3.1 Exponential Functions And Their Graphs
333202 0301.qxd21812/7/05Chapter 33.110:25 AMPage 218Exponential and Logarithmic FunctionsExponential Functions and Their GraphsWhat you should learn Recognize and evaluate exponential functions with base a. Graph exponential functionsand use the One-to-OneProperty. Recognize, evaluate, and graphexponential functions withbase e. Use exponential functions tomodel and solve real-lifeproblems.Why you should learn itExponential functions can beused to model and solve real-lifeproblems. For instance, inExercise 70 on page 228, anexponential function is used tomodel the atmospheric pressureat different altitudes.Exponential FunctionsSo far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two typesof nonalgebraic functions—exponential functions and logarithmic functions.These functions are examples of transcendental functions.Definition of Exponential FunctionThe exponential function f with base a is denoted byf x a xwhere a 0, a 1, and x is any real number.The base a 1 is excluded because it yields f x 1x 1. This is a constantfunction, not an exponential function.You have evaluated a x for integer and rational values of x. For example, youknow that 43 64 and 41 2 2. However, to evaluate 4x for any real number x,you need to interpret forms with irrational exponents. For the purposes of thistext, it is sufficient to think ofa 2(where 2 1.41421356)as the number that has the successively closer approximationsa1.4, a1.41, a1.414, a1.4142, a1.41421, . . . .Example 1Evaluating Exponential FunctionsUse a calculator to evaluate each function at the indicated value of x.Functiona. f x 2 xb. f x 2 xc. f x 0.6xSolution Graphing Calculator Keystrokes 3.1 ENTER2 ENTER2 Function Valuea. f 3.1 2 3.1b. f 2 3c. f 2 0.6 3 2.6 Comstock Images/AlamyValuex 3.1x x 32 3 2 ENTERDisplay0.11662910.11331470.4647580Now try Exercise 1.The HM mathSpace CD-ROM andEduspace for this text containadditional resources related to theconcepts discussed in this chapter.When evaluating exponential functions with a calculator, remember toenclose fractional exponents in parentheses. Because the calculator follows theorder of operations, parentheses are crucial in order to obtain the correct result.
333202 0301.qxd12/7/0510:25 AMPage 219Section 3.1Note that an exponential functionf x a x is a constant raisedto a variable power, whereas apower function g x x n is avariable raised to a constantpower. Use a graphing utilityto graph each pair of functionsin the same viewing window.Describe any similarities anddifferences in the graphs.a. y1 y2 x2b. y1 3x, y2 x3y219Graphs of Exponential FunctionsExploration2x,Exponential Functions and Their GraphsThe graphs of all exponential functions have similar characteristics, as shown inExamples 2, 3, and 5.Example 2Graphs of y axIn the same coordinate plane, sketch the graph of each function.a. f x 2xb. g x 4xSolutionThe table below lists some values for each function, and Figure 3.1 shows thegraphs of the two functions. Note that both graphs are increasing. Moreover, thegraph of g x 4x is increasing more rapidly than the graph of f x 2x.g(x) 4x16x 3 2 10122x18141244x1641161214141614Now try Exercise 11.1210The table in Example 2 was evaluated by hand. You could, of course, use agraphing utility to construct tables with even more values.864f(x) 2x2 4 3 2 1 2FIGUREExample 3Graphs of y a –xx1234In the same coordinate plane, sketch the graph of each function.a. F x 2 x3.1b. G x 4 xSolutionG(x) 4 xThe table below lists some values for each function, and Figure 3.2 shows thegraphs of the two functions. Note that both graphs are decreasing. Moreover, thegraph of G x 4 x is decreasing more rapidly than the graph of F x 2 x.y161412x10864 2 10123121414116181642 x4214 x1641F(x) 2 x 4 3 2 1 2FIGURE3.2x1234Now try Exercise 13.In Example 3, note that by using one of the properties of exponents, the functions F x 2 x and G x 4 x can be rewritten with positive exponents.F x 2 x 11 2x2xand G x 4 x 1 14x 4x
333202 0301.qxd22012/7/05Chapter 33:30 PMPage 220Exponential and Logarithmic FunctionsComparing the functions in Examples 2 and 3, observe thatF x 2 x f x G x 4 x g x .andConsequently, the graph of F is a reflection (in the y-axis) of the graph of f. Thegraphs of G and g have the same relationship. The graphs in Figures 3.1 and 3.2are typical of the exponential functions y a x and y a x. They have oney-intercept and one horizontal asymptote (the x-axis), and they are continuous.The basic characteristics of these exponential functions are summarized inFigures 3.3 and 3.4.yNotice that the range of anexponential function is 0, ,which means that a x 0 for allvalues of x.y ax(0, 1)xFIGURE3.3yy a x(0, 1)xFIGUREGraph of y a x, a 1 Domain: , Range: 0, Intercept: 0, 1 Increasing x-axis is a horizontal asymptote ax 0 as x ContinuousGraph of y a x, a 1 Domain: , Range: 0, Intercept: 0, 1 Decreasing x-axis is a horizontal asymptote a x 0 as x Continuous3.4From Figures 3.3 and 3.4, you can see that the graph of an exponentialfunction is always increasing or always decreasing. As a result, the graphs passthe Horizontal Line Test, and therefore the functions are one-to-one functions.You can use the following One-to-One Property to solve simple exponentialequations.For a 0 and a 1, ax ay if and only if x y.Example 4Using the One-to-One Propertya. 9 3x 132 3x 12 x 11 xb. 1 x2One-to-One PropertyOriginal equation9 32One-to-One PropertySolve for x. 8 2 x 23 x 3Now try Exercise 45.
333202 0301.qxd12/7/0510:25 AMPage 221Section 3.1221Exponential Functions and Their GraphsIn the following example, notice how the graph of y a x can be used tosketch the graphs of functions of the form f x b a x c.Transformations of Graphs of Exponential FunctionsExample 5Each of the following graphs is a transformation of the graph of f x 3x.a. Because g x 3x 1 f x 1 , the graph of g can be obtained by shiftingthe graph of f one unit to the left, as shown in Figure 3.5.b. Because h x 3x 2 f x 2, the graph of h can be obtained byshifting the graph of f downward two units, as shown in Figure 3.6.c. Because k x 3x f x , the graph of k can be obtained by reflectingthe graph of f in the x-axis, as shown in Figure 3.7.d. Because j x 3 x f x , the graph of j can be obtained by reflecting thegraph of f in the y-axis, as shown in Figure 3.8.yy23g(x) 3x 1f (x) 3x12x 21 2FIGURE1f(x) 3 xh(x) 3 x 2 21Horizontal shiftFIGURE3.6Vertical shiftyy4213f(x) 3 xx 21 12k(x) 3 x 2FIGURE3.72 1x 13.5 12j(x) 3 xf(x) 3 x1x 2Reflection in x-axisFIGURE 13.812Reflection in y-axisNow try Exercise 17.Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the x-axisas a horizontal asymptote, but the transformation in Figure 3.6 yields a newhorizontal asymptote of y 2. Also, be sure to note how the y-intercept isaffected by each transformation.
333202 0301.qxd22212/7/0510:25 AMChapter 3Page 222Exponential and Logarithmic Functionsy3The Natural Base eIn many applications, the most convenient choice for a base is the irrationalnumber(1, e)e 2.718281828 . . . .2f(x) ( 1, e 1)This number is called the natural base. The function given by f x e x is calledthe natural exponential function. Its graph is shown in Figure 3.9. Be sureyou see that for the exponential function f x e x, e is the constant2.718281828 . . . , whereas x is the variable.ex(0, 1)( 2, e 2) 2FIGUREx 11Exploration3.9Use a graphing utility to graph y1 1 1 x x and y2 e in the sameviewing window. Using the trace feature, explain what happens to the graphof y1 as x increases.Example 6Use a calculator to evaluate the function given by f x e x at each indicatedvalue of x.a. x 2b. x 1c. x 0.25d. x 0.3y8f(x) 2e 0.24x7Solution654a.b.c.d.31x 4 3 2 1FIGUREEvaluating the Natural Exponential Function123Function Valuef 2 e 2f 1 e 1f 0.25 e0.25f 0.3 e 0.3Display0.13533530.36787941.28402540.7408182Now try Exercise 27.43.10Example 7y8a. f x 2e0.24x7Solution432g(x) 12 e 0.58x3.11x2341 3 2 10123f x 0.9741.2381.5732.0002.5423.2324.109g x 2.8491.5950.8930.5000.2800.1570.088x11b. g x 2e 0.58xTo sketch these two graphs, you can use a graphing utility to construct a table ofvalues, as shown below. After constructing the table, plot the points and connectthem with smooth curves, as shown in Figures 3.10 and 3.11. Note that the graphin Figure 3.10 is increasing, whereas the graph in Figure 3.11 is decreasing.5 4 3 2 1Graphing Natural Exponential FunctionsSketch the graph of each natural exponential function.6FIGUREGraphing Calculator Keystrokesex 2 ENTERex 1 ENTERex 0.25 ENTERex 0.3 ENTERNow try Exercise 35.
333202 0301.qxd12/7/0510:25 AMPage 223Section 3.1Use the formula 223ApplicationsExplorationA P 1 Exponential Functions and Their Graphsrnntto calculate the amount in anaccount when P 3000,r 6%, t 10 years, andcompounding is done (a) by theday, (b) by the hour, (c) by theminute, and (d) by the second.Does increasing the number ofcompoundings per year result inunlimited growth of the amountin the account? Explain.One of the most familiar examples of exponential growth is that of an investmentearning continuously compounded interest. Using exponential functions, you candevelop a formula for interest compounded n times per year and show how itleads to continuous compounding.Suppose a principal P is invested at an annual interest rate r, compoundedonce a year. If the interest is added to the principal at the end of the year, the newbalance P1 isP1 P Pr P 1 r .This pattern of multiplying the previous principal by 1 r is then repeated eachsuccessive year, as shown below.Year0123.Balance After Each CompoundingP PP1 P 1 r P2 P1 1 r P 1 r 1 r P 1 r 2P3 P2 1 r P 1 r 2 1 r P 1 r 3.Pt P 1 r ttTo accommodate more frequent (quarterly, monthly, or daily) compoundingof interest, let n be the number of compoundings per year and let t be the number of years. Then the rate per compounding is r n and the account balance aftert years is A P 1 rnnt.Amount (balance) with n compoundings per yearIf you let the number of compoundings n increase without bound, the processapproaches what is called continuous compounding. In the formula for ncompoundings per year, let m n r. This produces rn P 1 rmr 1mA P 1 m 1 72.7182804692.718281693 e P 1 P 1 ntAmount with n compoundings per yearmrtSubstitute mr for n.mrt1mSimplify.m rt.Property of exponentsAs m increases without bound, the table at the left shows that 1 1 m m eas m . From this, you can conclude that the formula for continuouscompounding isA Pert.Substitute e for 1 1 m m.
333202 0301.qxd22412/7/0510:25 AMChapter 3Page 224Exponential and Logarithmic FunctionsFormulas for Compound InterestBe sure you see that the annualinterest rate must be written indecimal form. For instance, 6%should be written as 0.06.Activities1. Sketch the graphs of the functionsf x e x and g x 1 e x on thesame coordinate system.76rnnt2. For continuous compounding: A Pe rtCompound InterestExample 8a. quarterly.b. monthly.c. continuously.g4f3x 4 3 2 1 1. For n compoundings per year: A P 1 A total of 12,000 is invested at an annual interest rate of 9%. Find the balanceafter 5 years if it is compoundedy5After t years, the balance A in an account with principal P and annualinterest rate r (in decimal form) is given by the following formulas.12342. Determine the balance A at the endof 20 years if 1500 is invested at6.5% interest and the interest iscompounded (a) quarterly and (b)continuously.Answer: (a) 5446.73 (b) 5503.953. The number of fruit flies in an experimental population after t hours isgiven by Q t 20e0.03t, t 0.a. Find the initial number of fruit fliesin the population.b. How large is the population offruit flies after 72 hours?Answer: (a) 20 flies (b) 173 fliesGroup ActivityThe sequence 3, 6, 9, 12, 15, . . . is given byf n 3n and is an example of lineargrowth. The sequence 3, 9, 27, 81, 243, . . .is given by f n 3n and is an exampleof exponential growth. Explain thedifference between these two types ofgrowth. For each of the followingsequences, indicate whether thesequence represents linear growth orexponential growth, and find a linear orexponential function that represents thesequence. Give several other examples oflinear and exponential growth.1 1 1 1 1a. 2, 4, 8, 16, 32, . . .b. 4, 8, 12, 16, 20, . . .2 48 10c. 3, 3, 2, 3, 3 , 4, . . .d. 5, 25, 125, 625, . . .Solutiona. For quarterly compounding, you have n 4. So, in 5 years at 9%, thebalance is A P 1 rnntFormula for compound interest 12,000 1 0.0944(5)Substitute for P, r, n, and t. 18,726.11.Use a calculator.b. For monthly compounding, you have n 12. So, in 5 years at 9%, thebalance is A P 1 rnnt 12,000 1 Formula for compound interest0.091212(5) 18,788.17.Substitute for P, r, n, and t.Use a calculator.c. For continuous compounding, the balance isA Pe rtFormula for continuous compounding 12,000e0.09(5)Substitute for P, r, and t. 18,819.75.Use a calculator.Now try Exercise 53.In Example 8, note that continuous compounding yields more than quarterlyor monthly compounding. This is typical of the two types of compounding. Thatis, for a given principal, interest rate, and time, continuous compounding willalways yield a larger balance than compounding n times a year.
333202 0301.qxd12/7/0510:25 AMPage 225Section 3.1Example 9Plutonium (in pounds)P10987654321( 12( t/24,100(24,100, 5)P 10(100,000, 0.564)t50,000100,000Years of decayFIGURERadioactive DecayIn 1986, a nuclear reactor accident occurred in Chernobyl in what was then theSoviet Union. The explosion spread highly toxic radioactive chemicals, such asplutonium, over hundreds of square miles, and the government evacuated the cityand the surrounding area. To see why the city is now uninhabited, consider themodelRadioactive DecayP 10225Exponential Functions and Their Graphs3.12 12t 24,100which represents the amount of plutonium P that remains (from an initial amountof 10 pounds) after t years. Sketch the graph of this function over the intervalfrom t 0 to t 100,000, where t 0 represents 1986. How much of the 10pounds will remain in the year 2010? How much of the 10 pounds will remainafter 100,000 years?SolutionThe graph of this function is shown in Figure 3.12. Note from this graph thatplutonium has a half-life of about 24,100 years. That is, after 24,100 years, halfof the original amount will remain. After another 24,100 years, one-quarter of theoriginal amount will remain, and so on. In the year 2010 t 24 , there will stillbe P 1012 24 24,100 100.000995912 9.993 poundsof plutonium remaining. After 100,000 years, there will still be P 1012 100,000 24,100 10124.1494 0.564 poundof plutonium remaining.Now try Exercise 67.Writing About MathematicsSuggestion:One way your students might approachthis problem is to create a table, covering x -values from 2 through 3, foreach of the functions and compare thistable with the given tables. If thismethod is used, you might considerdividing your class into groups of threeor six and having the groups assign oneor two functions to each member. Theyshould then pool their results and workcooperatively to determine that eachfunction has a y-intercept of 0, 8 .Another approach is a graphicalone: the groups can create scatter plotsof the data shown in the table andcompare them with sketches of thegraphs of the given functions. Considerassigning students to groups of four andgiving the responsibility for sketchingthree graphs to each group member.WRITING ABOUTMATHEMATICSIdentifying Exponential Functions Which of the following functions generated thetwo tables below? Discuss how you were able to decide. What do these functionshave in common? Are any of them the same? If so, explain why.a. f1 x 2(x 3)b. f2 x 8 12 c. f3 x 12 d. f4 x 12 7e. f5 x 7 2xf. f6 x 8 2xxx(x 3)x 10123x 2 1012g x 7.5891115h x 3216842Create two different exponential functions of the forms y a b x and y c x dwith y-intercepts of 0, 3 .
333202 0301.qxd22612/7/0510:25 AMChapter 33.1Page 226Exponential and Logarithmic FunctionsThe HM mathSpace CD-ROM and Eduspace for this text contain step-by-step solutionsto all odd-numbered exercises. They also provide Tutorial Exercises for additional help.ExercisesVOCABULARY CHECK: Fill in the blanks.1. Polynomials and rational functions are examples of functions.2. Exponential and logarithmic functions are examples of nonalgebraic functions, also called functions.3. The exponential function given by f x e x is called the function, and the base eis called the base.4. To find the amount A in an account after t years with principal P and an annual interest rate r compoundedn times per year, you can use the formula .5. To find the amount A in an account after t years with principal P and an annual interest rate r compoundedcontinuously, you can use the formula .PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.In Exercises 1– 6, evaluate the function at the indicatedvalue of x. Round your result to three decimal places.Function1. f x Value2. f x 2.3xx 233. f x 5xx 24. f x 3 5x3x 105. g x 5000 2x x 1.56. f x 200 1.2 12xx 24y 215. f x 16. f x 4x 3 318. f x 4x, g x 4x 119. f x 2x, g x 5 2 x20. f x 10 x, g x 10 x 3y(b) x 667721. f x 2 , g x 2 4422. f x 0.3x, g x 0.3x 5x2x 24 2yx24625. y y(d)6644 27. f x 2x9. f x 2 x2x4 4 2 28. f x 2x 110. f x 2x 2In Exercises 23–26, use a graphing utility to graph theexponential function.23. y 2 x 224. y 3 x 23x 2 126. y 4x 1 2In Exercises 27–32, evaluate the function at the indicatedvalue of x. Round your result to three decimal places.Function2 214. f x 6x2 x 16(c) 413. f x 6 x17. f x 3 x, g x 3x 42 4112. f x 2 In Exercises 17–22, use the graph of f to describe thetransformation that yields the graph of g.In Exercises 7–10, match the exponential function with itsgraph. [The graphs are labeled (a), (b), (c), and (d).](a) x111. f x 2 xx 5.63.4xIn Exercises 11–16, use a graphing utility to construct atable of values for the function. Then sketch the graph ofthe function.2x4Value27. h x e xx 3428. f x e xx 3.229. f x 2e 5xx 1030. f x x 2401.5e x 231. f x 5000e0.06xx 632. f x x 20250e0.05x
333202 0301.qxd12/7/0510:25 AMPage 227Section 3.1In Exercises 33–38, use a graphing utility to construct atable of values for the function. Then sketch the graph ofthe function.33. f x e x34. f x e x35. f x 36. f x 2e 0.5x3e x 437. f x 2e x 2 438. f x 2 e x 5In Exercises 39– 44, use a graphing utility to graph theexponential function.39. y 1.08 5x40. y 1.085x41. s t 42. s t 3e 0.2t2e0.12t43. g x 1 e x44. h x e x 2In Exercise 45–52, use the One-to-One Property to solve theequation for x.45. 3x 1 2747. 2x 2 49.e3x 22 351. ex46. 2x 3 16132 e3 e2x48. 1550.e2x 152. exx 12 6 125 e4 e5xCompound Interest In Exercises 53–56, complete thetable to determine the balance A for P dollars invested atrate r for t years and compounded n times per year.n12412365ContinuousAExponential Functions and Their Graphs22762. Trust Fund A deposit of 5000 is made in a trust fundthat pays 7.5% interest, compounded continuously. It isspecified that the balance will be given to the college fromwhich the donor graduated after the money has earnedinterest for 50 years. How much will the college receive?63. Inflation If the annual rate of inflation averages 4% overthe next 10 years, the approximate costs C of goods orservices during any year in that decade will be modeled byC t P 1.04 t, where t is the time in years and P is thepresent cost. The price of an oil change for your car ispresently 23.95. Estimate the price 10 years from now.64. Demand The demand equation for a product is given by p 5000 1 44 e 0.002xwhere p is the price and x is the number of units.(a) Use a graphing utility to graph the demand function forx 0 and p 0.(b) Find the price p for a demand of x 500 units.(c) Use the graph in part (a) to approximate the greatestprice that will still yield a demand of at least 600 units.65. Computer Virus The number V of computers infected bya computer virus increases according to the modelV t 100e4.6052t, where t is the time in hours. Find (a) V 1 ,(b) V 1.5 , and (c) V 2 .66. Population The population P (in millions) of Russiafrom 1996 to 2004 can be approximated by the modelP 152.26e 0.0039t, where t represents the year, with t 6corresponding to 1996. (Source: Census Bureau,International Data Base)53. P 2500, r 2.5%, t 10 years(a) According to the model, is the population of Russiaincreasing or decreasing? Explain.54. P 1000, r 4%, t 10 years(b) Find the population of Russia in 1998 and 2000.55. P 2500, r 3%, t 20 years(c) Use the model to predict the population of Russia in2010.56. P 1000, r 6%, t 40 yearsCompound Interest In Exercises 57– 60, complete thetable to determine the balance A for 12,000 invested atrate r for t years, compounded continuously.t1020304050A67. Radioactive Decay Let Q represent a mass of radioactiveradium 226Ra (in grams), whose half-life is 1599 years.The quantity of radium present after t years is1 t 1599.Q 25 2 (a) Determine the initial quantity (when t 0).(b) Determine the quantity present after 1000 years.(c) Use a graphing utility to graph the function over theinterval t 0 to t 5000.57. r 4%58. r 6%59. r 6.5%60. r 3.5%61. Trust Fund On the day of a child’s birth, a deposit of 25,000 is made in a trust fund that pays 8.75% interest,compounded continuously. Determine the balance in thisaccount on the child’s 25th birthday.68. Radioactive Decay Let Q represent a mass of carbon14 14C (in grams), whose half-life is 5715 years. The quan1 t 5715.tity of carbon 14 present after t years is Q 10 2 (a) Determine the initial quantity (when t 0).(b) Determine the quantity present after 2000 years.(c) Sketch the graph of this function over the interval t 0to t 10,000.
333202 0301.qxd22812/7/05Chapter 310:25 AMPage 228Exponential and Logarithmic FunctionsSynthesisModel It69. Data Analysis: Biology To estimate the amount ofdefoliation caused by the gypsy moth during a givenyear, a forester counts the number x of egg masses on140 of an acre (circle of radius 18.6 feet) in the fall. Thepercent of defoliation y the next spring is shown in thetable. (Source: USDA, Forest Service)Egg masses, xPercent of defoliation, y02550751001244819699True or False? In Exercises 71 and 72, determine whetherthe statement is true or false. Justify your answer.71. The line y 2 is an asymptote for the graph off x 10 x 2.72. e 271,801.99,990Think About It In Exercises 73–76, use properties of exponents to determine which functions (if any) are the same.73. f x 3x 2g x 3x 9h x g x 22x 6h x 64 4x 1 x9 3 75. f x 16 4 x g x A model for the data is given by74. f x 4x 1276. f x e x 3 1 x 24g x e3 xh x 16 2 2x 100y .1 7e 0.069xh x e x 377. Graph the functions given by y 3x and y 4x and use thegraphs to solve each inequality.(a) Use a graphing utility to create a scatter plot of thedata and graph the model in the same viewingwindow.(b) Create a table that compares the model with thesample data.(c) Estimate the percent of defoliation if 36 egg masses1are counted on 40acre.(d) You observe that 23 of a forest is defoliated thefollowing spring. Use the graph in part (a) to1estimate the number of egg masses per 40acre.70. Data Analysis: Meteorology A meteorologist measuresthe atmospheric pressure P (in pascals) at altitude h (inkilometers). The data are shown in the table.(a) 4x 3x78. Use a graphing utility to graph each function. Use thegraph to find where the function is increasing anddecreasing, and approximate any relative maximum orminimum values.Pressure, P05101520101,29354,73523,29412,1575,069A model for the data is given by P 107,428e 0.150h.(a) Sketch a scatter plot of the data and graph the model onthe same set of axes.(b) Estimate the atmospheric pressure at a height of8 kilometers.(b) g x x23 x(a) f x x 2e x79. Graphical Analysis 0.5xf x 1 Use a graphing utility to graphxg x e0.5andin the same viewing window. What is the relationshipbetween f and g as x increases and decreases withoutbound?80. Think About It(a) 3xAltitude, h(b) 4x 3x(b)3x 2Which functions are exponential?(c) 3x(d) 2 xSkills ReviewIn Exercises 81 and 82, solve for y.81. x 2 y 2 25 82. x y 2In Exercises 83 and 84, sketch the graph of the function.83. f x 29 x84. f x 7 x85. Make a Decision To work an extended applicationanalyzing the population per square mile of the UnitedStates, visit this text’s website at college.hmco.com. (DataSource: U.S. Census Bureau)
218 Chapter 3 Exponential and Logarithmic Functions What you should learn Reeo aczgnind evaluate expo-nential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize,evaluate, and graph exponential functions with base e. Ue espxaonient l functions
9-2 Exponential Functions Exponential Function: For any real number x, an exponential function is a function in the form fx ab( ) x. There are two types of exponential functions: Exponential Growth: fx ab b( ) x, where 1 Exponential Decay: fx ab b( ) , where 0 1
Unit 6 Exponential and Logarithmic Functions Lesson 1: Graphing Exponential Growth/Decay Function Lesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functions Why do I need to Learn This? Many real life applications involve exponential functions.
348 Chapter 7 Exponential and Logarithmic Functions 7.1 Lesson WWhat You Will Learnhat You Will Learn Graph exponential growth and decay functions. Use exponential models to solve real-life problems. Exponential Growth and Decay Functions An exponential function has the form y abx, where
The Natural Logarithmic and Exponential The Natural Logarithmic and Exponential and Exponential Function FunctionFunctions sss: . Differentiate and integrate exponential functions that have bases other than e. Use exponential functions to model compound interest and exponential
Chapter 3 Exponential & Logarithmic Functions Section 3.1 Exponential Functions & Their Graphs Definition of Exponential Function: The exponential function f with base a is denoted f(x) a x where a 0, a 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use
218 Chapter 3 Exponential and Logarithmic Functions What you should learn Recognize and evaluate expo-nential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate, and graph exponential functions with base e. Use exponential
Section II: Exponential and Logarithmic Functions Unit 1: Introduction to Exponential Functions Exponential functions are functions in which the variable appears in the exponent. For example, fx( ) 80 (0.35) x is an exponential function si
AutoCAD education, and apply them right away to your fi rst real drawing. Let us take a look at the Layers Properties Manager. This is AutoCAD’s Layers dialog box, where everything important related to layers happens. Open a new fi le and, if you decide to use toolbars, also bring up the Layer toolbar. We take a closer look at that toolbar in Section 3.3 , but for now you need only the .
