3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate

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1 1 5.1 and 5. Eponenial Funcions Form I: Y Pa, a 1, a > 0 P is he y-inercep. (0, P) When a > 1: a = growh facor = 1 + growh rae The equaion can be wrien as The larger a is, he seeper he graph is. Y P( 1 r) When 0 < a < 1: a = decay facor = 1-decay rae The equaion can be wrien as Y P( 1 r) The closer a is o zero, he seeper he graph. Horizonal Asympoe a y = 0. **This can change wih ransformaions. Domain: All real numbers. Range: (0, ) **This can change wih ransformaions. Eample: Evaluae 3 g( ) a 4 a) = 1 b) Does g() model eponenial growh or decay? Find he rae.

2 To graph eponenial Funcions: Mehod I: Make a able of values. E: Graph y 4 3. Idenify wiher he equaion models eponenial growh or decay. Find he rae.

3 3 Mehod II: Transformaions. Y Pa c d (Form ) Reflecions of Y Pa Y Pa 1 P a Reflec over he y-ais. Y Y Pa Reflec over he -ais. Y Pa 1 P a Reflec over he -ais and he y-ais Horizonal Shifs Verical Shifs Pay aenion o moving he asympoe!

4 4 Eample: Graph h ( ) Eample: Wrie he equaion of he graph below.

5 5 In 5. Y Pa, le base a =.7188 = e. (Form 3) Naural Eponenial Funcion: c Y Pe. Transformaions: Y Pe d. Eample: Graph 1 h( ) e 3. Concep: The concenraion (in milligrams) of drug in he human body as he iniial amoun of 100mg dissipaes over ime (in hours) is graphed below. a) Esimae when half of he drug will be gone. Half Life: The amoun of ime i akes for half a quaniy o decay. b) Esimae when wo half-lives will occur. Doubling Time: The amoun of ime required for a quaniy o double in size.

6 6 Form 4: Y Pa n n Y P () for doubling ime; n = doubling inerval. 1 n n Y P P(.5) P() n for half life; n = half life inerval. Commen: The unis on n and mus mach! Eample: A breed of mouse was inroduced ono a small island wih an iniial populaion of 30 mice. a) Scieniss esimae ha he mouse populaion doubles every year. Find a funcion ha models he number of mice afer years. b) Scieniss esimae ha he mouse populaion halves every hree years. Find a funcion ha models he number of mice afer years.

7 7 Form 5: A( ) P 1 r n n A = fuure value P = principal (presen value) r = ineres rae as a decimal (APR) n = compounding inerval = ime in years Eample: $500 is invesed a an ineres rae of 3.75% per year, compounded quarerly, find he value of he invesmen afer: a) 10 years b) 10 monhs c) 10 days Eample: Find he presen value of $10,000 if ineres is paid a a rae of 9% per year, compounded semiannually, for 3 years.

8 8 Growh Facor/Growh Rae: r Consider A( ) P 1 P( a) P(1 r) n n APR = APY = Eample: Find he APY for $500 invesed earning 3.75% ineres per year, compounded quarerly.

9 9 Form 6: r Y Pe. r < 0, eponenial decay r e = decay facor r > 0, eponenial growh r e = growh facor Eample: A sky diver jumps from a heigh above he ground. The air resisance is proporional o he diver s velociy, and he consan of proporionaliy is 0.. I can be shown ha he downward velociy 0. of he sky diver a ime is V( ) 80(1 e ), where is in seconds and V() is in fee per second. a) Find he iniial velociy. b) Graph he velociy, and use his o find he erminal velociy.

10 Eponenial Form Log Form y a log a y Common log: Base y log10 y Naural log: Base e y e ln e y Cancellaion Properies 1. log a a e) log a a ln e e) lne. log a a e) ln e 3. ln(1)=0 The -inercep is (1, 0) unless here are ransformaions. log (1) 0 a Eample: Wrie log in eponenial form. Eample: Wrie in log form.

11 11 Eample: Evaluae each saemen. a) log9 81 b) 1 ln e Eample: Find. a) log 16 4 b) log 5 0.5

12 1 Poin (a, b) on Inverses Y a becomes (b, a) on y log a. Y log. Eample: Skech Y 3. Use his o graph 3

13 13 Reflecions Domain of f() = log ( ) b g consiss of all -values such ha g() > 0. Eample: Find he domain of Y 4 log ( 8 4 ).

14 14 Eample: Le f() = log5 and g() =. Find he domain of ( f g)( ).

15 15 Transformaions of Y log b( a c) d a > 1 epands he graph 0< a < 1 conracs he graph (, y) -> (, ay) Visual: (-c) shifs he graph righ on he -ais. (+c) shifs he graph lef on he -ais. (,y) -> ( c, y) +d shifs he graph up on he y-ais. -d shifs he graph down on he y-ais. (,y) -> (, y d) Eample: Graph Y log 5( 1) 1.

16 Change of base formula log b log log a a b To change from base b o base 10: log log b log b To change from base b o base e: ln log b ln b Use his on he calculaor! Eample: Evaluae log 4 15 on he calculaor.

17 17 Rules of Logs: 1. log b( AB) log b( A) log b( B) A. log b( ) log b( A) log b( B) B p 3. log ( A ) p log ( A ) b b Eample: Epand 10 log ( 1) Eample: Combine log( ) [log log( 6) ]. 3

18 Mehods o solve Equaions Type I: Variable in a log 1. Conrac he log. Rewrie he log in erms of eponens 3. Simplify and solve for he variable. Eample: Solve he equaion log ( ). Eample: Find he inverse of f() = log 5 ( 1). Type II: Variable in one eponen. 1. Isolae he base and eponen on one side of he equaion. Take he log of boh sides of he equaion. 3. Simplify and solve for he variable. Eample: Solve each equaion. a) e b) e 4 4e 1 0

19 19 Eample: Find he inverse of g ( ). Eample: Find he ime required for an invesmen of $5,000 o grow o $8,000 a an ineres rae of 7.5%, compounded quarerly. Type III: Two of he same bases, each wih an eponen. 1. Rewrie he equaion in he form. Se M = N 3. Solve for he variable. b M b N Eample: Solve he equaion Type IV: Two differen bases, each wih an eponen. 5 1 Eample: Solve he equaion 7.

20 0 Type V: One variable is normal and he oher variable is conained in a log or eponen. 1. Se he equaion = 0.. Solve graphically. Eample: Solve each equaion. a) 1 3 b) log( 1)

21 1 5.6 Recall Form 6: r Y Pe. r < 0, eponenial decay r e = decay facor r > 0, eponenial growh r e = growh facor We re going o sar applying his o word problems, yeah!! ;> Eample: The populaion of a cerain species of fish has growh rae of 1.% per year. I is esimaed ha he fish populaion in 000 was 1 million. a) Wrie an equaion using he eponenial model b) Esimae he populaion in 005. c) In wha year will he populaion reach 15 million? P ) r ( Pe, where is years afer 000. Commen: You could have been presened wih he same siuaion as in he above eample, bu given a graphical descripion insead of a verbal descripion.

22 Form 7: Eponenial growh/decay model r Y Pe ; r ln n Eample: The half life of cesium-137 is 30 years. Suppose we have a 10gram sample. a) Model he daa using m ) r ( Pe. b) How long will i ake for grams of cesium-137 o remain?

23 3 Eample: If 50 mg of radioacive elemen decays o 00 mg in 48 hours, find he half-life. Eample: Afer 3 days, a sample of radon- decayed o 58% of is original amoun. a) Find he half-life. b) How long does i ake o reach % of he original amoun?

24 4.1 and. Types of Symmery 1. X-Ais Symmery: Replace y by (-y).. Y-Ais Symmery: Replace by (-). 3. Origin Symmery: Replace (, y) by (-, -y). Eample: Deermine he ype of symmery shown by he equaion, if any. Y

25 5 Sandard Form of a Circle: ( r h) ( y k). Cener a (h, k). Radius r. Eample: Find he cener, he radius, and graph y y. Eample: Wrie he equaion of a circle wih a cener a (-1, 5) and passing hrough he poin (-4, -6). Eample: Wrie he equaion of a circle when he endpoins P(-1, 1) and Q(5, 9) are on he diameer of he circle.

26 6 Eample: Skech he region bound by he se { (, y) 1 y 9} Eample: Find he area of he region ha lies ouside he circle y 4y 1 0. y 4 bu inside he circle

27 7 1. Visual: Two Scenarios:

28 8 Eample: Find he verices, he foci, he lengh of he minor ais, and he lengh of he major ais. Graph on he calculaor and by hand. 4 5y 100 Eample: Wrie he equaion for he scenario below: Foci (0, 3) and Verices (0, 5)

29 9 Commen: In he equaion 1, le a = b: a y b Eccenriciy (e) Measures he srech of he ellipse. e c a measure o he focus measure o he vere 0 < e < 1 e 0, e 1, Eample: Wrie he equaion of an ellipse when he eccenriciy is he lengh of he major ais is 4. 3 he foci are on he y-ais, and

30 Think of he ellipse wih fied poins. Wha if he second poin could slide along a fied line? Pick a poin P such ha Parabola Se of all poins in a plane equidisan from a fied poin F (Focus) and a fied line l (direcri) ha also lies in he plane.

31 31 Sandard Form of a Parabola wih a: I. Verical Ais 4 py p = disance from he focus o he vere p > 0, CCU p < 0, CCD Small p indicaes ha he vere is close o he focus and we ge a parabola. Small p indicaes ha he vere is far from he focus and we ge a parabola.

32 3 Sandard Form of a Parabola wih a: I. Horizonal Ais y 4 p or on calculaor: Laus recum: Line running hrough he focus wih endpoins on he parabola, perpendicular o he ais. Widh of he parabola = focal diameer = magniude of he laus recum = 4p Eample: Find he focus, direcri, and focal diameer. Graph by hand and on he calculaor. a) y b) 7y 0

33 33 Eample: Wrie he equaion of he parabola given: a) Vere a he origin and direcri y = 6. b) Focal diameer = 8 and focus on he negaive y-ais. c) Eample: In a suspension bridge he shape of he suspension cables is parabolic. The bridge has owers 600 meers apar and he lowes poin of he suspension cables is 150 meers below he op of he owers. Find he equaion of he parabolic par of he cable. Eample: A reflecor for a saellie dish is parabolic in cross secion, wih receiver a he focus F. The reflecor is 1 foo deep and 0 fee wide from rim o rim. How far is he receiver from he vere of he parabolic reflecor?

34 Concep: Hyperbola wih a I. Horizonal Transverse Ais a: Disance from he cener o he vere. b: Disance perpendicular o he ransverse ais. c: disance from he cener o he focus. c a b.

35 35 Seps o graph: 1. Draw he reference recangle.. Draw he asympoes. 3. The hyperbola hugs he asympoes and passes hrough he verices. General Equaion: ransverse ais conjugae ais a b 1 Horizonal Transverse Ais: a y b 1 II. Verical Transverse Ais

36 36 Eample: Graph each equaion. y a) b) 9 16 y 1 Eample: Wrie an equaion given he informaion. a) verices ( 0, 6) asympoe a y 1 3 b) Foci a ( 0, 1) lengh of he ransverse ais = 1

37 Shifed Conics -h moves he graph righ on he -ais +h moves he graph lef on he -ais y k moves he graph up on he y-ais y+k moves he graph down on he y-ais. Circle ( h) ( y k) r cener a (h, k) Ellipse ( h) ( y k) a b 1 cener a (h, k) foci ( h c, k ) verices ( h a, k ) ( 3) Eample: Graph ( y 3) Label he cener, foci, and verices. Sae he lengh of he major and minor aes.

38 38 Parabola ( h) 4 p( y k ) ( y k) 4 p( h ) Vere (h, k) Vere (h, k) focus (h+p, k) focus (h, k+p) direcri y = k p direcri = h p Eample: Graph ( y 5) 6 1. Sae he vere, focus, and direcri.

39 39 Hyperbola ( h) ( y k) a b 1 ( y k) ( h) a b 1 Cener (h, k) Cener (h, k) Verices (h a, k) Verices (h, k a) foci (h c, k) foci (h, k c) asympoes ( y k) b a ( h) asympoes ( y k) ( a b h) ( y 1) Eample: Graph ( 3) 1. 5 Sae he cener, foci, verices, and he equaion of he asympoes.

40 40 Eample: Wrie an equaion for he graph below.

41 41 Shifed Conics A Cy D Ey F 0, A and C are no boh zero. Circle: A = C Ellipse: A and C have he same sign. Hyperbola: A and C have opposie signs. Commen: Degenerae conics can resul. Degenerae conics are lines, inersecing lines, and poins. I is also possible ha he conic does no eis. Eample: Complee he square and deermine wheher he equaion represens a circle, ellipse, hyperbola, parabola, or degenerae conic. a) y b) y y 1 c) 16 4( y )