Exponential Functions
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1 WS07.0 Eponentil Functions Aim: To study the properties of eponentil functions nd lern the fetures of their grphs Section A Activity : The Eponentil Function, f ( ).. For f ( ) : The bse of f ( ) is (ii) The eponent of f ( ) is Wht is vrying in the function f ( )? Wht is constnt in the function f ( )?. For f ( ) : Wht re the possible inputs i.e. vlues for (the domin)? Nturl numbers Irrtionl numbers Integers Rtionl numbers 3. Set up tble of vlues nd drw the grph of f ( ) X y f( ) Describe the grph of f ( ) : 6 Rel numbers on your whitebord: Is it stright line? (ii) Is y f( ) incresing or decresing s increses? From the tble bove, find the verge rte of chnge over different intervls. For emple from to nd to 3. Wht do you notice? Describe how the curvture/rte of chnge is chnging. 4
2 5. For f ( ) : Wht re the possible outputs (rnge) for f ( ). (ii) Is it possible to hve negtive outputs? Eplin why? Wht hppens to the output s decreses? Is n output of 0 possible? Why do you think this is? (v) Wht re the implictions of this for the -intercept of the grph? (vi) Wht is the y-intercept of the grph of f ( )? 5
3 Section A Activity : The Eponentil Function, g ( ) 3.. For g ( ) 3 : The bse of g ( ) 3 is (ii) The eponent of g ( ) 3 is Wht is vrying in the function g ( ) 3? Wht is constnt in the function g ( ) 3?. For g ( ) 3 : Wht re the possible inputs i.e. vlues for (the domin)? Nturl numbers Irrtionl numbers Integers Rtionl numbers 3. Set up tble of vlues nd drw the grph of g ( ) 3 X 3 y g( ) In reltion to the grph of g ( ) 3 : Rel numbers on your whitebord: Is it stright line? (ii) Is y g( ) incresing or decresing s increses? From the tble bove, find the verge rte of chnge over different intervls. For emple from to nd to 3. Wht do you notice? Describe how the curvture/rte of chnge is chnging. 6
4 5. For g ( ) 3 : Wht re the possible outputs (rnge) for g ( ) 3. (ii) Is it possible to hve negtive outputs? Eplin why? Wht hppens to the output s decreses? Is n output of 0 possible? Why do you think this is? (v) Wht re the implictions of this for the -intercept of the grph? (vi) Wht is the y-intercept of the grph of g ( ) 3? 7
5 Section A Activity 3: Compre the grph of f ( ) with the grph of g ( ) 3.. How re they similr nd how do they differ?. Consider the reltions (, y), y, y nd (, ),, 3 Are they functions? Eplin. y y y. 3. Wht nme do you think is given to this type of function nd why do you think it is given this nme? Section A Activity 4: Understnd the chrcteristics of f( ),.. Wht is the domin of f( ),?. In reltion to the grph of f( ),. Is it stright line? (ii) Is y f( ) incresing or decresing s increses? (v) Does it hve mimum vlue? Does it hve minimum vlue? Describe how its curvture/rte of chnge is chnging. 3. Wht is the rnge of f( ),? 4. Wht is the intercept of the grph f( ),? 5. Wht is the y intercept of the grph f( ),? 8
6 Section B Activity : The Eponentil Function, f ( ).. For (ii) f ( ) : The bse of The eponent of f ( ) is f ( ) Wht is vrying in the function f ( )? Wht is constnt in the function f ( )?. For f ( ) : Wht re the possible inputs i.e. vlues for (the domin)? Nturl numbers Irrtionl numbers Integers Rtionl numbers is Rel numbers 3. Set up tble of vlues nd drw the grph of f ( ) on your whitebord: X y f( )
7 4. In reltion to the grph of f ( ) : Is it stright line? (ii) Is y f( ) incresing or decresing s increses? From the tble bove, find the verge rte of chnge over different intervls. For emple from to nd to 3. Wht do you notice? Describe how the curvture/rte of chnge is chnging. 5. For f ( ) : Wht re the possible outputs (rnge) for f ( ). (ii) Is it possible to hve negtive outputs? Eplin why? Wht hppens to the output s decreses? Is n output of 0 possible? Why do you think this is? (v) Wht re the implictions of this for the -intercept of the grph? (vi) Wht is the y-intercept of the grph of f ( )? 0
8 Section B Activity : The Eponentil Function, g ( ). 3. For (ii) g ( ) 3 : The bse of The eponent of g ( ) 3 is g ( ) 3 Wht is vrying in the function g ( ) 3? Wht is constnt in the function g ( ) 3?. For g ( ) 3 : Wht re the possible inputs i.e. vlues for (the domin)? Nturl numbers Irrtionl numbers Integers Rtionl numbers is Rel numbers 3. Set up tble of vlues nd drw the grph of g ( ) 3 below on your whitebord: X y g( )
9 4. In reltion to the grph of g ( ) : 3 (ii) Is it stright line? Is y incresing or decresing s increses? From the tble bove, find the verge rte of chnge over different intervls. For emple from to nd to 3. Wht do you notice? Describe how the curvture/rte of chnge is chnging. 5. For g ( ) : 3 Wht re the possible outputs (rnge) for g ( ). 3 (ii) Is it possible to hve negtive outputs? Eplin why? Wht hppens to the output s decreses? Is n output of 0 possible? Why do you think this is? (v) Wht re the implictions of this for the -intercept of the grph? (vi) Wht is the y-intercept of the grph of g ( ) 3?
10 Section B Activity 3: Compre the grph of f ( ) with the grph of g ( ). 3. How re they the sme nd how do they differ?. Consider the reltions (, y), y, y nd (, y), y, y 3. Are they functions? Eplin. 3. Wht nme do you think we give to this type of function nd why do you think it is given this nme? Section B Activity 4: Understnd the chrcteristics of f( ), 0.. Wht is the domin of f( ), 0? In reltion to the grph of. f( ), 0. Is it stright line? (ii) Is y f( ) incresing or decresing s increses? Does it hve mimum vlue? Does it hve minimum vlue? Describe how its curvture/rte of chnge is chnging. 3. Wht is the rnge of f( ), 0? 4. Wht is the intercept of the grph f( ), 0? 5. Wht is the y intercept of the grph f( ), 0? 3
11 Note: For ll the following, you should ssume tht the domin is. Section C Activity : Compre the grph of f( ) with the grph of f( ).. How re the grphs similr?. How re the grphs different? 3. Rewrite k f( ) in the form f( ). 4. Wht trnsformtion mps the grph of f( ) onto the grph of f( )? Section C Activity : Compre the grph of g( ) 3 with the grph of g( ). 3. How re the grphs similr?. How do the grphs differ? k 3. Rewrite g( ) in the form g( ) Wht trnsformtion mps the grph of g( ) 3 onto the grph of g( )? 3 4
12 Section C Activity 3: Now I see.... If f( ),,, then the properties of the eponentil function re:. If f( ),,, then the fetures of the eponentil grph re: 3. If f( ),, 0, then the properties of the eponentil function re: 4. If f( ),, 0, then the fetures of the eponentil grph re: 5
13 Section C Activity 4: Which of the following equtions represent eponentil functions? Function f ( ) Is it n eponentil Function? Yes/No Reson f( ) f ( ) ( ) f ( ) (3) f ( ) f( ) 3( ) f ( ) (0.9) 6
14 Problem Solving Questions on Eponentil Functions Note: Etension Activities re required to strengthen students bilities in the following res from the syllbus: Level Syllbus Pge JCHL ( ) f nd f( ) 3, where,. Pge 3 LCFL ( ) f nd f( ) 3, where,. Pge 3 LCOL f( ) b, where, b,. Pge 3 LCHL f( ) b, where, b,. Pge 3. A cell divides itself into two every dy. The number of cells C fter D dys is obtined from the function: C D () Drw grph of the function for 0 D 6. (b) Find the number of cells fter 5 dys.. The vlue of mobile phone M (in cents) fter T yers cn be obtined from the following function: T M k,where k is constnt. () Drw grph of the function for 0 T 6. (b) Find the vlue of k given tht the vlue of the mobile phone fter 3 yers is 00. (c) Find the vlue of the phone fter 7 yers. 3. The number of bcteri B in smple fter strting n eperiment for m minutes is given by: B 50(3) 0.04m () Find the number of bcteri in the smple t the strt of the eperiment. (b) Find the number of bcteri in the smple fter strting the eperiment for 3 hours. 4. The grph of f( ) k is shown: () Find the vlue of k nd. (b) Hence find the vlue of f ( ) when 8. 7
15 Q5. Olive finds tht the number of bcteri in smple doubles every 5 hours. Originlly there re 8 bcteri in the smple. Complete the tble below: Number of hours (hrs.) Number of bcteri (b) () Epress b in terms of h. (b) Find the number of bcteri in the smple fter 3 hours. (c) How mny hours lter will the number of bcteri be more thn 00. Q6. When microwve oven is turned on for minutes the reltionship between the temperture C inside the oven is given by C ( ) (0.9) where 0. () Find the vlue of C (0). (b) Eplin the mening of C (0). (c) Cn the temperture inside the microwve oven rech 550C? Answers h 5 Q (b) 3,768 cells, Q (b) 800 (c) 6.5, Q3 () 50 (b) 36,0 bcteri, Q4 () k, 3 (b) 3,, Q5 (b) b 8() (c) 48 bcteri (d) 8. hrs., Q6 () 0 o C (c) No 8
16 WS07.03 Looking t Eponentil Functions in Another Wy ph Compound interest Absorption of drugs in the body Google Rnking of pges? Sound levels Astronomicl clcultions Simplifying grphicl nlysis Compring erthquke sizes 9
17 Activity Mking the Most of Euro Invest for yer t 00% compound Interest. Investigte the chnge in the finl vlue, if the nnul interest rte of 00% is compounded over smller nd smller time intervls. The interest rte i per compounding period is clculted by dividing the nnul rte of 00% by the number of compounding periods per yer. Compounding period t Finl vlue, F P( i), where i is the interest rte for given compounding period nd t is the number of compounding periods per yer. Clculte F correct to 8 deciml plces. Yerly i Every 6 mths. i Every 3 mths. i Every mth. i Every week. i Every dy. i Every hour. i Every minute. i F F ( ).5 Every second. i Wht if the compounding period ws millisecond 9 (0 s)? Wht difference would it mke? 3 6 (0 s), microsecond (0 s) or nnosecond Will F ever rech 3? How bout.8? 0
18 Activity Further Eplortion of Eponentil Functions. How long will it tke for sum of money to double if invested t 0% compound interest rte compounded nnully?. 500 mg of medicine enters ptient s blood strem t noon nd decys eponentilly t rte of 5% per hour. Write n eqution to epress the mount remining in the ptient s blood strem t fter t hours. (ii) Find the time when only 5 mg of the originl mount of medicine remins ctive. 3. y 0 0 () 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 nd third columns (c) Using the tble, nd your knowledge of indices, crry out the following opertions of multipliction nd division in the second sequence, linking the nswer to numbers in the first sequence (ii) 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
19 Using Different Bses , ,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 3(3 ) 4(4 ) 5(5 ) 6(6 ) 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 Formul nd Tbles Pge Sén gus rtim Indices nd rithms p q pq y y y y p q pq p q pq q 0 0 p p q q p q q p q b b p p p p b b p p p y q y ( ) b b
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