Exponential and Logarithmic Functions
|
|
- Chrystal Dean
- 7 years ago
- Views:
Transcription
1 Eponential and Logarithmic Functions 3 3. Eponential Functions and Their Graphs 3. Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3. Eponential and Logarithmic Equations 3.5 Eponential and Logarithmic Models In Mathematics Eponential functions involve a constant base and a variable eponent. The inverse of an eponential function is a logarithmic function. In Real Life Eponential and logarithmic functions are widel used in describing economic and phsical phenomena such as compound interest, population growth, memor retention, and deca of radioactive material. For instance, a logarithmic function can be used to relate an animal s weight and its lowest galloping speed. (See Eercise 95, page.) Juniors Bildarchiv / Alam IN CAREERS There are man careers that use eponential and logarithmic functions. Several are listed below. Astronomer Eample 7, page 0 Pschologist Eercise 3, page 53 Archeologist Eample 3, page 58 Forensic Scientist Eercise 75, page 5
2 Chapter 3 Eponential and Logarithmic Functions 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Monke Business Images Ltd/Stockbroker/PhotoLibrar What ou should learn Recognize and evaluate eponential functions with base a. Graph eponential functions and use the One-to-One Propert. Recognize, evaluate, and graph eponential functions with base e. Use eponential functions to model and solve real-life problems. Wh ou should learn it Eponential functions can be used to model and solve real-life problems. For instance, in Eercise 7 on page, an eponential function is used to model the concentration of a drug in the bloodstream. Eponential Functions So far, this tet has dealt mainl with algebraic functions, which include polnomial functions and rational functions. In this chapter, ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. These functions are eamples of transcendental functions. Definition of Eponential Function The eponential function f with base a is denoted b f a where a > 0, a, The base a is ecluded because it ields f. This is a constant function, not an eponential function. You have evaluated a for integer and rational values of. For eample, ou know that 3 and. However, to evaluate for an real number, ou need to interpret forms with irrational eponents. For the purposes of this tet, it is sufficient to think of a (where.35) as the number that has the successivel closer approimations a., a., a., a., a.,.... Eample and is an real number. Evaluating Eponential Functions Use a calculator to evaluate each function at the indicated value of. Function Value a. b. c. f f f Solution Function Value Graphing Calculator Kestrokes Displa a. f ENTER 0.9 b. f ENTER c. f ENTER Now tr Eercise 7. > > > When evaluating eponential functions with a calculator, remember to enclose fractional eponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result.
3 Section 3. Eponential Functions and Their Graphs 7 Graphs of Eponential Functions The graphs of all eponential functions have similar characteristics, as shown in Eamples, 3, and 5. Eample Graphs of a You can review the techniques for sketching the graph of an equation in Section.. g() = f() = 3 3 FIGURE 3. G() = F() = FIGURE In the same coordinate plane, sketch the graph of each function. a. f b. g Solution The table below lists some values for each function, and Figure 3. shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of g is increasing more rapidl than the graph of f. Now tr Eercise 7. The table in Eample was evaluated b hand. You could, of course, use a graphing utilit to construct tables with even more values. Eample 3 Graphs of a In the same coordinate plane, sketch the graph of each function. a. F b. G Solution The table below lists some values for each function, and Figure 3. shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of G is decreasing more rapidl than the graph of F. Now tr Eercise 9. In Eample 3, note that b using one of the properties of eponents, the functions F and G can be rewritten with positive eponents. F and G
4 8 Chapter 3 Eponential and Logarithmic Functions Comparing the functions in Eamples and 3, observe that F f and G g. Consequentl, the graph of F is a reflection (in the -ais) of the graph of f. The graphs of G and g have the same relationship. The graphs in Figures 3. and 3. are tpical of the eponential functions a and a. The have one -intercept and one horizontal asmptote (the -ais), and the are continuous. The basic characteristics of these eponential functions are summarized in Figures 3.3 and 3.. Notice that the range of an eponential function is 0,, which means that a > 0 for all values of. = a (0, ) Graph of a, a > Domain:, Range: 0, -intercept: 0, Increasing -ais is a horizontal asmptote a 0 as. Continuous FIGURE 3.3 (0, ) = a Graph of a, a > Domain:, Range: 0, -intercept: 0, Decreasing -ais is a horizontal asmptote a 0 as. Continuous FIGURE 3. From Figures 3.3 and 3., ou can see that the graph of an eponential function is alwas increasing or alwas decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Propert to solve simple eponential equations. For a > 0 and a, a a if and onl if. One-to-One Propert Eample Using the One-to-One Propert a Original equation 9 3 One-to-One Propert Solve for. b Now tr Eercise 5.
5 Section 3. Eponential Functions and Their Graphs 9 In the following eample, notice how the graph of a the graphs of functions of the form f b ± a c. can be used to sketch Eample 5 Transformations of Graphs of Eponential Functions You can review the techniques for transforming the graph of a function in Section.7. Each of the following graphs is a transformation of the graph of f 3. a. Because g 3 f, the graph of g can be obtained b shifting the graph of f one unit to the left, as shown in Figure 3.5. b. Because h 3 f, the graph of h can be obtained b shifting the graph of f downward two units, as shown in Figure 3.. c. Because k 3 f, the graph of k can be obtained b reflecting the graph of f in the -ais, as shown in Figure 3.7. d. Because j 3 f, the graph of j can be obtained b reflecting the graph of f in the -ais, as shown in Figure 3.8. g() = f() = 3 f() = 3 h() = 3 FIGURE 3.5 Horizontal shift FIGURE 3. Vertical shift f() = 3 3 k() = 3 j() = 3 f() = 3 FIGURE 3.7 Reflection in -ais FIGURE 3.8 Reflection in -ais Now tr Eercise 3. Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the -ais as a horizontal asmptote, but the transformation in Figure 3. ields a new horizontal asmptote of. Also, be sure to note how the -intercept is affected b each transformation.
6 0 Chapter 3 Eponential and Logarithmic Functions 3 (, e) f() = e (, e ) (0, ) (, e ) FIGURE 3.9 The Natural Base e In man applications, the most convenient choice for a base is the irrational number e This number is called the natural base. The function given b f e is called the natural eponential function. Its graph is shown in Figure 3.9. Be sure ou see that for the eponential function f e, e is the constant , whereas is the variable. Eample Evaluating the Natural Eponential Function Use a calculator to evaluate the function given b f e at each indicated value of. a. b. c. 0.5 d f() = e FIGURE g() = e FIGURE 3. Solution Function Value Graphing Calculator Kestrokes Displa a. f e e ENTER b. f e e ENTER c. f 0.5 e 0.5 e 0.5 ENTER.805 d. f 0.3 e 0.3 e 0.3 ENTER Now tr Eercise 33. Eample 7 Graphing Natural Eponential Functions Sketch the graph of each natural eponential function. a. f e 0. b. g e 0.58 Solution To sketch these two graphs, ou can use a graphing utilit to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figures 3.0 and 3.. Note that the graph in Figure 3.0 is increasing, whereas the graph in Figure 3. is decreasing f g Now tr Eercise.
7 Section 3. Eponential Functions and Their Graphs Applications One of the most familiar eamples of eponential growth is an investment earning continuousl compounded interest. Using eponential functions, ou can develop a formula for interest compounded n times per ear and show how it leads to continuous compounding. Suppose a principal P is invested at an annual interest rate r, compounded once per ear. If the interest is added to the principal at the end of the ear, the new balance is P P P Pr P r. This pattern of multipling the previous principal b r is then repeated each successive ear, as shown below. Year Balance After Each Compounding 0 P P P P r P P r P r r P r 3 P 3 P r P r r P r t P t P r t To accommodate more frequent (quarterl, monthl, or dail) compounding of interest, let n be the number of compoundings per ear and let t be the number of ears. Then the rate per compounding is r n and the account balance after t ears is A P r n nt. Amount (balance) with n compoundings per ear m 0 00,000 0,000 00,000,000,000 0,000,000 m m e If ou let the number of compoundings n increase without bound, the process approaches what is called continuous compounding. In the formula for n compoundings per ear, let m n r. This produces A P r n nt P r mr mrt P m mrt P m m rt. Amount with n compoundings per ear Substitute mr for n. Simplif. Propert of eponents As m increases without bound, the table at the left shows that m m e as m. From this, ou can conclude that the formula for continuous compounding is A Pe rt. Substitute e for m m.
8 Chapter 3 Eponential and Logarithmic Functions WARNING / CAUTION Be sure ou see that the annual interest rate must be written in decimal form. For instance, % should be written as 0.0. Formulas for Compound Interest After t ears, the balance A in an account with principal P and annual interest rate r (in decimal form) is given b the following formulas.. For n compoundings per ear: A P r n nt. For continuous compounding: A Pe rt Eample 8 Compound Interest A total of $,000 is invested at an annual interest rate of 9%. Find the balance after 5 ears if it is compounded a. quarterl. b. monthl. c. continuousl. Solution a. For quarterl compounding, ou have n. So, in 5 ears at 9%, the balance is A P r n nt, (5) $8,7.. Formula for compound interest Substitute for P, Use a calculator. and t. b. For monthl compounding, ou have n. So, in 5 ears at 9%, the balance is A P r n nt, (5) Formula for compound interest Substitute for P, and t. $8, Use a calculator. c. For continuous compounding, the balance is A Pe rt Formula for continuous compounding,000e 0.09(5) Substitute for P, r, and t. $8, Use a calculator. Now tr Eercise 59. In Eample 8, note that continuous compounding ields more than quarterl or monthl compounding. This is tpical of the two tpes of compounding. That is, for a given principal, interest rate, and time, continuous compounding will alwas ield a larger balance than compounding n times per ear. r, n, r, n,
9 Section 3. Eponential Functions and Their Graphs 3 Eample 9 Radioactive Deca The half-life of radioactive radium Ra is about 599 ears. That is, for a given amount of radium, half of the original amount will remain after 599 ears. After another 599 ears, one-quarter of the original amount will remain, and so on. Let represent the mass, in grams, of a quantit of radium. The quantit present after t ears, then, is 5 t 599. a. What is the initial mass (when t 0)? b. How much of the initial mass is present after 500 ears? Algebraic Solution a. 5 t Write original equation. Substitute 0 for t. 5 Simplif. So, the initial mass is 5 grams. b. 5 t Write original equation. Substitute 500 for t Simplif. Use a calculator. So, about 8. grams is present after 500 ears. Now tr Eercise 73. Graphical Solution Use a graphing utilit to graph 5 t 599. a. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 0, the value of is 5, as shown in Figure 3.. So, the initial mass is 5 grams. b. Use the value feature or the zoom and trace features of the graphing utilit to determine that when 500, the value of is about 8., as shown in Figure 3.3. So, about 8. grams is present after 500 ears FIGURE 3. FIGURE CLASSROOM DISCUSSION Identifing Eponential Functions Which of the following functions generated the two tables below? Discuss how ou were able to decide. What do these functions have in common? Are an of them the same? If so, eplain wh. a. f b. f c. f d. f e. f f. f g h 3 8 Create two different eponential functions of the forms a b and c d with -intercepts of 0, 3.
10 Chapter 3 Eponential and Logarithmic Functions 3. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. Polnomial and rational functions are eamples of functions.. Eponential and logarithmic functions are eamples of nonalgebraic functions, also called functions. 3. You can use the Propert to solve simple eponential equations.. The eponential function given b f e is called the function, and the base e is called the base. 5. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded n times per ear, ou can use the formula.. To find the amount A in an account after t ears with principal P and an annual interest rate r compounded continuousl, ou can use the formula. SKILLS AND APPLICATIONS In Eercises 7, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value f 0.9 f.3 f 5 f 3 5 g 5000 f In Eercises 3, match the eponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) (0, ) (0, ) (d) 3. f. f 5. f. f (0, ( (0, ) In Eercises 7, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 7. f 8. f 9. f 0. f. f. f 3 3 In Eercises 3 8, use the graph of f to describe the transformation that ields the graph of g f 3, f, f, f 0, f 7, f 0.3, g 3 g 3 g 3 3 g 0 g 7 g In Eercises 9 3, use a graphing utilit to graph the eponential function In Eercises 33 38, evaluate the function at the indicated value of. Round our result to three decimal places. Function Value 33. h e 3 3. f e f e f.5e f 5000e f 50e
11 Section 3. Eponential Functions and Their Graphs 5 In Eercises 39, use a graphing utilit to construct a table of values for the function. Then sketch the graph of the function. 39. f e 0. f e. f 3e. f e f e. f e 5 In Eercises 5 50, use a graphing utilit to graph the eponential function s t e 0.t 8. s t 3e 0.t 9. g e 50. h e In Eercises 5 58, use the One-to-One Propert to solve the equation for e 3 e 3 5. e e 57. e 3 e 58. e e 5 COMPOUND INTEREST In Eercises 59, complete the table to determine the balance A for P dollars invested at rate r for t ears and compounded n times per ear. n 35 Continuous A 59. P $500, r %, t 0 ears 0. P $500, r 3.5%, t 0 ears. P $500, r %, t 0 ears. P $000, r %, t 0 ears COMPOUND INTEREST In Eercises 3, complete the table to determine the balance A for $,000 invested at rate r for t ears, compounded continuousl. t A 3. r %. r % 5. r.5%. r 3.5% 7. TRUST FUND On the da of a child s birth, a deposit of $30,000 is made in a trust fund that pas 5% interest, compounded continuousl. Determine the balance in this account on the child s 5th birthda. 8. TRUST FUND A deposit of $5000 is made in a trust fund that pas 7.5% interest, compounded continuousl. It is specified that the balance will be given to the college from which the donor graduated after the mone has earned interest for 50 ears. How much will the college receive? 9. INFLATION If the annual rate of inflation averages % over the net 0 ears, the approimate costs C of goods or services during an ear in that decade will be modeled b C t P.0 t, where t is the time in ears and P is the present cost. The price of an oil change for our car is presentl $3.95. Estimate the price 0 ears from now. 70. COMPUTER VIRUS The number V of computers infected b a computer virus increases according to the model V t 00e.05t, where t is the time in hours. Find the number of computers infected after (a) hour, (b).5 hours, and (c) hours. 7. POPULATION GROWTH The projected populations of California for the ears 05 through 030 can be modeled b P 3.9e t, where P is the population (in millions) and t is the time (in ears), with t 5 corresponding to 05. (Source: U.S. Census Bureau) (a) Use a graphing utilit to graph the function for the ears 05 through 030. (b) Use the table feature of a graphing utilit to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California eceed 50 million? 7. POPULATION The populations P (in millions) of Ital from 990 through 008 can be approimated b the model P 5.8e 0.005t, where t represents the ear, with t 0 corresponding to 990. (Source: U.S. Census Bureau, International Data Base) (a) According to the model, is the population of Ital increasing or decreasing? Eplain. (b) Find the populations of Ital in 000 and 008. (c) Use the model to predict the populations of Ital in 05 and RADIOACTIVE DECAY Let Q represent a mass of radioactive plutonium 39 Pu (in grams), whose halflife is,00 ears. The quantit of plutonium present after t ears is Q t,00. (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 75,000 ears. (c) Use a graphing utilit to graph the function over the interval t 0 to t 50,000.
12 Chapter 3 Eponential and Logarithmic Functions 7. RADIOACTIVE DECAY Let Q represent a mass of carbon C (in grams), whose half-life is 575 ears. The quantit of carbon present after t ears is (a) Determine the initial quantit (when t 0). (b) Determine the quantit present after 000 ears. (c) Sketch the graph of this function over the interval t 0 to t 0, DEPRECIATION After t ears, the value of a wheelchair conversion van that originall cost $30,500 7 depreciates so that each ear it is worth 8 of its value for the previous ear. (a) Find a model for V t, the value of the van after t ears. (b) Determine the value of the van ears after it was purchased. 7. DRUG CONCENTRATION Immediatel following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After t hours, the concentration is 75% of the level of the previous hour. (a) Find a model for C t, the concentration of the drug after t hours. (b) Determine the concentration of the drug after 8 hours. EXPLORATION TRUE OR FALSE? In Eercises 77 and 78, determine whether the statement is true or false. Justif our answer. 77. The line is an asmptote for the graph of f Q 0 t 575. e 7,80 99,990 THINK ABOUT IT In Eercises 79 8, use properties of eponents to determine which functions (if an) are the same. 79. f f g 3 9 g h 9 3 h 8. f 8. f e 3 g g e 3 h h e Graph the functions given b 3 and and use the graphs to solve each inequalit. (a) < 3 (b) > 3 8. Use a graphing utilit to graph each function. Use the graph to find where the function is increasing and decreasing, and approimate an relative maimum or minimum values. (a) f e (b) g GRAPHICAL ANALYSIS Use a graphing utilit to graph and e in the same viewing window. Using the trace feature, eplain what happens to the graph of as increases. 8. GRAPHICAL ANALYSIS Use a graphing utilit to graph f 0.5 and in the same viewing window. What is the relationship between f and g as increases and decreases without bound? 87. GRAPHICAL ANALYSIS Use a graphing utilit to graph each pair of functions in the same viewing window. Describe an similarities and differences in the graphs. (a) (b) 3, 3, 88. THINK ABOUT IT Which functions are eponential? (a) 3 (b) 3 (c) 3 (d) 89. COMPOUND INTEREST Use the formula A P r n nt g e 0.5 to calculate the balance of an account when P $3000, r %, and t 0 ears, and compounding is done (a) b the da, (b) b the hour, (c) b the minute, and (d) b the second. Does increasing the number of compoundings per ear result in unlimited growth of the balance of the account? Eplain. 90. CAPSTONE The figure shows the graphs of, e, 0,, e, and 0. Match each function with its graph. [The graphs are labeled (a) through (f).] Eplain our reasoning. b a c 0 8 PROJECT: POPULATION PER SQUARE MILE To work an etended application analzing the population per square mile of the United States, visit this tet s website at academic.cengage.com. (Data Source: U.S. Census Bureau) d e f
SECTION 5-1 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationExponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014
Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,
More informationExponential equations will be written as, where a =. Example 1: Determine a formula for the exponential function whose graph is shown below.
.1 Eponential and Logistic Functions PreCalculus.1 EXPONENTIAL AND LOGISTIC FUNCTIONS 1. Recognize eponential growth and deca functions 2. Write an eponential function given the -intercept and another
More informationM122 College Algebra Review for Final Exam
M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant
More informationTHE POWER RULES. Raising an Exponential Expression to a Power
8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationPartial Fractions. and Logistic Growth. Section 6.2. Partial Fractions
SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationAlgebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED
Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8.
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationSection 0.3 Power and exponential functions
Section 0.3 Power and eponential functions (5/6/07) Overview: As we will see in later chapters, man mathematical models use power functions = n and eponential functions =. The definitions and asic properties
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationShake, Rattle and Roll
00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More information6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH
6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions
More informationSTUDENT TEXT AND HOMEWORK HELPER
UNIT 4 EXPONENTIAL FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles Allan E. Bellman Basia Hall William G. Handlin, Sr. Dan Kenned Stuart J. Murph Grant Wiggins Boston, Massachusetts
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic
More informationLinear Inequality in Two Variables
90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationPre Calculus Math 40S: Explained!
Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph
More informationWhy should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationIndiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.
STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationExponential, Logistic, and Logarithmic Functions
5144_Demana_Ch03pp275-348 1/13/06 12:19 PM Page 275 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic
More informationFind the Relationship: An Exercise in Graphing Analysis
Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationMidterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationExponential Functions, Logarithms, and e
chapter 3 Starry Night, painted by Vincent Van Gogh in 889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Eponential Functions, Logarithms, and e This chapter focuses
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More information135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More informationEQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM
. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationDirect Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship
6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationChapter 4: Exponential and Logarithmic Functions
Chapter 4: Eponential and Logarithmic Functions Section 4.1 Eponential Functions... 15 Section 4. Graphs of Eponential Functions... 3 Section 4.3 Logarithmic Functions... 4 Section 4.4 Logarithmic Properties...
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationThe Slope-Intercept Form
7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph
More informationThe numerical values that you find are called the solutions of the equation.
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
More informationFunctions and Their Graphs
3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More information2.5 Library of Functions; Piecewise-defined Functions
SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationPolynomial and Rational Functions
Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationComplex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No real-number solutions. From the definition, it follows that i 2 1.
7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More information4.2 Applications of Exponential Functions
. Applications of Eponential Functions In this section ou will learn to: find eponential equations usin raphs solve eponential rowth and deca problems use loistic rowth models Eample : The raph of is the
More informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions
More information2.3 TRANSFORMATIONS OF GRAPHS
78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he
More informationMath Review. The second part is a refresher of some basic topics for those who know how but lost their fluency over the years.
Math Review The Math Review is divided into two parts: I. The first part is a general overview of the math classes, their sequence, basic content, and short quizzes to see if ou are prepared to take a
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved.
3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.
More information3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533
Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationEllington High School Principal
Mr. Neil Rinaldi Ellington High School Principal 7 MAPLE STREET ELLINGTON, CT 0609 Mr. Dan Uriano (860) 896- Fa (860) 896-66 Assistant Principal Mr. Peter Corbett Lead Teacher Mrs. Suzanne Markowski Guidance
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationCollege Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science
College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 91 Outline 1 Exponential
More informationSECTION 2-2 Straight Lines
- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More information