Interest. Investigate the change in the final value, if the annual interest rate of 00% is compounded over smaller and smaller time intervals. The interest rate i per compounding period is calculated by dividing the annual rate of 00% by the number of compounding periods per year. Compounding period t Final value, F P( i), where i is the interest rate for a given compounding period and t is the number of compounding periods per year. Calculate F correct to 8 decimal places. Yearly i Every 6 mths. i Every 3 mths. i Every mth. i Every week. i Every day. i Every hour. i Every minute. i F F ( ).5 Every second. i What if the compounding period was millisecond 9 (0 s)? What difference would it make? 3 6 (0 s), microsecond (0 s) or nanosecond Will F ever reach 3? How about.8? 0

21 Activity Further Eploration of Eponential Functions. How long will it take for a sum of money to double if invested at 0% compound interest rate compounded annually?. 500 mg of a medicine enters a patient s blood stream at noon and decays eponentially at a rate of 5% per hour. (i) Write an equation to epress the amount remaining in the patient s blood stream at after t hours. (ii) Find the time when only 5 mg of the original amount of medicine remains active. 3. y 0 0 (a) Describe the type of sequence formed by the numbers in the first column (b) Describe the type of sequence formed by the numbers in the second and third columns (c) Using the table, and your knowledge of indices, carry out the following operations of multiplication and division in the second sequence, linking the answer to numbers in the first sequence. (i) (ii) (iii) y y 3 3 8, , , ,777, , ,554, , ,08, , ,7, , ,435, , ,870,9 0 0,048, ,073,74,84,097,5 3 3,47,483,648 49, ,94,967,96

22 Using Different Bases , ,96 4 0, ,04 5 3,5 5 7, , , , ,656 6,000,000 7,87 7 6, ,5 7 79, ,000, , , ,65 8,679, ,000, , ,44 9,953,5 9 0,077,696 9,000,000, ,049 0,048, ,765, ,466,76 0 0,000,000,000 log 3(3 ) log 4(4 ) log 5(5 ) log 6(6 ) log 0(0 ) , ,96 4 0, ,04 5 3,5 5 7, , , , ,656 6,000,000 6,87 7 6, ,5 7 79, ,000, , , ,65 8,679, ,000, , ,44 9,953,5 9 0,077,696 9,000,000, ,049 0,048, ,765, ,466,76 0 0,000,000,000 0 Formula and Tables Page Séana agus logartaim Indices and logarithms p q pq a a a log y log log y a y log y a a p q a pq p q pq q a a log a 0 a log a 0 a a p p a q q p q a q p q a a a ab a b p p p p a a b b p p p a a a log log log y a a a qlog log a log a a y log ( a ) a a loga log b log log a a a b

23 Switching between Eponential and logarithmic forms of Equations 4. Evaluate the epression below forming an equation Write the equivalent eponential form of the equation formed from the first column 5. Eponential form of an equation log log Write the equivalent log form of the equation in the previous column log () 0 0 log 8 log e e log ( 4) b 6. Cut out the equilateral triangles and using the rules for logs, line up sides having equivalent epressions to make a heagon. Note: This jigsaw was created using Tarsia, a free software package for creating numerous types of jigsaw and matching eercises. 3

24 7A. Evaluate each of the following: log (3 ) log (64) log (3) log () log (7 9) log (43) log (7) log (9) log (5 5) log (5) log (5) log (5) log 6 log (64) log (6) log 6 6 What pattern seems to hold? Can you write a rule for log b ( y) in terms of log b ( ) and log b ( y )? 7B. Evaluate each of the following: log (64 4) log (6) log (64) log (4) log (6 6) log (36) log (6) log (6) log 0(00 000) log 0 0 log (00) log (000) 0 0 log (5 5) log () log ( 5) log (5) What pattern seems to hold? Can you write a rule for log b in terms of log b and log b y? y 7C. Evaluate each of the following: 3 log (8) log (5) log (56) log (6) 4 log (0) log (0,000) 0 0 log (7) log (79) 3 3 3log (8) 3 log (56) 4log (0) 4 0 log (7) 3 What pattern seems to hold? Can you write a rule for log in terms of log? b y b 4

25 8. From discrete to continuous f ( ) Use the graph to estimate: (i) log (6) (ii) log (39.4) 5

26 9. Drawing the graph of the inverse of f ( ) f ( ) (a) Fill in the table below and hence draw the graph of g( ) f ( ). f ( ) (, y) g( ) log ( ) (, y) 4, 4 4, 0 0,, 4, , 8, 4 (b) What is the relationship between f( ) and g( ) log ( )? (c) Eplain why the relation g( ) log ( ), is a function. 6

27 (d) For g( ) log ( ) : (i) Identify the base of g( ) log ( ). (ii) What is varying for the function g( ) log ( ). (iii) What is constant for the function g( ) log ( ). (e) For g( ) log ( ) : (i) What is the domain? (ii) What is the range? (f) In relation to the graph of g( ) log ( ) : (i) (ii) (iii) Is it a straight line? Is y increasing or decreasing as increases? Describe how the rate of change varies as increases. (g) For g( ) log ( ) : (i) Where does the graph cross the ais? (ii) What happens to the output as decreases between 0 and? (iii) What is the y intercept of the graph of g( ) log ( ). (iv) What is the relationship between the y ais and the graph of g( ) log ( ). 7

28 0. Using the graph below, sketch and label the graphs of the following functions: h( ) 0, k( ) log ( ), l( ) e and m( ) ln( ). 0 f ( ) g( ) log ( ). True or false discussion: Equation Equivalent eponential form T/F Correct equation (if false) log 8 4 log log 5 log 0 log log 64 log 4 log 6 log3 4 8 log8 log6 log 4 log 8 log 5. Give possible numbers or variables for the blanks in the equations below. (i) log 3 (vi) log log 3 (ii) log (vii) log log (viii) log log log (iii) log log 7 (i) log log (iv) log log (v) log log log 8

29 3. (i) What is the relationship betweenlog b( ) and log ( )? (ii) Cut out the following and match each graph to its function. b j( ) log ( ) 0 w( ) log ( ) 0 n( ) ln( ) p( ) log ( ) r( ) log ( ) u( ) log ( ) e 9

30 4. LCFL 0 Paper Q7 (b) A scientist is growing bacteria in a dish. The number of bacteria starts at and doubles every hour. (i) Complete the table below to show the number of bacteria over the net five hours Time in hours Number of bacteria (in thousands) 0 (ii) Draw a graph to show the number of bacteria over the five hours. (iii) Use your graph to estimate the number of bacteria in the dish after hours. (iv) The scientist is growing the bacteria in order to do an eperiment. She needs at least bacteria in the dish to do the eperiment. She started growing the bacteria at 0:00 in the morning. At what time is the dish of bacteria ready for the eperiment? 5. M and M are the magnitudes of two earthquakes on the Richter magnitude scale. If A and A are their corresponding amplitudes, measured at equal distances from the A earthquakes, then M M log0 A. An earthquake rated 6.3 on the Richter magnitude scale in Iran on 6 th Dec 003 killed 40,000 people. The earthquake on Banda Aceh on 6 th Dec 004 was rated at 9. on the Richter magnitude scale. How many times greater in amplitude of ground motion was the earthquake in Banda Aceh compared to the earthquake on Iran? 6. ph is a measure of the acidity. The ph is defined as follows: ph H H log 0[ ] where [ ] is the hydrogen ion concentration in an aqueous solution. o A ph of 7 is considered neutral. For bases: ph 7. For acids: ph 7. [at 5 C] 5 A substance has [ H ].7 0 moles/litre. Determine the ph and classify the substance as an acid or a base. 7. Sound levels are measured in decibels (db). If we are comparing two sound levels, B and B measured in db, I B B 0log0 I The sound levels at The Who concert in 976 at a distance of 46 m in front of the speakers was measured as B = 0 db. What is the ratio of the intensity I of the band sound at that spot compared to a jackhammer operating at a sound level of B = 9 db at the same spot ,000 was deposited at the beginning of January 005 into an account earning 7% compound interest annually. When will the investment be worth 00,000? Verify and justify that the following two formulae give the same answer. F 40,000(.07) F 40,000e Comment on that answer in each case. t t 30

31 WS07.04 Activity : A to G Transformations of the Quadratic. Fill in the tables for the activity you are completing.. On your white boards, using the same aes and scales, plot the graphs of the given functions for your activity. Label your graphs clearly. (i.e. all the graphs for activity A should be on the same graph. LOOK AT THE X AND Y COUPLES TO HELP YOU SCALE YOUR AXES ACTIVITY A ACTIVITY B f( ) (, y) f( ) (, y) g( ) (, y) g( ) (, y) p( ) 3 (, y) p( ) 3 (, y) k( ) 0.5 (, y) k( ) 0.5 (, y) Activity A By considering the graph of f( ) what effect does a have on g( ) af( ) a. Activity B By considering the graph of f( ) what effect does a have on g( ) af( ) a. 3

32 ACTIVITY C ACTIVITY D f( ) (, y) f( ) (, y) g( ) (, y) g( ) (, y) p( ) 3 (, y) p( ) 3 (, y) k( ) 4 (, y) k( ) 4 (, y) Activity C By considering the graph of f( ) what effect does c have on g( ) f( ) c c. Activity D By considering the graph of f( ) what effect does c have on g( ) f( ) c c. 3

33 ACTIVITY E ACTIVITY F f( ) (, y) f( ) (, y) g( ) ( ) (, y) g( ) ( ) (, y) p( ) ( 3) (, y) p( ) ( 3) (, y) k( ) ( 0.5) (, y) k( ) ( 0.5) (, y) Activity E By considering the graph of f( ) what effect does a have on g( ) f( a) ( a ). Activity F By considering the graph of f( ) what effect does a have on g( ) f( a) ( a ). 33

34 ACTIVITY G f( ) (, y) g( ) ( ) (, y) p( ) ( ) (, y) k( ) ( ) (, y) Activity G By considering the graph of f( ) what effect do a and c have on g( ) f( a) c ( a) c? 34

35 Activity : Review of Graphs of Functions Write down the function which represents the graphs shown on the slides. Graph f( ) f ( ) Function Local Maimum/Minimum 35

36 Activity 3: Different forms of the Quadratic y 4 5 (, y). Fill in the tables opposite y ( 5)( ) (, y) y ( ) 9 (, y) Plot the points and draw the graph for each of the functions in the table. 3. What do you notice about all the graphs and all of the three functions you have plotted in this activity? 4. What items of information from each of the functions can help us if sketching the graph of a function? 36