Exponents and Exponential Functions

Transcription

1 Chapter Eponents and Eponential Functions Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. Prerequisite Skills for the chapter Eponents and Eponential Functions. : eponent, ase. An epression that represents repeated multiplication of the same factor is called a power ,.,...,.,.. % %...% %...%.%.. % %.. Let e the input, or independent variale, and let e the output, or dependent variale. Notice that each output is more than the corresponding input. So, a rule for the function is. To graph the function, plot a point for each ordered pair. Draw a line through the points. Lesson. Appl Eponent Properties Involving Products Investigating Algera Activit for the lesson Appl Eponent Properties Involving Products Eplore Epression Epression as repeated multiplication ( ) ( ) () () [() ()] [() () ()] () ( ) Epression Numer of factors Simplified epression () () () Sample answer: The eponent of the simplified epression is equal to the sum of the eponents in the first column. Eplore Epression Epanded Epression ( ) ( ) ( ) [() ] () () () () (a ) (a ) (a ) (a ) Epression Epression as repeated multiplication ( ) ( ) ( ) [() ] ()()()()()()()() (a ) (a a a) (a a a) (a a a) Epression Numer of factors Simplified Epression ( ) [() ] () (a ) a Sample answer: The eponent of the simplied epression is equal to the product of the eponents in the first column. Draw Conclusions.. () () () (). m m m m. (). [() ] () (). c () c c. If a is a real numer and m and n are positive integers, then a m a n a m n.. If a is a real numer and m and n are positive integers, then (a m ) n a mn. Guided Practice for the lesson Appl Eponent Properties Involving Products... () () () ().. ( ). [() ] () (). (n ) n n. [(m ) ] (m ) (m ). ( ). (n) ( n) () n n. (m n) ( m n) m n m n. ( ) ( ). Aout, or,,, ees were studied in Idaho. Eercises for the lesson Appl Eponent Properties Involving Products Skill Practice. The order of magnitude of the quantit,, people is the power of nearest the quantit, or people.. When powers have the same ase, their product is the ase raised to the sum of the eponents... Algera Worked-Out Solution Ke

2 . +. ()() ()() () () ()() () () (n)(n) () + n + () + n +. ()()() () (). ( ) +. [() ) + + n + () + n ] () () [() ] () + () +.. (. ( + ) + n n. ( + ) +. The eponents were multiplied instead of added;. ( + ) + c + c + c c + c + c c c. (() + ) () +. B; ()() () (). +. D;. +. z + z + z z () z. a + a + a a a. ( ) ( ) + +. [( )] ( ) + ( ). [(d ) ] + (d ). + (d ). () () +. () ( + ). ( ) + + () +? +. (m )? + m () + m + a +? + + a a +m?? + + a +? + a a + m + m m a +? a +? m. d + (d ) d + + d + d + + d +?? + + d. people. ( )( ) d. ( ) ( ) () ( ) () () () (p ) (.p ) ( + p ) + (. + p ) ( + p) + (. + p) + () + (.) + p + p. ( z )(z) [() z + ]( z ) + + z z + + z p z. () p (a)? + a a. +? + z? + z + +? + z? + z + ( )?? + +. (z?) z.. (pq) + p + q pq?. () + +? Algera Worked-Out Solution Ke. (s)(r st)(rst) s + () + r + + s + t + () + r + + s + t + s + () + r + s + t + + r + s + t ()()() + s + r + t rst Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.

3 . Sample answer: (), +, ( ). (a + )n (a + ) + (a + ) (a + ), with a + (s) t s t. + + t (t) s t t s t t + as a factor n times. The associative and commutative properties of multiplication allow ou to rewrite this epression as a + a a with each factor appearing n times. This epression is equal to an + n. + st t st Prolem Solving ules cm ules in quart. quart cm. Step Numer of new ranches. ()() m grains of sand ( ft) grains of sand. ft Length of new ranch Numer of atoms,, atoms. ( ounces) atoms ounce. a. ( )(). ( nanometers)() nanometers. ( meters) + m. a. ()() ft. V ()()() () ft Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. c. The volume will e times as great ecause in the formula for volume, the radius is squared. So, V ()( + )() Lesson. Appl Eponent Properties Involving Quotients Guided Practice for the lesson Appl Eponent Properties Involving Quotients. (). () () () Luminosit of Canopus (watts). Luminosit of Sirius(watts) Canopus is aout times as luminous as Sirius. Eercises for the lesson Appl Eponent Properties Involving Quotients Skill Practice. (). () () () (). () () ().. (). (). a a. () () (). (). () () () ; ;. The length of the new ranch added at Step is ase raised to the difference of the eponents. ft.. When powers have the same ase, their quotient is the ()(). In the power, is the ase and is the eponent. ()(). a. Gold (ounces) Algera Worked-Out Solution Ke

4 () + m m n + () + m n m n. C; ( ) () + (). + () + + ( ). The quotient of powers propert was used incorrectl. The eponents should e sutracted, not added; z. z + z z z z (). D; () p p. q q j j. k k a a. (). () ().? (). () ()? () a a a. () + (c) c c c. d d (d ) d (a) a a a. ()??. + p? p p p? () ( ) () () () Algera Worked-Out Solution Ke c. d c d (c)? (d )? c d?c? d c d?? ( f )(g) f g. fg f g?? f g f g f g??? p? p?? + p p..? m n m + f g? Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. (m) m. + + m (n) m n

5 st st st (st). + + st st st st st st Solve the sstem st st + s t s t ( ) Sustitute revised Eqn into Eqn. + st + st s t st m n. m mn + n () (mn) (mn) + (m) (n) mn mn + m n m n mn Prolem Solving. a. mn mn m n mn + () ( ) +. + () () Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. +. Sample answer: ; ; m a a n Given a a an am m m Identit propert of multiplication. Multipl fractions. an m a Quotient of powers propert. an m new squares Equation +. times more squares. $ per capita GDP m. km + m km m m/sec sec min h min da h r das sec ø, r sec... ø. times less right., times greater. a. gigates. megates c. Multipl the numer of tes. Quiz for the lessons Appl Eponent Properties Involving Products and Appl Eponent Properties Involving Quotients. () (). () () () +. side length. ()() () () ( + ) + steps. Sustitute into Eqn. +. Let m < n. Revised Equation Equation.. + Algera Worked-Out Solution Ke

6 . ( ). (). ( ). ( ) ( ).. (). w v w v.. ( ) l acres, aout pounds Lesson. Define and Use Zero and Negative Eponents Investigating Algera Activit for the lesson Define and Use Zero and Negative Eponents Eplore Eponent, n Value of n Eponent, n Value of n Each time the eponent is decreased, the value is divided the ase. Eponent, n Power, n Draw Conclusions. n n n Eponent, n Power, n. a. Power, n Eponent Power, n Eponent Guided Practice for the lesson Define and Use Zero and Negative Eponents.. ().. ().. ( ) ()(). () () () () ()... () The order of magnitude of the mass of a proton is gram. Eercises for the lesson Define and Use Zero and Negative Eponents Skill Practice. Sample answer: I would use the product of powers propert ecause the epression simplifies to. B the definition of zero eponent,.. Sample answer: The definition of negative eponents is defined onl for nonzero ases... () ().. () ().. ().... Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. Algera Worked-Out Solution Ke

7 . undefined. undefined. + (). D; ( + + ) () () a. False, a () a; a. +. (). a a. False, a ; a a a a. and, an is greater than, and an, an. When n, an a and an a, so an an. When n., n,, so an is greater than, an is etween and, and an. an... For, a,, when n,, n., so an is a whole numer, an is a fraction, and an. an. When n, an a and an a, so an an. When n., n,, so an is a fraction, an is a whole numer, and an, an.. ( ) (). + + Prolem Solving () grains of salt. is, not. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved (g). (g) g g. () grains of rice g. kg g kg g () times larger. (h) (h) ()h h. mn mn s r. r s.... (z) z (z). (d) + d d (d) (). () g liter liter. red lood cells +. () + () + (). (fg).. a. For a., when n,, n., so an is etween. () countereample: not. a a. True, a. () + not countereample: () The entire sample contained aout red lood cells.. No; mass of giant fan palm + g g kg g kg, not kg.. a. Numer of folds Fraction of original area n. where n is the numer of folds. (). D; Algera Worked-Out Solution Ke

8 (). a. t ) ( ( ) / /. / + / + / ()/ / + / / / + /. sec (/) (/) cm + sec. sec cm cm cm / (/) sec. a. I.Pd.P() P (.)() (.) P. (/) /. watts P. I.(.)d. ()/()/ ()(/) (/) d ()/ () c. The intensit is divided. / l l. a. BTU Ï + Your stereo uses BTUs in ear.. ()/()/()/ ()(/) (/) (/) l. BTU BTU (). For,,, /, /; for, / /; + / for., /. /. Samples: + /.. pound of sulfur dioide is added to the air. Etension for the lesson Define and Use Zero and Negative Eponents. / (/) + (/) (Ï ). / (/) + () (/) (Ï ).. / (/) + (/) (Ï ) / Ï. / + / (/) (/) / (/) (/) / /. Algera Worked-Out Solution Ke. times faster. a. V s in.. Power of a quotient propert.. a. Because V Ĭr, the order of magnitude of the volume of the droplet is ()()() cm.. r + Because V Ĭr, the order of magnitude of the c. Divide the volume of the raindrop the volume of the droplet. Mied Review of Prolem Solving for the lessons Appl Eponent Properties Involving Products, Appl Eponent Properties Involving Quotients, and Define and Use Zero and Negative Eponents volume of the raindrop is ()()() cm.. / (/) + () (/) (Ï ) ; / and /.. / / Ï / and () droplets Quotient of powers propert watts. + m Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.

9 Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.. a. in. in. in. in.. () in. c. () in.. a. Sample answer: How man milliseconds are in a gigasecond?. Sample answer: How man megaseconds are in a gigasecond? Lesson. Write and Graph Eponential Growth Functions Guided Practice for the lesson Write and Graph Eponential Growth Functions. The -values are multiplied for each increase of in, so a. When, a. So, a rule for the function is.. Domain: all real numers Range: all positive real numers. The graph of is a vertical shrink of the graph of.. The graph of is a vertical shrink with a reflection in the -ais of the graph of.. C,(.) t,(.) ø, In, the value of the car was aout $,.. a( r) t (.) (.) ø. You will have $. in ears. Eercises for the lesson Write and Graph Eponential Growth Functions Skill Practice. In the eponential growth model a( r) t, the quantit r is called the growth factor.. The eponential function a (where a > ) represents eponential growth when >.. is a vertical stretch of. The -values for are times the corresponding -values for.. The -values are multiplied for each increase of in, so a. When, a. So, a rule for the function is.. The -values are multiplied for each increase of in, so a. When, a. So, a rule for the function is.. The -values are multiplied for each increase of in, so a. When, a. So, a rule for the function is.. The -values are multiplied for each increase of in, so a. When, a. So, a rule for the function is.. Sample answer: If the function is linear, for each increase of in, the corresponding -values increase a set amount. If the function is eponential, for each increase of in, the corresponding -values are multiplied a set amount. Algera Worked-Out Solution Ke

10 .. Range: > Range: >. (.).... Range: > Range: >. (.)..... Range: > Range: >. Range: >. (.).... Range: >. (.)... Algera Worked-Out Solution Ke. Range: >. Range: > Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved..

11 . Range: > Range: > as a decimal. P a( r)t.(.).(.) ø. In the price of a pound of flour was aout $... + Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved..... The graph of + is a vertical stretch of the graph of.. + The graph of + is a vertical stretch with a reflection in the -ais of the graph of. The graph of + is a vertical shrink of the graph of. The graph of + is a vertical stretch with a reflection in the -ais of the graph of The graph of. + is a vertical stretch of the graph of. The graph of + is a vertical stretch of the graph of.. +. The error is that the percent increase was not written.. +. The graph of (.) + is a vertical shrink of the graph of The graph of + is a vertical shrink with a reflection in the -ais of the graph of. The graph of + is a vertical shrink of the graph of. Algera Worked-Out Solution Ke

12 . + Eponential: An increase in means multiplication in, so a +. When, a. So, +.. The graphs are the same. f () g() The graph of + is a vertical shrink with a reflection in the -ais of the graph of... + a. ()(.) $.. ()(.) $. The graph of + a( r)t. Prolem Solving is a vertical shrink with a reflection in the -ais of the graph of. c. ()(.) $. d. ()(.) $.. a. r %. a (million) c (.)t c (.)t, where c is the numer of computers (in millions) and t is the numer of ears since.. c (.)t (.). There will e aout,,, computers in use worldwide in a. r %. a,,.. The graph of. + is a vertical stretch with a reflection in the -ais of the graph of. g,,(.)t g,,(.)t, where g is the numer of gas grills and t is the numer of ears since.. g,,(.)t,,(.),,. Aout,, gas grills were shipped in.. a. Tree : A (.)t. a. A (.)t Tree : A (.)t. A (.)t. Intersection = Y= c ; ecause, must equal.. %;laate-c-l-t Sample answer: A growth rate of % would create a growth factor of, which would represent the tripling of the population ever ear.. Linear: Slope Point-slope form: ( ) Algera Worked-Out Solution Ke Tree Tree Intersection X=. Y=. In aout. ears the trees will have the same asal area.. Yes; Sample answer: For each increase of feet in length, the cost is multiplied. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.

13 . C; The eponential model is a etter approimation of actual U.S. population for the time period -. Time Blogs months, months,, months,, months,,. () r nt. A P (.) n $.. () r nt. A P (.) n. () r nt. A P n growth factor... (.) growth rate. $.. a. initial amount. million.... $.. Dail; in an account that is compounded dail, each da ou earn interest on oth the principal and the interest that was accrued on the previous das.. Use the intersect feature on our graphing calculator to determine that the graphs intersect when <.. So, the douling time is aout ears. To check, note that the calculator gives a value for. of., or aout. Domain: Range:. million. million Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. c. When, or in, the numer of Internet users worldwide was aout million. nan---t..(.) When, ø hertz.,,,,. a. growth per ear,, Prolem Solving Workshop for the lesson Write and Graph Eponential Growth Functions. a. (.)t (.)t. (.)() $. c. ; Using the fill down feature of the spreadsheet, ou can see that when t,.. So the intercit us fare was $. in.. The error is that the growth rate is written in place of the growth factor. The function should e (.)t.. a. T.(.)t T.(.)t,. A linear model is,,,.. The growth was,. people per ear.. Using the eponential regression feature on a graphing calculator,,,(.). The population growth was aout.% each ear. c. Linear model at :,,,.(),, Eponential model at :,,(.),,. Linear model at :,,,.(),, Eponential model at :,,(.),,. Algera Worked-Out Solution Ke

14 Months since Ma,, t Numer of transistors, T ( millions)... pieces is multiplied.. c. there are pieces of arn at stage.. a. Yes; For each increase of in the stage; the length of each piece is multiplied Aout. million transistors in a CPU were released the compan in Novemer.. a. V,(.)t V,(.)t.. a. Yes; For each increase of in the stage, the numer of c. units The length of each new piece of arn at stage units. is Guided Practice for the lesson Write and Graph Eponential Deca Functions. Yes; The -values are multipied for each increase of Years since, t Value, V (dollars), the form a +. When, a. So,,. in, so the tale represents an eponential function of. (.)...,,...., Domain: all real numers Range: all positive real numers When t, or in, the value of the home was aout $,. Lesson. Write and Graph Eponential Deca Functions Investigating Algera Activit for the lesson Write and Graph Eponential Deca Functions Stage Numer of pieces Length of each new piece Algera Worked-Out Solution Ke. + (.)... The graph of + (.) is a vertical stretch of the graph of (.). Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved..

15 .. (, ) (, ) Range: > The graph represents eponential deca. The -intercept is, so a, a. +. A function rule is (.). There will e aout. million acres left in. Eercises for the lesson Write and Graph Eponential Deca Functions Skill Practice. The deca factor in the eponential deca model a( r)t is r. Range: >. function falls from left to right while the graph of an eponential growth function rises from left to right. Range: >.. The -values are multiplied for each increase of in Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved., so the tale represents an eponential function of the form a +, when, a. So, +.. The -values are multiplied for each increase of in, so the tale represents an eponential function of the form a +. When, a. So, +.. The -values are multiplied for each increase of in, so the tale represents an eponential function of the form a +. When, a. Range: >., so the function is not eponential. Range: >. The -values are increased for each increase of in.... Range: >. (.) So, +.. Sample answer: The graph of an eponential deca. P (.)t (.) ø.. Range: >. (.).. Range: > Algera Worked-Out Solution Ke

16 . +. (.)... The graph of + is a vertical stretch of the graph of.. + Range: >. (.).... Range: > The graph of + is a vertical shrink of the graph of.. (.) Range: > The graph of + is a vertical shrink of the graph. (.) of Range: >..... The graph of. + is a vertical shrink of the graph of.. D; When, a The function is + (.).. + a vertical stretch of the graph of. Algera Worked-Out Solution Ke The graph of. + is a vertical stretch of the The graph of + is graph of. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. +

17 . + reflection in the -ais of the graph of. a vertical stretch with a. reflection in the -ais of the. graph of. + reflection in the -ais of the graph of.. A; (.) Let : (.) -intercept: (, ). C; (.) vertical stretch with a reflection in the -ais Let : (.) of the graph of. -intercept: (, ). +. B; (.) -intercept:, The graph of + is Let : (.).. + The graph of. + is a vertical stretch with a The graph of + is a Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. is a vertical shrink with a The graph of + is The graph of. +. a( r)t a vertical shrink with a initial amount a, reflection in the -ais of the deca rate r. deca factor r.. graph of.,(.)t. D; deca factor deca rate. +. r r.. The error is that the deca rate was placed where the deca factor should e. The equation should e: a( r)t,(.)t,(.)t The graph of + is a. The graph represents eponential deca ecause it falls vertical shrink with a reflection in the -ais of the graph of. a. from left to right. The -intercept is, so a r. The function rule is (.) Algera Worked-Out Solution Ke

18 . The graph represents eponential deca ecause it falls from left to right. The -intercept is, so a. Prolem Solving. Let represent the value of the cell phone and t represent a the numer of ears since purchase.. (.)t (.)t (.). The value of the cell phone after ears is $. The function rule is (.).. a. Initial amount a,. The graph represents eponential growth ecause it rises Deca rate r. from left to right. The -intercept is, so a. Deca Factor r.. a. Let B represent the numer of ats and t represent the. numer of ears since.,(.)t,(.) ø,. The function rule is (.). There were aout, ats in.. a. m() is a vertical shrink of f ().. No; in, the oat will e worth. n() is a vertical stretch with a reflection in the -ais of f(). Selling the oat for $ will e selling the oat for less than what it s worth. is multiplied for each increase of in, so. When, a. A function rule is.. (, ), (, ). a. Initial amount Deca factor. Rounds completed Teams remaining is multiplied for each increase of in, so. a c. a a A function rule is..,,, is multiplied for each increase of in, so. a A function rule is.. To find t, divide the numer of das,, the half-life,. Then A (.)t (.).. ; t The amount left after das is. grams.. The graphs are the same graph. + + g () Algera Worked-Out Solution Ke. a. Deca factor. c. The distance etween the nut and the first fret is d F.(.) G (.) ø. inches. a + () After round there will e teams left in the tournament. Deca Rate.. f (). d.(.) ø. inches a The distance etween the th and th frets is d.(.) ø. inches.. Let represent the remaining alance and t represent the numer of months since purchase. (.)t (.)t After the rd month, the remaining alance is (.).. If the student us the computer without paing interest, he should pa the remaining alance $. after the rd month. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. c. p() is a vertical translation of unit up of f().. (, ),, B (.) $..

19 . a. st athlete (.) (.) nd athlete.(.).(.) (.).(.). + c. At aout, ø.. So the first athlete will e aout ears old when her maimal ogen consumption is equal to. liters per minute. Quiz for the lessons Define and Use Zero and Negative Eponents, Write and Graph Eponential Growth Functions and Write and Graph Eponential Deca Functions. () + () () + + (). + () () + Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. () (). () () + () + ( ) () + +. (.) () +. (z) + z z (z)... () Algera Worked-Out Solution Ke

20 Position, Term,. (.)..... The difference of consecutive terms are: a a a a a a The common difference is, so the sequence is arithmetic.. Let represent the value of the coin and t represent the Position, Term, numer of ears since purchase. t t (.) (.) When t : (.). The value of the coin after ears is $... The ratios of consecutive terms are: a a a,, a a a The common ratio is, so the sequence is geometric. Position, Term,. The ratios of consecutive terms are: a a a,, a a a The common ratio is, so the sequence is geometric. Position, Term,. The difference of consecutive terms are: a a a a a a The common difference is, so the sequence is arithmetic. Algera Worked-Out Solution Ke. The ratios etween consecutive terms are: a a a,, a a a The common ratio is, so the sequence is geometric. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. First Etension for the lesson Write and Graph Eponential Deca Functions

21 Position, Term,. The differences etween consecutive terms are: a a Second Etension for the lesson Write and Graph Eponential Deca Functions.,,,, a a.,,,, a a The common difference is, so the sequence is arithmetic. Position, Term,.,,,,.,,,, laate-km-alb-a laate-km-alb-a.,,,,.,,,,. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. laate-km-alb-a a. r a a. r a an + n an ()n a + a (), a laate-km-alb-a.,,,,.,,,,. r a laate-km-alb-a laate-km-alb-a n an + a + a. a a a a.,,,, a laate-km-alb-a laate-km-alb-a The ratios of consecutive terms are constant so the series is geometric where r. an + n n Position, Term, laate-km-alb-a Algera Worked-Out Solution Ke

22 . a,. Sample answer: Let represent the value of the house an an and t represent the numer of quarters since. a(.)t a(.)t an.an When t,. a,, a(.) an an. a, an an. a, an an. a, an an. a, an.an,.a,. a The value of the house can e modeled,(.)t. A house with a value of $, at the end of would have a value of aout $, at the end of.. a. Let represent the amount of mone in the account and t represent the numer of ears since the $ was deposited,. a, an an. a., an an.. a, a, an an an ;,. a, a, an (an )(an );,,. a, a, a, an an an an ; a( r)t (.)t (.)t.. a, an an ; acteria. This is the eplicit rule with n sustituted for n. Mied Review of Prolem Solving for the lessons Write and Graph Eponential Growth Functions and Write and Graph Eponential Deca Functions. a. Let represent the amount of medication in patient s loodstream (in milligrams) and let t represent the time since the medication was taken. The amount of medication is halved ever hours, so (.)t /.. (.)/ (.).... a, a, an an an ;, an. a, a, an an ;,.. a, a, an an an ;,,.,.,,,,,,,,,,,, c. No, after ears the musician will onl have $... a. The graph rises from left to right, so it represents eponential growth.. The -intercept is,, so a,. a,,. A function rule is,(.). c.,(.),(.),. The usiness is worth $,. after ears. After hours, the patient s loodstream will have. mg of the medication.. a. The -intercept is,, so a,. a,,. A function rule is,(.).. The deca factor, r, for the truck is.. So, r.. r The deca rate is., or %.. Deca factor r. r r. Algera Worked-Out Solution Ke Chapter Review for the chapter Eponents and Eponential Functions. The function (.) t is an eponential deca function, and the ase. is called the deca factor.. Sample answer: A tale represents a linear function if the output values change the addition of the same numer. A tale represents an eponential function if the output values change the multiplication of the same numer.. The function (.) represents eponential deca ecause the ase,., is less than.. The function (.) represents eponential growth ecause the ase,., is greater than. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.. a,

23 . The function (.) represents eponential growth. (.) ecause the ase,., is greater than.. +. () () (). z + z + z z ()... z. ( ) + Domain: all real numers. F () G () + (). F ( ) G + ( ) Range: all positive real numers ( ).. ( + ) (). ( ) kilograms. (.) (). () () () +. m m. n n. () Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.. + p p p p... Domain: all real numers Range: all positive real numers. r r r r r s s s r r s s r Domain: all real numers. The order of magnitude of the mean personal income in Montana in was $. Range: all positive real numers kilogram. + () nanogram kilogram femtogram femtogram nanogram. The graph of + is a vertical stretch with a reflection in the -ais of the graph of.. The graph represents eponential growth. The -intercept is, so a. a A function rule is. + Domain: all real numers Range: all positive real numers Algera Worked-Out Solution Ke

24 . The graph represents eponential deca. The -intercept is, so a.. (r ) + r r + r r r a r r. A function rule is.. Let V represent the value of the car (in dollars) and t represent the numer of ears since the car was purchased. V a( r) t,(.) t,(.) t Domain: all real numers Range: all positive real numers Susitute for t. V,(.)t,(.) ø. The approimate value of the car in ears is $.. Chapter Test for the chapter Eponents and Eponential Functions. ( + ) ()() () () +... () + The graph of. is a. + + vertical shrink of the graph of.. F G () + () (). () () (). t + t t t s s. t t. tes frames min. + + frame min hr. ( p) p p. (). + z z z z There are aout tes of data in hour of an animated film.. ( ) +. c a. (). a. Let represent the earl salar and t represent the numer of ears since accepting the jo. c c a + (),(.) t a,(.)t a.,(.)t,(.),. d + d d d (d). + + d d dc c c c c. + ( ) + () ()() + Algera Worked-Out Solution Ke The emploee s salar after ears is $,.. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.

25 () ( ) a. P (.) a Initial amount: Deca rate: + Deca factor: r. r. (u) (u) u u z z (u ) (z ) u. u.u u z u z u z + c. At a meters, the atmospheric pressure is aout ( ) ( ) ( ). ( ) ( ) a a. + a + a + a a a (a). atmosphere. Etra Practice for the chapter Eponents and Eponential Functions. +. () + () () (). () (). () +. [()] () + (). ( + ) + ( ). + ( ) m + m m m. n + n + n n n. ( ) +. ( Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved. ). () () + +. (d ) + d + (d ) + d + d + d d. (s) (s ) () + (s) + + (s) + s + + s s.. () (c) (c) c c t. + t t t t t p p. ( ) + ( ). ( ) + ( ) a a a a + c. c. (d e) (d e) + (d ) + (e) d e e d. t. u tu. (z) + ()z (z) + () + z z g () g g (e)g. (e) e +e +e e e g ,,. Algera Worked-Out Solution Ke

26 ..., (. ). ( + ).. (. )(. ) (. +.) ( + ). (. ) ( + )..... ( ) ( + ) Algera Worked-Out Solution Ke. Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.

27 (.) Copright Houghton Mifflin Harcourt Pulishing Compan. All rights reserved.. + (.) The -value are multipled for each increase of in, so the tale represents an eponential in the form a with. The value when is, so a. The tale represents the eponential function +. Algera Worked-Out Solution Ke