Review Packet for Math 623 Final Exam

Transcription

1 Review Packet for Math 623 Fial Exam Attached is review material for the 2007 Math 623 Fial Exam, which covers chapters 1-4 i the UCSMP Fuctios, Statistics, ad Trigoometry textbook. The material for each chapter icludes a chapter overview, a list of thigs a studet should be able to do, sample problems o chapter material, ad more detailed otes about the chapter.

2 Chapter 1 Overview of Makig Sese of Data This chapter presets differet statistical measures for data sets with just oe variable (ot relatios betwee two or more variables). There are measures of cetral tedecy (mea, media, mode) ad measures of spread (rage, iterquartile rage, stadard deviatio, ad variace). The chapter also reviews a variety of methods for presetig data, icludig histograms, frequecy tables, stem-ad-leaf plots, ad box-ad-whisker plots. You should be able to: 1. Apply the capture-recapture method 2. Iterpret ad create pie charts, tables, scatter plots, bar graphs, ad stemplots 3. Calculate average rates of chage 4. Calculate measures of ceter: mea, media, mode 5. Calculate measures of spread: rage, iterquartile rage, stadard deviatio, variace 6. Iterpret ad create 5-umber summaries ad box-ad-whisker plots for datasets. 7. Iterpret ad create frequecy tables ad histograms 8. Use sigma otatio 1

3 Problems Problem 1: A biologist captures ad tags 80 frogs from aroud a etwork of pods. He the releases them ad captures a secod batch of birds, 90 this time. Of the 90, 12 have tags. What is a estimate of the frog populatio? Problem 2 Fid the average rate of chage of the populatio i Bosto betwee 1900 ad Is the rate of chage less tha or greater tha the rate of chage from 1850 to 1970 (Data is i sectio 1-2/1-3 below. A-Block Stem F-Block Problem 3: Fid the miimum, 1 st quartile, media, 3 rd quartile ad maximum scores ad the rage for A-block. Problem 4: The followig are the yearly icomes (i $ thousads) of employees at a small compay: 40, 33, 55, 22, 55, 38, 41, 51, 45, 280. Calculate the mea, media, ad mode of the set. Compare the mea ad media. If they are sigificatly differet, explai why. Number Height i of Players Iches Problem 5: The table to the left provides iformatio about the height of the players o the varsity basketball team at a local high school. First, calculate the total umber of players. Use that to calculate the mea, media, ad mode. Problem 6 A teacher has give each of the 20 studets i her math class a ID umber from 1 to 20. If x 1 represets the score o a test of studet 1 ad x 2 represets the score of studet 2, etc., use sigma otatio to represet the mea score of the studets i this class. Problem 7 Do the 5-umber summary for this set of data ad draw a boxplot: 12, 13, 15, 16, 16, 19, 22, 23, 25, 29, 33, 39, 40, 44 Problem 8 2

4 Fid the stadard deviatio ad variace of this set of data: 5,12, 22, 4, 7. If we replace 22, 4, ad 5 with 10, 10, ad 11, what will happe to the mea? What will happe to the stadard deviatio? Chapter 1 Review Material 1.1 Collectig Data The set of all idividuals or objects that you wat to study is called the populatio. The characteristic that you wat to study (height, weight, IQ, etc.) is called a variable. It may ot be possible to study the etire populatio. I that case, you study oly a part of the whole populatio, a subset called a sample. Whe samples are take radomly from the etire populatio that meas that every member of the populatio has a equal chace of beig chose. Capture-recapture method: this is a method of collectig a sample of objects, aimals, or people from a populatio. For birds, for example, certai birds, radomly selected, are captured ad tagged. They are the released back ito their habitat. A secod group of birds is radomly selected. By coutig the umber of tagged birds i the secod group, you ca estimate the populatio you studyig: umber of tagged aimals i sample umber of tagged aimals i populatio = where P is the umber of aimals i sample P populatio you are studyig. For example, say you capture ad tag 50 birds ad the release them back ito the wild. A short period later, you capture aother group of 60 birds. If 10 of those 60 are tagged, you ca estimate the populatio of birds overall is 10/60 = 50/P. Crossmultiply to get 10P = 3000 or P = 300. That is your estimate of the bird populatio. 1-2 ad 1-3: Displays of Data You should be able to read ad iterpret tables ad a variety of graphs. Lie, Bar, ad Scatter Plot Graphs Oe type of data set is called time-series data, which shows the chage i a variable over time. Time-series data ca be show i a variety of formats, icludig: 3

5 Populatio (i thousads) Populatio (i thousad ) Populatio (i thousad ) A B C Year Populatio (i thousads) Above is a table showig the populatio of Bosto from 1850 to 1988 (time-series data) ad three meas of represetig the data: A) bar graph B) lie graph ad C) scatter plot. Both B ad C are coordiate graphs where poits ad the coected (B) or ot coected (A). These graphs also eable us to calculate average rates of chage betwee ay two poits o the graph. For example, the average rate of chage of the populatio of Bosto betwee 1850 ad 1970 is calculated by fidig the chage i the y-variable (populatio) ad dividig by the 641! chage i the x-variable (years) which, i this case, gives us = = 3.37 which, i 1970! this case, meas a average chage of 3.37 thousad or per year. This is equal to the slope of the lie segmet that would coect those two poits o the graph.. Stem-ad-Leaf Plots (or Stemplots) A stemplot takes a set of data ad orgaizes it by the umber i the tes digit. For example, cosider the followig set of test scores: 31, 57, 95, 92, 84, 73, 81, 71, 62, 80, 69, 59, 74, 73. Stems Leaves The stemplot puts the tes digit for each score i the Stems colum, arraged from lowest to highest, ad puts each oes digit i the leaves colum. Notice that the score 73 is repeated so it appears twice i the stemplot. Leaves are arraged from left to right lowest to highest. If there were a score of 105, it would have a stem of 10. With a stemplot, it is easy to fid the maximum score (95), the miimum score (31) ad the rage (95-31 = 64). Notice as well how easy it is to see that the score of 31 is ot clustered with the other scores ad therefore represets a outlier (scores which are very differet from the rest). 4

6 A-Block Stems F-Block A back-to-back stemplot has the stem i the ceter ad two sets of leaves, each arraged with the lowest umber earest the stem ad the umbers risig as you move away from the stem. The back-to-back plot eables you to compare two sets of data. To the left is a plot comparig two classes: 1-4 Measures of Ceter Measures of Ceter or Measures of Cetral Tedecy are meat to represet typical scores. Mea: This is the arithmetic average of the data. Fid the sum of the data ad divide by the umber of items i the data set. For the F-block class i the stemplot above, the average would be = Media: If you arrage all the data poits i umerical order ad take the middle poit, the value of that data poit is the media. If there are a odd umber of data poits, you will have oe middle score which will be the media. If you have a eve umber of data poits, you will have two middle poits. Just take the average of those two poits to fid the media For the F-block class above, there are 14 scores, so there are two scores i the middle, 73 ad 73. The media is the average of these two scores, 73. For the A-block class, there are 15 scores, so there is oe middle score, 71, ad that is the media. Mode: Less commoly used to represet the cetral tedecy of a set of data, the mode is the value that occurs the most frequetly i the set. For example, for the F-block class, the mode is 73 because that is the most commoly occurrig score (it is the oly that occurs twice). For the A-block class, there are two modes: 88 ad 70, each of which occur twice. Meas ad Sigma Notatio If x 1, x 2, x 3, x 4, x is a set of umbers, the you ca represet the sum of that set of umbers as! xi. Sice the mea of a set of umbers is the sum divided by the umber of items i the x= 1 5

7 set, you ca use the sigma otatio to represet the mea as well. The mea of the set referred to! x= 1 earlier is Problem 6 x i A teacher has give each of the 20 studets i her math class a ID umber from 1 to 20. If x 1 represets the score o a test of studet 1 ad x 2 represets the score of studet 2, etc., use sigma otatio to represet the mea score of the studets i this class. 1-5 Quartiles, Percetiles, ad Box Plots We will ow start lookig at measures that represet the spread or distributio of data rather tha the cetral tedecy. Rage, which we have see earlier, measures the differece betwee the highest ad the lowest values i a set, givig a picture of how spread out the data items are. Quartiles Quartiles are the values that divide a ordered set ito 4 subsets of equal size. Fid the quartiles as follows: 2 d quartile the media of the data 1 st quartile the media divides the data set ito two parts: umbers below (but ot icludig) the media ad umbers above (but ot icludig) the media. The 1 st quartile is the media of the set of umbers below the media. 3 rd quartile media of the set of umbers above the media of the etire set. Example: If the ordered (put i order) set is the followig: 2,4, 5, 7, 9, 11, 13, 14, 16, 19, 22, 22, 25, 30, 40, what are the 1 st, 2 d, ad 3 rd quartiles? 2 d quartile: 14 2 d quartile is the media. Sice there are 15 items i the set, 7 fall ito the lower set ad 7 ito the upper set ad the media is the 8 th umber or 16 for this set. 1 st quartile: 7 1 st quartile is the media of the 7 umbers below 14 (the media) 3 rd quartile: 22 3 rd quartile is the media of the 7 umbers above 14 (the media) The iterquartile rage or IQR is the differece betwee the 1 st ad 3 rd quartiles. I this case, that would be 22-7 or 15. The IQR tells you the upper ad lower values for the middle 50 percet of the data. The 3 quartiles plus the miimum ad maximum of the set together give you the 5-umber summary of the data. Box Plot or Box-ad-Whiskers Plot This is a visual represetatio of the 5-umber summary. For the above set, the plot would look like this: 6

8 Miimum Box Whiskers st Qrt. Media 3 rd Qrt Maximum The box refers to the box that starts at the 1 st quartile value (8) ad exteds to the 3 rd quartile value (22). The legth of the box is 22-8 = 14 which is the IQR. 50 percet of the scores are betwee 8 ad 22. The vertical bar all the way to the left represets the miimum score (2) ad the bar all the way to the right represets the maximum score (40). The horizotal lies o either side of the box represet the whiskers. The legth of the whole figure, from 2 to 40 is 38 ad that represets the rage. The vertical lie iside the box is the media. Problem 7 Do the 5-umber summary for this set of data ad draw a boxplot: 12, 13, 15, 16, 16, 19, 22, 23, 25, 29, 33, 39, 40, Histograms 1-8 Variace ad Stadard Deviatio Stadard Deviatio ad Variace are two additioal measures of spread. Both relate to the mea of a set of data ad both are based o the deviatio, or differece of each data value from the mea. Here is the algorithm for calculatig the variace ad stadard deviatio for a data set with umbers: 1. Calculate the mea of the data 2. Fid the deviatio (differece) of each value from the mea 3. Square each deviatio ad add the squares. 4. Divide the sum of the squared deviatios (step 3) by -1 (ot by ). This is the variace. 5. Fid the square root of the variace. This is the stadard deviatio. Example: Data set cotais elemets 3, 4, 7, 10, 12. The mea is ( )/5 = 7.2 Number Deviatio from the Mea Deviatio Squared = =

9 = = = = = = = = Sum = 60.8 Divide 60.8 by (5-1) = 60.8/4 = 15.2 = Variace of the set The stadard deviatio =!15.2 = We ca also also write the variace ad the stadard deviatio usig sigma otatio: for the set x..., 1, x2, x, we say that the mea = x. The variace = s The stadard deviatio = s = " i= 1 ( x! x) i! 1 2 = " 2 i= 1 ( x! x) i! 1 Two sets may have the same mea but substatially differet stadard deviatios. That meas that the set with the higher stadard deviatio is more spread out. 2 8

10 Chapter 2: Overview of Fuctios ad Models This chapter itroduces bivariate data, which mas that it ivolves relatioships betwee two variables. Frequetly, oe of the variables is time as you look at how oe variable (populatio, average icome, etc.) chage over time. The chapter looks at several types of mathematical models liear ad quadratic. These provide mathematical descriptios of real situatios, usually ivolvig some simplificatio. These models ca be used to make estimatios ad predictios. Studets should be able to: 1. Recogize whether relatioships are fuctios; use fuctio otatio 2. Idetify domai ad rage of fuctios, icludig absolute value, quadratic, liear, step 3. Iterpret ad develop liear models (y=mx + b) to fit real world situatios 4. Create scatter plots, fid ceter of gravity, ad lie of best fit 5. Uderstad limitatios of data ad iterpolatio ad extrapolatio 6. Work with step fuctios 7. Iterpret correlatio measures (r ad r 2 ) 8. Solve ad graph quadratic equatios 9. Fid quadratic models to fit data 9

11 Problem 1 Which of the above graphs represet fuctios? Problem 2 f(x) = x ad g(x) = 12 x. a) Evaluate f(-3) + f(3). b) Evaluate g(-4)*f( 5) c) Fid the domai ad rage of f(x) ad g(x). Problem 3 Create a liear model to represet this situatio: there is a liear relatioship betwee how log it takes to react to pai ad the distace of the ijury from the brai. The reactio time to pai 100 cetimeters from the brai is 6.8 millisecods. The reactio time to pai 170 cetimeters from the brai is 7.5 millisecods. Problem 4 Plot the followig data poits, fid the ceter of gravity ad fid a lie of best fit that goes through the ceter of gravity x y Problem 5 For the quadratic model y = 3x 2 2x 5 a) Fid the y-itercept b) Fid the x-itercepts c) Fid the vertex d) Fid y whe x = 4 10

12 2-1 The Laguage of Fuctios. This sectio itroduces the idea of a fuctio as a particular type of relatioship betwee two sets of ordered pairs. Fuctio: Ordered pairs i which each first elemet is paired with exactly oe secod elemet. A secod defiitio is a correspodece betwee two sets A ad B i which each elemet of A correspods to exactly oe elemet of B. A fuctio assigs each x value to oe y value. If a relatio. assigs two differet y values to a value of x, the it is ot a fuctio. You ca tell from the graph of a relatio if it is a fuctio. For it to be a fuctio, it must pass the vertical lie test. This meas that the curve caot itersect a vertical lie i more tha oe place (see problem 1) The set of first elemets if the Domai of the fuctio ad the set of secod elemets is the Rage. Geerally, you may assume that the domai of a fuctio is the set of all real umbers for which the fuctio is defied, uless some other domai is explicitly stated. I may cotexts the secod umber i the ordered pair depeds o the first umber i the pair. The first umber the is called the idepedet variable ad the secod is the depedet variable. Fuctio otatio: the symbol f(x) is read f of x ad is also called Euler otatio. The otatio is particularly useful whe there are multiple fuctios i a situatio ad each fuctio ca the be assiged its ow ame. The umber or variable i the paretheses is called the argumet. The fuctio f(x) = 2x 7 meas that you take the argumet, multiply it by 2 ad the subtract 7. Example 1 f(x)=2x 3 ad g(x) = x a) Evaluate f(3) + g(2): f(3) meas that you replace x with 3 i the expressio 2x 3. This gives you 2(3) 3 = 3. g(2) = = 6. f(3) + g(2) = = 9. b) Fid the domai ad rage of f(x) ad g(x). For f(x), the domai is the set of real umbers because the fuctio is defied for all values of x. The rage will also be the set of real umbers. For g(x), the domai is agai the set of all real umbers. The rage, however, will be restricted to y>=2 because x 2 is always zero or positive ad whe you add 2, the result will always be 2 or greater. 11

13 For the fial: be able to idetify the domai ad rage of a fuctio: There are several issues that come up with domais: -- The fuctio may ot be defied for every value of x. Examples iclude: f(x) = 1/x where the fuctio is ot defied whe x = 0 g(x) =!x where the fuctio is ot defied whe x<0 h(x) = 1/(x-3) where the fuctio is similar to f(x) but i this case is ot defied for x=3. -- The fuctio may have real world costraits that limit the domai. j(x) = the total cost if you are buyig x CDs at $10 per CD. The domai is restricted to positive itegers because you caot buy a fractio of a CD or a egative umber of CDs. There are also issues that arise with respect to the rage: -- The rage may be limited because the fuctio oly produces certai values: f(x) = floor(x) meas that you roud dow to the ext lowest iteger. The rage is limited to itegers. g(x) = x 2 the rage is limited to oegative umbers because x 2 ca ot be egative. h(x) = 1/x the rage excludes 0 because 1/x ca ever equal Liear Models Be able to develop a equatio to describe a situatio: A carpet istaller charges $100 plus $2 per square foot: C = total cost; x = square feet of carpetig: C = 2x Be able to take the equatio from above ad solve give a value of x or give a value of C -- If there are 150 sq feet of carpetig, what is the cost: C = 2(150)+100 = $ If the total cost of the istallatio is $624, how may sq. feet of carpetig were istalled? 624 = 2x = 100! 524 = 2x! 262 = x Be able to fid a equatio give two poits or a poit ad the slope: fid equatio of lie that passes through (2,5) ad (4,12). -- Slope = (12-5)/(4-2) = 7/2 = Poit Slope form: y-5 = 3.5(x-2) or y-12 = 3.5(x-4). Either is the equatio of the lie -- Slope Itercept form: y=mx+b where m is the slope ad b is the y-itercept. Covertig y-5=3.5(x 2)! y 5 = 3.5x 7! y = 3.5x 2. Be able to explai the meaig of the slope ad the y-itercept i a particular word problem: i the carpeter above, the slope is the charge per square foot of carpet; the y-itercept ($100) is the setup fee charged by the carpet istaller eve before ay carpet is laid dow. Uderstad the differece betwee iterpolatio ad extrapolatio: (p. 89) 12

14 2-3 Lie of Best Fit Be able to calculate ceter of gravity of a set of coordiates (calculate the mea of the x coordiates ad the mea of the y coordiates). Those two calculatios will give you the coordiates of the ceter of gravity Kow that a lie of best fit must pass through the ceter of gravity i a liear model. Be able to use that lie of best fit to iterpolate ad to extrapolate (predict withi the rage of date you have ad outside the rage of your data). Be able to explai the limitatios of extrapolatio ad provide a example Kow that the sum of the squares of the deviatios from the mea of a give set is a measure of the spread of the data i the set. Be able to calculate this (pp ). 2-4 Step Fuctios Kow how to work with step fuctios: how may CDs ca you purchase with $54 if each CD costs $7.50 Kow the meaigs of cotiuous, discrete, roudig up, ceilig fuctio 2-5 Correlatio Correlatio coefficiet measures the stregth of the liear relatioship betwee two variables. Perfect correlatio = 1 meas that all poits fall o the same lie. Positive correlatio: depedet variable goes up as idepedet variable goes up. Both fall at the same time. 2-6 Quadratic Models Memorize quadratic formula: if ax 2 + bx + c = 0, the! ±! 2a 2 b b 4ac If b 2-4ac<0, x has o real value. If b 2-4ac=0, the x = -b/2a. Be able to use Newto s equatio for the height of a object after it has bee throw ito the air (p. 113) Be able to use a quadratic formula that describes a situatio to predict. Be able to iterpret r Fidig Quadratic Models Be able to take 3 poits o a lie or o a quadratic curve ad derive the equatio (p ) 13

15 Chapter 3 Overview of Trasformatios of Fuctios ad Data This chapter focuses o trasformatios of graphs ad of sets of data. Two types of trasformatio are explored: traslatios (addig a costat to oe or both variables) ad scale chages (multiplyig oe or both variables by a costat). The chapter also reviews the symmetry of fuctios with respect to the y-axis or to the origi. Fially, Chapter 3 looks at how to combie fuctios ad to fid the iverses of fuctios. Studets should be able to: 1. Sketch quickly the paret graphs of y = x 2, y=x 3, y=1/x, y= 1/x 2, y= x, y = greatest iteger i x (step fuctio), ad y =!x 2. Sketch traslatios of paret graphs, describe the traslatio usig traslatio otatio, ad fid the equatio of the traslated curve 3. Describe the impact o measures of cetral tedecy ad measures of spread of addig a costat to each item i a set of data. 4. Sketch scales chages of paret graphs, describe the scale chage usig S(x,y) otatio, ad fid the equatio of the trasformed curve 5. Describe the impact o measures of cetral tedecy ad measures of spread of multiplyig each item i a data set by a costat. 6. Recogize ad prove symmetry of fuctios (showig that a fuctio is odd, eve, or either). 7. Evaluate compositios of fuctios 8. Fid iverses of fuctios ad idetify if two fuctios are iverses of each other. 14

16 5 Problem 1 y 5 y 5 y x x x a b c Sketch the graphs of a) y=x 2 b) y = 1/x c) y = x Problem 2 Fid the equatios of the images of fuctios a, b, ad c above uder the followig traslatios. Also, sketch each image above o the same graph as the image s paret fuctio. For curve a): T(x,y) = (x-1,y+1) For curve b): T(x.y) = (x 4, y) For curve c): T(x,y) = (x + 2, y 2) Problem 3 The salaries at Alpha Dog Food Ic., have the followig statistical measures: Mea = $45,000; Media = $50,000; Rage = $100,000; Iterquartile Rage = $20,000; Stadard Deviatio = $11,000; Variace = $121,000. If everyoe at the compay is give a $1000 bous at the ed of the year, fid the effect of that bous o each of the statistical measure listed above. 5 Problem 4 Graph each paret fuctio uder the scale chage listed below the blak graph. y 5 y 5 y x x x

17 y = x 3 S(x,y) = (x/2, 2y) y = 1/x2 S(x,y) = (2x, -2y) y = x! " # $ S(x,y) = (2x,y/2) Problem 5 Usig the same data as i problem 3, assume that istead of a $1,000 bous, each perso i the compay gets a 2% raise. What would be the effect of that icrease of each of the statistical measures give? Problem 6 Idetify each of the followig fuctios as odd, eve, or either: a) y = x 3 2x b) y = (x-2) 2 + x c) y = 3x 4-3 Problem 7 f(x) = 3x 2 2 g(x) = 3 - x a) Fid f(g(4)) b) Fid g f(0) c) Show algebraically that f(g(x)) does ot equal g(f(x)). Problem 8 Fid the iverses of f(x) ad g(x) from problem 7. Are the iverses fuctios? 16

18 Chapter 3 Details Graph Traslatios ad Scale Chages Whe a parabola, the graph of y =x 2, is shifted 3 uits to the left, it create a image parabola where the x-coordiate of each poit is 3 greater tha the x-coordiate of the correspodig poit o the origial parabola. The ew parabola also has a equatio. The poit 4,16 is o the origial parabola. The image of that poit is 7,16. Whereas the poit 4,16 worked with the equatio y = x2 (because 16 = 4 2 ), the poit 7,16 does ot fit that equatio. Istead, y is ow the square ot of x, but of x-3 because y stayed the same ad is still the square of 4 but x is ow 7. This will be true of every poit o the origial parabola ad its image poit o the image parabola. Accordigly, the equatio of the ew parabola is y = (x-3) 2. If the shift of f(x) is to the right 3 uits, the equatio of the image fuctio will be y = f(x-3). If the shift is 3 to the left, the image fuctio equatio is y = f(x+3). Similarly, if there is a shift up of 2, the equatio of the image fuctio if y 2 = f(x), etc. This works the same with scale chages: If each x-coordiate is doubled, the image equatio is y= f(x/2), etc. See the attached page for a series of examples of traslatios ad scale chages of graphs. Traslatios ad Scale Chages of Data Whe the same costat is added to each item i a data set, that same costat should be added to all the measures of cetral tedecy of that dataset: mea, media, ad mode. If the mea of a set is 10 ad 5 is added to each member of the set, the the mea of the adjusted set is 15 (10 + 5). However, all measures of spread rage, iterquartile rage, stadard deviatio, ad variace, remai uchaged. Whe each item i the set is multiplied by the same costat, the you must also multiply the measures of cetral tedecy by that same costat. For example, if each item i a set with a mea of 11 ad a media of 12 is multiplied by 3, the the mea of the traslated data will be 3 x 11 = 33 ad the mea of the traslated data will be 3 x 12 = 36. Ulike with traslatios of data, scale chages of data also chage the measures of spread rage, iterquartile rage, ad stadard deviatio will all be multiplied by 3 i the example earlier i the paragraph. However, variace, which is the square of the stadard deviatio, will be icreased ot by 3 but by 3x3 or 9. 17

19 Examples: Traslatios ad Scale Chages o Graphs Paret Fuctio Trasformed Equatio Geometric Descriptio of the Trasformatio Trasformatio Formula (usig T(x,y) or S(x,y) y = x y = x + 3 Shifted Left 3 T(x,y) = (x 3, y) y = x y + 4 = x Shifted dow 4 T(x,y) = (x, y-4) y = x2 y = (x - 2) 2 Shifted Right 2 T(x,y) = (x+2,y) y = x 3 y + 2 = (x+3) 2 Shifted Left 3 ad Dow 2 T(x, y) = (x 3, y 2) y = x 1 y = x 1 +1 Shifted up 1 T(x,y) = (x, y+1) y = x y = x 3 Stretch vertically by a factor of 3 y = x 2 y = (2x) 2 Shrik horizotally by factor of 2 y = x y = 3 x y = x 1 y 1 = 2 2x Shrik horizotally by a factor of 3 Stretch vertically by a factor of 2, shrik horizotally by factor of 2 y = x 3 2(y+1)= 3(x 3) 3 vertically dow 1; shruk vertically by a factor of 2 ad Shift horizotally right 3 ad horizotally by factor of 3 S(x,y) = (x, 3y) & x # S(x, y) = $, y! % 2 "! x " S(x,y) = #, y $ % 3 & S(x,y) = (x/2, 2y) Odd ad Eve Fuctios Eve Fuctios: f(-x) = f(x). Graphically, this meas that the fuctio is symmetrical aroud the y-axis: if reflected across the y-axis, it reflects oto itself. Examples: Ay fuctio with oly eve expoets or absolute values with o horizotal shifts (y = x 2, y = x 4 x 2 + x, etc.) Odd Fuctios: f(-x) = -f(x). Graphically, this meas that the fuctio is symmetrical aroud the origi (0,0). It ca be rotated 180 aroud that poit ad come back oto itself. 18

20 Alteratively, the fuctio ca be reflected first across the y-axis ad the across the x-axis back oto itself. Examples: Ay fuctio with oly odd expoets with o horizotal shifts: y = x 3, y = x 5 x, etc. Compositio of Fuctios Whe there are two fuctios f(x) ad g(x) ad the outputs of oe fuctio become the iputs of the other, that is called a compositio of fuctios. For example, if f(x) = 2x 4 ad g(x) = 3x, the f(g(5)) meas that first g(5) is calculated, = 3 x 5 = 15. The 15 becomes the iput for f(x) ad f(15) = 26. Therefore, f(g(x)) = 26. This compositio ca also be writte f g(x) which is read f followig g of x. The other otatio, f(g(x)), Euler s otatio, is read f of g of x. Iverse Fuctios Two sets of ordered pairs are iverses of each other if the x coordiates of oe set are the y- coordiates of the other set ad vice versa. The graphs of the two sets of ordered pairs will be reflectios of each other across the lie y=x. To calculate the iverse of a fuctio, say y = 3x + 2, simply swap the x ad y values ad solve for y. This gives us x = 3y + 2. Subtract 2 from both sides to get x 2 = 3y. x! 2 Divide both sides to get y =. This is the iverse of y = 3x + 2. I this case, both are 3 fuctios, although that is ot always the case. Two fuctios, f(x) ad g(x), are iverse fuctios if ad oly if (f(g(x)) = x for all x i the domai of g ad g(f(x)) = x for all x i the domai of f. Whe f ad g are iverse fuctios, the f=g -1 19

21 Chapter 4: Overview of Power, Expoetial, ad Logarithmic Fuctios I previous chapters, we examied liear ad quadratic fuctios. I this chapter, we will explore th root fuctios (square root, cube root, etc.); ratioal power fuctios ad properties of expoets; ad expoetial fuctios of the type y=ab x, where b>1 meas growth ad b<1 meas decay. We will also itroduce the fuctios that are the iverse of the expoetial fuctios: logarithmic fuctios. The chapter also itroduces some applicatios of these equatio types. You should be able to: 1. Evaluate expressios with fractioal expoets ad radicals. 2. Recogize ad use the Product of Powers Property, Power of a Product Property, Quotiet of Powers Property, ad Power of a Quotiet Property, Zero Expoet Theorem, ad Negative Expoet Theorem (see sectio 4-2 below) 3. Graph expoet fuctios ad idetify the domai ad rage. 4. Fid a expoetial model to fit data. 5. Evaluate logarithms i differet bases 6. Use logarithmic equatios as iverses of expoetial equatios ad solve for variables i expoets. 7. Use ad evaluate atural logarithms 8. Use atural logarithms i problems; create models usig atural logarithms 9. Uderstad ad work with the properties of logarithms (4-7) 20

22 Chapter 4 Problems 1. Evaluate a. 625 b. 32 c. 16 d. 8 e. f State the domai ad rage of y = 10 x 3. Fid the iverse of each equatio: a. 5 3 a. y = x b. y = x 4. I 1987, the populatio of Chia was approximately 1,062,000,0000. If the populatio is growig at a rate of 0.9% per year, what would the populatio be i 1995? Show the model you used. 5. If the populatio of a city is growig at a costat rate ad was 750,000 i 1990 ad had reached 1,000,000 by 2000, what is its rate of growth per year? 6. Evaluate: a. Log 2 8 b. Log 8 2 c. Log 25 d. Log 5 25 e. Log Carbo-14 has a half-life of 5700 years. a. Use a cotiuous growth model to fid the aual rate of growth. b. If a orgaism had 15 grams of grams of C-14 at death ad its fossil cotaied 1 gram of C- 14, how old was the fossil? 21

23 4.1 th Root Fuctios th root of a umber x is a solutio r to the equatio r =x. Whe is a iteger>=2, r is a th root of x if ad oly if r =x. 2 3 = 8 meas that 2 is the 3 rd root of 8. Sice 3 4 =81 meas that 3 is the 4 th root of 81. The phrase th root of x ca apply to ay x, real or complex. The symbol 1 x is used oly whe x is a oegative real umber. The radical otatio 1 x meas x whe x is oegative but also has a use whe x is egative ad is a odd positive iteger greater tha 1. x ad 1 x are iverse fuctios o the domai x: x>= Ratioal Power Fuctios For all positive itegers m ad ad base x>0 m 1 m x = ( x ) = ( x) m 1 m m m x = ( x ) = ( x ) Example: = (8 ) = 2 = 128 = (8 ) = = 128 Power Postulates 22

24 x " x = x m m+ ( xy) x x m m! = x (x # 0) = x y $ x % x & ' = ( y 0) # ( y ) y 0 x = 1 ( x! 0)! 1 x = x 4.3 Expoetial Fuctios A expoetial equatio, y=ab x, ca be defied whe a"0, b>0, ad b"1 To the left is the graph of y = 3(2 x ) ad y = 3(0.5 x ). The two are reflectios of each other across the y-axis. For both fuctios, the domai is the set of all real umbers ad the rage is the set of positive real umbers. 3(2 x ) is a situatio where y icreases as x icreases. For 3(0.5 x ), y decreases as x icreases Expoetial equatios describe ay situatio where growth or decay is at a costat rate. Frequetly, these rates are expressed as percets: the populatio of the New Zealad grew at a average aual rate of 3% from 1980 to This meas that, o average, the populatio of oe year was equal to 1.03 times the populatio of the previous year. By 1985, the populatio had grow from the 1980 populatio to be 1.03 x 1.03 x 1.03 x 1.03 x 1.03 times as great, or times the 1980 populatio. Whe the growth rate, or b, is less tha 1, the situatio is oe of decay. For example, suppose that you start with 100 grams of a radioactive material that decays at a rate of 50 percet per year. At the ed of x years, you will have 100(0.5) x. If, for example, you wat to kow how much is left after 8 years, substitute 8 for x to get 100(0.5) 8 which works out to be 0.39 grams. 4-4 Fidig Expoetial Models 23

25 Suppose that whe you graph some give data, it forms a curve that appears to be expoetial. How do you fid the equatio? As follows: 1. Pick two poits. Suppose that you have 74 isects after 3 days ad 108 isects after 5 days ad you suspect that the growth is expoetial. 2. Create two equatios of the y=ab x : This gives you 74 = ab 3 ad 108 = ab 5 3. Divide the secod equatio by the first to get (108/74) o oe side ad (ab 5 /ab 3 ). This works out to = b 2 so b = Plug this value of b back ito the equatio to fid the value of a which turs out to be The equatio the is y = 41.97(1.208) x. By pluggig i differet values for x, you ca predict how may isects there will be after 10 days or 100 days or for ay value of x. 4-5 Logarithmic Fuctios Although we ca fid y for differet values of x i the expoetial equatio, we do ot kow how to create a iverse equatio that would permit us to solve for x give a certai value of y. We defie the iverse of y=b x to be x = b y, but how do we solve for y i this secod equatio? We defie y to be the logarithm of x to the base b if ad oly if x = b y. Examples: Log meas logarithm of 125 base 5 or as a expoetial equatio, if x = Log 5 125, that meas that 5 x = 125 so therefore x = 3. Log 8 2 = 1/3 because 8 equals 2 to the 1/3 power. Whe a log is writte without a explicit base, that meas the base is 10. Log 100 = 2 because 10 2 = e ad Natural Logarithms 1 The umber e is defied as the e = lim (1 + ) As gets higher ad higher, the expressio!" approaches the value of e, approximately For a variety of mathematical reasos, it makes sese to use this umber e as the base for logarithms i a umber of situatios. The atural logarithm of x is writte as l x. Whe a situatio is chagig cotiuously (populatio growth, bacterial divisio, etc.) it makes sese to use a model of cotiuous growth: 24

26 The amout of a substace after t years of growth at the rate r is give by the expressio A(t) = Pert, where P is the iitial amout, r is the rate of growth ad t is the umber of years (or other time period) that have passed. Example: If the populatio of a city has bee growig at the rate of 5% aually for 5 years, from a iitial populatio of 2.5 millio, what will be the curret populatio? Sice the populatio has bee growig cotiuously, the cotiuous growth model will give us a slightly more accurate model tha the expoetial model: A(5) = 2.5e.05*5 This gives 2.5 * e 0.25 = 3.21 millio. Usig Logarithms to Solve for Variables Suppose that we have a situatio modeled by the equatio y = 3(1.25) x Suppose that we wat to kow the value of x whe y has a value of 10. We must use logarithms to solve that problem: = 3(1.25) x 2. Divide both side by 3 to get 10/3=(1.25) x 3. Take log of both sides: log(10/3) = log(1.25) x 4. The properties of logarithms eables us to covert the right side of the equatio to x(log 1.25) 5. Divide both sides by Log 1.25 to get ((Log 10/3)/Log 1.25) = x = Whe x = 5.395, y will equal 10. Usig the Natural Logarithm Suppose that we kow that a city of 500,000 is growig at the rate of 2 percet per year ad we wat to kow whe the populatio will reach 1,000,000. We would proceed as follows: 1,000,000 = 500,000e.02t where t is the umber of years. -- First, we divide both sides by 500,000 to get 2 = e.02t -- Secod, we take the logarithm of both sides: l(2) = l(e.02t ) = t le.02 =.02t -- Solvig, we get 0.693=.02t ad therefore t = 34.6 years. 4-7 Properties of Logarithms These properties of logarithms follows from properties of expoets explored earlier i the chapter: For ay base b, log b 1 = 0 For ay base b ad for ay positive real umbers x ad y, log b (xy) = log b x + log b y For ay base b ad for ay positive real umbers x ad y, log b (x/y) = log b x log b y 25

27 For ay base b, ay positive real umber x ad ay real umber p, log b x p = p log b x For all values of a, b, ad c for which logarithms exist, log b log a = log c c a b 26