. Eponential Growth and Deca Functions COMMON CORE Learning Standards HSA-SSE.B.3c HSF-IF.C.7e HSF-IF.C.8b HSF-LE.A. HSF-LE.B.5 Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing calculator to evaluate an eponential function. For eample, consider the eponential function f ) =. Function Value Graphing Calculator Kestrokes Displa f 3.) = 3. 3. ENTER 0.9 f 3 ) = /3 3 ) ENTER.5870 Identifing Graphs of Eponential Functions Work with a partner. Match each eponential function with its graph. Use a table of values to sketch the graph of the function, if necessar. a. f ) = b. f ) = 3 c. f ) = d. f ) = ) e. f ) = 3 ) f. f ) = ) A. B. C. D. E. F. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. Characteristics of Graphs of Eponential Functions Work with a partner. Use the graphs in Eploration to determine the domain, range, and -intercept of the graph of f ) = b, where b is a positive real number other than. Eplain our reasoning. Communicate Your Answer 3. What are some of the characteristics of the graph of an eponential function?. In Eploration, is it possible for the graph of f ) = b to have an -intercept? Eplain our reasoning. Section. Eponential Growth and Deca Functions 95
. Lesson What You Will Learn Core Vocabular eponential function, p. 9 eponential growth function, p. 9 growth factor, p. 9 asmptote, p. 9 eponential deca function, p. 9 deca factor, p. 9 Previous properties of eponents Graph eponential growth and deca functions. Use eponential models to solve real-life problems. Eponential Growth and Deca Functions An eponential function has the form = ab, where a 0 and the base b is a positive real number other than. If a > 0 and b >, then = ab is an eponential growth function, and b is called the growth factor. The simplest tpe of eponential growth function has the form = b. Core Concept Parent Function for Eponential Growth Functions The function f ) = b, where b >, is the parent function for the famil of eponential growth functions with base b. The graph shows the general shape of an eponential growth function. The -ais is an asmptote of the graph. An asmptote is a line that a graph approaches more and more closel. 0, ) f) = b b > ) The graph rises from left to right, passing, b) through the points 0, ) and, b). The domain of f ) = b is all real numbers. The range is > 0. If a > 0 and 0 < b <, then = ab is an eponential deca function, and b is called the deca factor. Core Concept Parent Function for Eponential Deca Functions The function f ) = b, where 0 < b <, is the parent function for the famil of eponential deca functions with base b. The graph shows the general shape of an eponential deca function. The graph falls from left to right, passing through the points 0, ) and, b). f) = b 0 < b < ) 0, ), b) The -ais is an asmptote of the graph. The domain of f ) = b is all real numbers. The range is > 0. 9 Chapter Eponential and Logarithmic Functions
Graphing Eponential Growth and Deca Functions Tell whether each function represents eponential growth or eponential deca. Then graph the function. a. = b. = ) a. Step Identif the value of the base. The base,, is greater than, so the function represents eponential growth. Step Make a table of values. 0 3 8 3, 8) 8 = Step 3 Plot the points from the table., ), Step Draw, from left to right, a smooth curve that, ) begins just above the -ais, passes through, 0, ) the plotted points, and moves up to the right. b. Step Identif the value of the base. The base,, is greater than 0 and less than, so the function represents eponential deca. Step Make a table of values. 3 0 8 Step 3 Plot the points from the table. Step Draw, from right to left, a smooth curve that begins just above the -ais, passes through the plotted points, and moves up to the left. 3, 8), ) 8, ) 0, ) =,, Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the function represents eponential growth or eponential deca. Then graph the function.. =. = 3 ) 3. f ) = 0.5). f ) =.5) Eponential Models Some real-life quantities increase or decrease b a fied percent each ear or some other time period). The amount of such a quantit after t ears can be modeled b one of these equations. Eponential Growth Model = a + r) t Eponential Deca Model = a r) t Note that a is the initial amount and r is the percent increase or decrease written as a decimal. The quantit + r is the growth factor, and r is the deca factor. Section. Eponential Growth and Deca Functions 97
Solving a Real-Life Problem REASONING QUANTITATIVELY The percent decrease, 5%, tells ou how much value the car loses each ear. The deca factor, 0.85, tells ou what fraction of the car s value remains each ear. The value of a car in thousands of dollars) can be approimated b the model = 50.85) t, where t is the number of ears since the car was new. a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in the value of the car. c. Estimate when the value of the car will be $8000. a. The base, 0.85, is greater than 0 and less than, so the model represents eponential deca. b. Because t is given in ears and the deca factor 0.85 = 0.5, the annual percent decrease is 0.5, or 5%. 30 c. Use the trace feature of a graphing calculator to determine that 8 when t = 7. After 7 ears, = 50.85) the value of the car will be about $8000. X=7 Y=8.07 5 5 X 7 8 9 0 X= Y.53..89.78.88.988 7.00 Writing an Eponential Model In 000, the world population was about.09 billion. During the net 3 ears, the world population increased b about.8% each ear. a. Write an eponential growth model giving the population in billions) t ears after 000. Estimate the world population in 005. b. Estimate the ear when the world population was 7 billion. a. The initial amount is a =.09, and the percent increase is r = 0.08. So, the eponential growth model is = a + r) t Write eponential growth model. =.09 + 0.08) t Substitute.09 for a and 0.08 for r. =.09.08) t. Simplif. Using this model, ou can estimate the world population in 005 t = 5) to be =.09.08) 5. billion. b. Use the table feature of a graphing calculator to determine that 7 when t =. So, the world population was about 7 billion in 0. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? In Eample, the value of the car can be approimated b the model = 50.9) t. Identif the annual percent decrease in the value of the car. Estimate when the value of the car will be $8000.. WHAT IF? In Eample 3, assume the world population increased b.5% each ear. Write an equation to model this situation. Estimate the ear when the world population was 7 billion. 98 Chapter Eponential and Logarithmic Functions
Rewriting an Eponential Function The amount in grams) of the radioactive isotope chromium-5 remaining after t das is = a0.5) t/8, where a is the initial amount in grams). What percent of the chromium-5 decas each da? = a0.5) t/8 Write original function. = a[0.5) /8 ] t Power of a Power Propert a0.9755) t Evaluate power. = a 0.05) t Rewrite in form = a r) t. The dail deca rate is about 0.05, or.5%. Compound interest is interest paid on an initial investment, called the principal, and on previousl earned interest. Interest earned is often epressed as an annual percent, but the interest is usuall compounded more than once per ear. So, the eponential growth model = a + r) t must be modified for compound interest problems. Core Concept Compound Interest Consider an initial principal P deposited in an account that pas interest at an annual rate r epressed as a decimal), compounded n times per ear. The amount A in the account after t ears is given b A = P + r n ) nt. Finding the Balance in an Account You deposit $9000 in an account that pas.% annual interest. Find the balance after 3 ears when the interest is compounded quarterl. With interest compounded quarterl times per ear), the balance after 3 ears is A = P + r n ) nt = 9000 + 0.0 ) 3 90.. Write compound interest formula. P = 9000, r = 0.0, n =, t = 3 Use a calculator. The balance at the end of 3 ears is $90.. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. The amount in grams) of the radioactive isotope iodine-3 remaining after t hours is = a0.5) t/3, where a is the initial amount in grams). What percent of the iodine-3 decas each hour? 8. WHAT IF? In Eample 5, find the balance after 3 ears when the interest is compounded dail. Section. Eponential Growth and Deca Functions 99
. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY In the eponential growth model =..5), identif the initial amount, the growth factor, and the percent increase.. WHICH ONE DOESN T BELONG? Which characteristic of an eponential deca function does not belong with the other three? Eplain our reasoning. base of 0.8 deca factor of 0.8 deca rate of 0% 80% decrease Monitoring Progress and Modeling with Mathematics In Eercises 3 8, evaluate the epression for a) = and b) = 3. 3.. 5. 8 3. 7. 5 + 3 8. In Eercises 9 8, tell whether the function represents eponential growth or eponential deca. Then graph the function. See Eample.) 9. = 0. = 7. = ) 3. = 3). = 8). = 5) 5. =.). = 0.75) 7. = 0.) 8. =.8) ANALYZING RELATIONSHIPS In Eercises 9 and 0, use the graph of f) = b to identif the value of the base b. 9., 3, 3) 0, ) 0., 5, 5) 0, ). MODELING WITH MATHEMATICS The value of a mountain bike in dollars) can be approimated b the model = 000.75) t, where t is the number of ears since the bike was new. See Eample.) a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in the value of the bike. c. Estimate when the value of the bike will be $50.. MODELING WITH MATHEMATICS The population P in thousands) of Austin, Teas, during a recent decade can be approimated b = 9.9.03) t, where t is the number of ears since the beginning of the decade. a. Tell whether the model represents eponential growth or eponential deca. b. Identif the annual percent increase or decrease in population. c. Estimate when the population was about 590,000. 3. MODELING WITH MATHEMATICS In 00, there were approimatel 33 million cell phone subscribers in the United States. During the net ears, the number of cell phone subscribers increased b about % each ear. See Eample 3.) a. Write an eponential growth model giving the number of cell phone subscribers in millions) t ears after 00. Estimate the number of cell phone subscribers in 008. b. Estimate the ear when the number of cell phone subscribers was about 78 million. 300 Chapter Eponential and Logarithmic Functions
. MODELING WITH MATHEMATICS You take a 35 milligram dosage of ibuprofen. During each subsequent hour, the amount of medication in our bloodstream decreases b about 9% each hour. a. Write an eponential deca model giving the amount in milligrams) of ibuprofen in our bloodstream t hours after the initial dose. b. Estimate how long it takes for ou to have 00 milligrams of ibuprofen in our bloodstream. JUSTIFYING STEPS In Eercises 5 and, justif each step in rewriting the eponential function. 5. = a3) t/ Write original function. = a[3) / ] t a.08) t = a + 0.08) t. = a0.) t/3 Write original function. = a[0.) /3 ] t a0.) t = a 0.5358) t 7. PROBLEM SOLVING When a plant or animal dies, it stops acquiring carbon- from the atmosphere. The amount in grams) of carbon- in the bod of an organism after t ears is = a0.5) t/5730, where a is the initial amount in grams). What percent of the carbon- is released each ear? See Eample.) 8. PROBLEM SOLVING The number of duckweed fronds in a pond after t das is = a30.5) t/, where a is the initial number of fronds. B what percent does the duckweed increase each da? 33. = a 3 ) t/0 3. = a 5 ) t/ 35. = a) 8t 3. = a 3 ) 3t 37. PROBLEM SOLVING You deposit $5000 in an account that pas.5% annual interest. Find the balance after 5 ears when the interest is compounded quarterl. See Eample 5.) 38. DRAWING CONCLUSIONS You deposit $00 into three separate bank accounts that each pa 3% annual interest. How much interest does each account earn after ears? Account Compounding quarterl monthl 3 dail Balance after ears 39. ERROR ANALYSIS You invest $500 in the stock of a compan. The value of the stock decreases % each ear. Describe and correct the error in writing a model for the value of the stock after t ears. = amount) Initial factor) Deca t = 5000.0) t 0. ERROR ANALYSIS You deposit $50 in an account that pas.5% annual interest. Describe and correct the error in finding the balance after 3 ears when the interest is compounded quarterl. A = 50 +.5 ) 3 A = $533.9 In Eercises 9 3, rewrite the function in the form = a + r) t or = a r) t. Then state the growth or deca rate. 9. = a) t/3 30. = a) t/ 3. = a0.5) t/ 3. = a0.5) t/9 In Eercises, use the given information to find the amount A in the account earning compound interest after ears when the principal is $3500.. r =.%, compounded quarterl. r =.9%, compounded monthl 3. r =.83%, compounded dail. r =.%, compounded monthl Section. Eponential Growth and Deca Functions 30
5. USING STRUCTURE A website recorded the number of referrals it received from social media websites over a 0-ear period. The results can be modeled b = 500.50) t, where t is the ear and 0 t 9. Interpret the values of a and b in this situation. What is the annual percent increase? Eplain.. HOW DO YOU SEE IT? Consider the graph of an eponential function of the form f ) = ab., ) 0, ),, 50. REASONING Consider the eponential function f ) = ab. f + ) a. Show that = b. f ) b. Use the equation in part a) to eplain wh there is no eponential function of the form f ) = ab whose graph passes through the points in the table below. 0 3 8 7 5. PROBLEM SOLVING The number E of eggs a Leghorn chicken produces per ear can be modeled b the equation E = 79.0.89) w/5, where w is the age in weeks) of the chicken and w. a. Determine whether the graph of f represents eponential growth or eponential deca. b. What are the domain and range of the function? Eplain. 7. MAKING AN ARGUMENT Your friend sas the graph of f ) = increases at a faster rate than the graph of g ) = when 0. Is our friend correct? Eplain our reasoning. 8 g 0 0 8. THOUGHT PROVOKING The function f ) = b represents an eponential deca function. Write a second eponential deca function in terms of b and. 9. PROBLEM SOLVING The population p of a small town after ears can be modeled b the function p = 850.03). What is the average rate of change in the population over the first ears? Justif our answer. Maintaining Mathematical Proficienc Simplif the epression. Assume all variables are positive. Skills Review Handbook) 53. 9 5. 57. + 3 58. a. Identif the deca factor and the percent decrease. b. Graph the model. c. Estimate the egg production of a chicken that is.5 ears old. d. Eplain how ou can rewrite the given equation so that time is measured in ears rather than in weeks. 5. CRITICAL THINKING You bu a new stereo for $300 and are able to sell it ears later for $75. Assume that the resale value of the stereo decas eponentiall with time. Write an equation giving the resale value V in dollars) of the stereo as a function of the time t in ears) since ou bought it. Reviewing what ou learned in previous grades and lessons 3 55. 5. 8 ) + 59. + 5 0. 35 ) 3 30 Chapter Eponential and Logarithmic Functions