3.1 Exponential Functions and Their Graphs

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1 .1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic epressions. Eponential Function: a function of the form f a b ; a, b 1, b a: initial value (-intercept); f ab a b: base Evaluating an Eponential Function E1: Determine the value for f 4. Note: A common mistake is multipling the and 4. Remember the order of operations (eponents before multiplication). Or have students write the function as f f. f. f 1 Graphs of Eponential Functions 1 E: Plot points and graph the functions 1 and Eponential Growth Domain: Range: Intercepts: Asmptotes: End Behavior: Increasing/decreasing Continuous? Eponential Deca Page 1 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

2 Graphs of Eponential Functions: Graph is INCREASING when b1 & a ; graph is DECREASING when b 1 & a. Transformations of Eponential Functions: h a b k Vertical Stretch: a 1 Vertical Shrink: a 1 Reflection over -ais: a Reflection over -ais: Horizontal Translation: h Vertical Translation: k Horizontal Asmptote: k Investigate the following transformations: f( ) vs. f( ) 4 f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) E: Graph the eponential function Transformation of 1.. Rewrite as. Horizontal Translation: Reflect: Vertical Translation: -intercept: f Horizontal Asmptote: 1 E4: Graph f( ) (1 ) The Natural Base e: e= 1 lim(1 ) Page of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

3 E5: Use a calculator to evaluate the function f ( ) e at each indicated value of. a. =- b. =.5 c. =-.4 E6: Sketch the graph of each natural eponential function a. f ( ) e b. f ( ) e Properties of Eponents: E. 7 a.) a a b.) a a c.) a d.) a a e.) ( ab) f.) ( ) b g.) a Reflection: Page of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

4 .1 Eponential Functions and Their Graphs Eponential Model: ab Eponential Growth: a & b 1; b is the growth factor Eponential Deca: a & b 1; b is the deca factor Applications of the Eponential Model E1: CCSD s student population went from,4 in 1956 to 91,51 students in 5. Write an eponential function that represents the student population. Predict the population in 1. Let represent the ear 195 and represent the number of students. Substitute the given values into the eponential model ab and solve for a and b. Derive Compound Interest Formula and e: Page 4 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

5 r Formula for Compound Interest: A P1 A = balance r = annual interest rate n P = principal t = time in ears n = number of times interest is compounded each ear Note: Annuall = 1 time per ear Semiannuall = times per ear Quarterl = 4 times per ear Monthl = 1 times per ear Weekl = 5 times per ear nt Interest Compounded Continuousl: A Pe rt r A P1 n E4a.: Calculate the balance if $ is invested for 1 ears at 6% compounded weekl. nt A? P t r n b.) What if the interest is compounded continuousl? Isolating the change in the compounding period reveals a naturall occurring constant. Let the compounding period (n) be equal to some constant (m) multiplied b the rate: n mr. r 1 1 A P1 P1 P 1 mr m m mrt mrt m rt m m 1 1 Graph 1 and find lim 1. Make a table. m m m The values of are approaching.718. This is the approimate value of the transcendental number e. Natural Base e: 1 e lim Interest Compounded Continuousl: A Pe rt k Natural Eponential Function: f a e Graph of e : Page 5 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

6 E. 5 A total of $1, is invested at an annual interest rate of %. Find the balance after 4 ears if the interest is compounded continuousl. c Logistic Function: f c (constant): limit to growth (maimum) 1 a b The logistic function is used for populations that will be limited in their abilit to grow due to limited resources or space. Think About It: What would limit population growth? E5: Estimate the maimum population for Dallas and find the population for the ear 8 1,1,64 given the function Pt () that models the population from t 1 1.6e Maimum population: c 1,1,64 Population in 8: t P(18) Graph of a Logistic Function: Domain: 1 f 1 e Range: Alwas Increasing Horizontal Asmptotes: Page 6 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

7 -intercept: E6: Sketch the graph of the function. State the -intercept and horizontal asmptotes. 1.4 H.A.: 1 You Tr: Describe the transformations needed to draw the graph of graph. f 4. Sketch the QOD: Using a table of values, how can ou determine whether the have an eponential relationship? Page 7 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

8 . Logarithmic Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of an eponential, logistic or logarithmic function. 9. The student will evaluate eponential or logarithmic epressions. 9.5 The student will graph the inverse of an eponential or logarithmic function. Review: Solve the following for. Tr Eploration: Use our calculator to write 1997 as a power of 1. Note: 4 1 1, , , (close!) Now use our calculator to find LOG Common Logarithm: Given a positive number p, the solution to 1 5 log (Compare to above) p is called the base-1 logarithm of p, epressed as log1 p, or simpl as log p. (When no base is specified, it is understood to be base 1.) A LOGARITHM IS AN EXPONENT. Logarithm (base b): b. log b (Read as log base b of. ) for, b and b 1 if and onl if log b b Note: You cannot take the log of a negative number! E1: Rewrite each equation (eponential form) to logarithmic form Base (b) = Eponent = 4 16 Base (b) = Eponent = E: Rewrite each equation (logarithmic form) to eponential form. 1. log515 Base (b) = Eponent =. log1, 5 Base (b) = Eponent = Page 8 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

9 Evaluating Logarithms: a logarithm is an eponent E: Evaluate the logarithms. 1. log f, f log log f., f log log. f Calculator: Graphing Logarithmic Functions: a logarithmic function is the inverse of an eponential function E4: Use a table of values to sketch the graph of log. Discuss the characteristics of the graph and compare the graph to the graph of. Note: To create the table, it is helpful to rewrite the function as and choose values for / 1/9 1 1 Domain: Range: Intercepts: Asmptote: Increasing: End Behavior: lim f lim f The function log is the inverse of, so its graph is the reflection of over the line. Transformations of log b : k a h log b Vertical Stretch: a, a 1 Horizontal Stretch: a, a 1 Reflection over -ais: a Reflection over -ais: Horizontal Translation: h Vertical Translation: k Page 9 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

10 E5: Describe the transformations used to graph the function sketch the graph. Reflect shift shift log 5. Then Properties of Logarithms: A.) log 1 a because B.) log 1 a a because C.) log a a Note: Graph in red is the graph of log. because D.) If log log a a, then Natural Logarithmic Function: logarithm base e, log e, written ln Notations: 1. log is used to represent log1 (common logarithm). ln is used to represent log e (natural logarithm) Evaluating Logarithmic Epressions: log E6: Evaluate log 1 and.5 log 1.5 Let log 1 log4.1 1 Let. Rewrite in eponential form: log Rewrite in logarithmic form: Note: B the properties of inverses, we could have evaluated the above eamples without rewriting using the following: log a a and log a a Page 1 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

11 Solving Logarithmic Equations: 1. E7: Solve the equations for. log log 6 The bases are equal, so 7 7 Note: Both solutions work in the original equation.. log9 Rewrite eponentiall:. 4 ln e Rewrite eponentiall: Note: Remember that the base of ln is e. You Tr: Sketch the graph of function on the same grid with its inverse. f ln QOD: Can ou evaluate the log of a negative number? Eplain. Page 11 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

12 . Properties of Logarithms Sllabus Objective: 9. The student will appl the properties of logarithms to evaluate epressions, change bases, and re-epress data. Eploration: Use our calculator to find log and log. Evaluate the following logarithms on our calculator, then speculate how ou could calculate them using onl the values of log and/or log. log.1 and log log 6 log log8 log8.9. log log.176 Recall: Properties of Eponents b b b Because a logarithm is an eponent, the rules are the same! b b b b b Properties of Logarithms: Condensed Epanded log RS log R log S b b b log R log R log S log b b b b c R c log R b S E1: Use the properties of logarithms to epand the following epressions. a.) log 5 b.) ln z c.) log 1 d.) log Page 1 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

13 E: Use the properties of logarithms to condense the following epressions. a.) log log4 b.) log log 1 c.) ln ln 1 ln 1 Evaluating Logarithmic Epressions with Base b Let log b. Rewrite in eponential form: b. Take the log of both sides. log b log Use the properties of logs to solve for. log log b log log b Note: This will work for a logarithm of an base, including the natural log. Change of Base Formula: log a ln a logb a log b ln b E: Evaluate the epression log5 8. B the change of base formula,. Using the calculator, E4: Evaluate the epression log64. B the change of base formula,. Using the calculator, Note This did NOT require the use of a calculator! We know that So log Page 1 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

14 Graphing Logarithmic Functions on the Calculator E5: Graph log and log 5 on the same grid on the graphing calculator. Some calculators cannot tpe in a log base into the calculator, so we must rewrite the functions using the change of base formula. Our Calculator: Tpe green alpha f, no need to use change of base. log 1 1 log log log log 5 5 log Caution: The graph created b the calculator is misleading at the asmptote! You Tr: Epand the epression using the properties of logarithms. ln z. QOD: When is it appropriate to use the change of base formula? Eplain how to evaluate a logarithm of base b without the change of base formula. Page 14 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

15 .4 Solving Eponential and Logarithmic Equations Sllabus Objectives: 9.4 The student will solve eponential, logarithmic and logistic equations and inequalities. 9.6 The student will compare equivalent logarithmic and eponential equations. Strategies for Solving an Eponential Equation: Rewrite both sides with the same base Take the log of both sides after isolating the eponential E1: Solve the following eponential equations. a.) 4 Rewrite with base : Both sides have the same base, so the eponents must be equal: b.) 8 6 We cannot rewrite both sides with the same base, so take the log of both sides. c.) 7 e 5 Isolate the eponential: Take the natural log of both sides: Note: We could have determined that immediatel using the equation e 1. Strategies for Solving a Logarithmic Equation: Condense an logarithms with the same base using the properties of logs Rewrite the equation in eponential form Check for etraneous solutions E: Solve the following logarithmic equations. a.) ln ln5 Condense: Rewrite in eponential form: Solve for : Check: b.) log 65 4 Rewrite in ep. form: Take the log of both sides: Rewrite in eponential form: Note: We could have determined that 5 immediatel using the equation root of each side. 4 Page 15 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter 65, b taking the 4 th

16 c.) log log 8 Condense: Rewrite in eponential form: Solve for : Check: (remember can t take the log of a negative!) Note: You must check ever possible solution for etraneous solutions. All negative answers are not necessaril etraneous! E: Solve the equation log 4 1. Rewrite in eponential form: Solve for : 5 Check: E4: Solve the equation Rewrite using properties of logs: log 6. Rewrite in eponential form: Check: Challenge Problems: Use our arsenal of eponential and logarithmic properties! E5: Solve the equation 6 7. Page 16 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

17 E6: Solve the equation Take the log of both sides: Rewrite: Solve for : Check You Tr: Solve the equation e QOD: Compare and contrast the methods for solving eponential and logarithmic equations. Page 17 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

18 .5 Eponential and Logarithmic Models Sllabus Objective: 9.7 The student will solve application problems involving eponential and logarithmic functions. T t T T T e kt s S, where Newton s Law of Cooling: The temperature T of an object at time t is T s is the surrounding temperature and T is the initial temperature of the object. E1: A 5 F potato is left out in a 7 F room for 1 minutes, and its temperature dropped to 5 F. How man more minutes will it take to reach 1 F? Solve for k using the given information: T t T T T e s S kt Use k to solve for t: It takes about minutes for the potato to cool to 1 F. This is minutes longer. nt r Compound Interest: A P1 A = balance amount P = principal (beginning) amount n r = annual interest rate (decimal) n = # of times compounded in a ear t = time in ears E: How long will it take for an investment of $, at 6% compounded semi-annuall to reach $5? r A P1 n nt It will take about ears. Page 18 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

19 A Pe E: How long will it take for an investment of $, at 6% compounded continuousl to reach $5? rt It will take about ears. Annual Percentage Yield (APY): the rate, compounded annuall ( t 1), that would ield the same return nt r For A P1 n, APY 1 r n 1 n E4: An amount of $4 is invested for 8 ears at 5% compounded quarterl. What is the equivalent APY? You Tr: Determine the amount of mone that should be invested at 9% interest compounded monthl to produce a balance of $, in 15 ears. QOD: Wh is using the annual percentage ield a more fair wa to compare investments? Page 19 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

20 .6 Nonlinear Models Sllabus Objectives: 9.7 The student will solve application problems involving eponential and logarithmic functions. 9.4 The student will solve eponential, logarithmic and logistic equations and inequalities. t Eponential Model: Pt Pb Growth Model: b1 r; b is called the Growth Factor Deca Model: b1 r; b is called the Deca Factor P t : population at time t P : initial population E1: Write an eponential function that models the population of Smallville if the initial population was,85, and it is decreasing b.% each ear. Predict how long it will take for the population to fall to. P r, so b Pt Solve for t when Pt t P t Use It will take about ears. E: The population of ants is increasing eponentiall such that on da there are 1 ants, and on da 4 there are ants. How man ants are there on da 5? t Pb and write a sstem of equations with the given information: 1 Pb Solve the sstem b dividing the equations: P t Pb 4 1 Pb 4 b b 1.7 Pb 1 1 Pb 1 P 1 P P 1 t 1 P 5 ants Note: Students could have called da t to come up with the same solution. 5 Page of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

Eponential Regression E: Find an eponential regression for the population of Las Vegas using the table below. Then predict the population in 9.

the time it takes for half of a sample of a radioactive substance to change its state P t Use E4: The half-life of radioactive Strontium is 8 das.

21 Eponential Regression E: Find an eponential regression for the population of Las Vegas using the table below. Then predict the population in ,4 4,64 15,787 58,95 54,847 9: Radioactive Deca: the process in which the number of atoms of a specific element change from a radioactive state to a nonradioactive state Half-Life: the time it takes for half of a sample of a radioactive substance to change its state P t Use E4: The half-life of radioactive Strontium is 8 das. Write an equation and predict the amount of a 5 gram sample that remains after 1 das. t Pb : Solve for b when 1 P, P.5, and t 8. Writing a Logistic Function f E5: Find a logistic function that satisfies the given conditions: Initial value = 4; limit to growth = 1; passes through the point 1,1. c c 1, a 4 1 a b, 1,1 b? You Tr: Complete the table. Isotope Half-Life (ears) Initial Quantit Amount After 1 Years 14 C 57 5 grams 14 C 57.7 gram 6 Ra 16 1 grams Reflection: Eplain how to determine if an eponential function is a growth or deca model. Page 1 of 1 Precalculus Graphical, Numerical, Algebraic Larson Chapter

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential