1.3 Exponential Functions
|
|
- Abigayle Ellis
- 6 years ago
- Views:
Transcription
1 Section. Eponential Functions. Eponential Functions You will be to model eponential growth and decay with functions of the form y = k a and recognize eponential growth and decay in algebraic, numerical, and graphical representations. Eponential functions Rules of eponents Applications of eponential functions (growth and decay) Compound interest The number e Eponential Growth Table. shows the growth of $00 invested in 996 at an interest rate of 5.5%, compounded annually. Year TABLE. Savings Account Growth Amount (dollars) Increase (dollars) 00 00(.055) 00( (.055 = =.0 = 7. 00(.055) = y=x After the first year, the value of the account is always.055 times its value in the previous year. After n years, the value is y = 00 (.055) n. Compound interest provides an eample of eponential growth and is modeled by a function of the form y P a, where P is the initial investment (called the principal) and a is equal to plus the interest rate epressed as a decimal. The equation y P a, a > 0, a #, identifies a family of functions called eponential functions. Notice that the ratio of consecutive amounts in Table. is always the same:.0/05.0 = 7./.0 =.88/ This fact is an important feature of eponential curves that has widespread application, as we will see. EXPLORATION Eponential Functions 6] by [-, 6] (a) Graph the function y a for a =,, 5, in a [ 5, 5] by [, 5] viewing window. For what values of is it true that < < For what values of is it true that > > For what values of is it true that Graph the function y = (l/a) = a- for a Repeat parts for the functions in part 5. DEFINITION Eponential Function Let a be a positive real number other than. The function f() = a is the eponential function with base a. 6] by [-, 6] (b) Figure.0 A graph of (a) y = and The domain of f() = a is ( 00, 00) and the range is (0, 00). If a >, the graph of f looks like the graph of y = in Figure.0a. If 0 < a < l, the graph of f looks like the graph of y = - in Figure.0b.
2 Chapter Prerequisites for Calculus EXAMPLE Graph the function y = Graphing an Eponential Function ). State its domain and range. Figure. shows the graph of the function y. It appears that the domain is ( 00, 00) The range is (, 00) because () > 0 for all. Now Try Eercise. [-5, 5] by [-5, 5] Figure. The graph of y (Eample l) Y = (X ) -. EXAMPLE Find the zeros of f() Finding Zeros 5.5 graphically. Figure.a suggests that f has a zero between = and =, closer to. We can use our grapher to find that the zero is approimately.756 (Figure.b). Now Try Eercise 9. Eponential functions obey the rules for eponents. Rules for Eponents y = 5-.5X 5] by [-8, 8] (a) If a > 0 and b > 0, the following hold for all real numbers and y. = a +y a.. (a)y. a. b = (ab) 5. a b a Zero X=l.7S6U708 [-5, 5] by [-8, 8] (b) Figure (a) Agraphoff() 5.5. (b) Showing the use of the ZERO feature to approimate the zero of f. (Eample ) t/0 Eponential Decay Eponential functions can also model phenomena that produce a decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a nonradioactive state by emitting energy in the form of radiation. EXAMPLE Modeling Radioactive Decay Suppose the half-life of a certain radioactive substance is 0 days and that there are 5 grams present initially. When will there be only gram of the substance remaining? The number of grams remaining after 0 days is 5 5 The number of grams remaining after 0 days is 5 5 Intersection =V6.k856 [0, 80 by [-, 5] Figure. (Eample ) The function y = 5( /)t/0 models the mass in grams of the radioactive substance after t days. Figure. shows that the graphs of Yl = 5( /) t/0 and h = I (for I gram) intersect when t is approimately 6.. There will be gram of the radioactive substance left after approimately 6. days, or about 6 days 0.5 hours. Now Try Eercise.
3 Section. Eponential Functions 5 Compound interest investments, population growth, and radioactive decay are all eamples of eponential growth and decay. DEFINITIONS Eponential Growth, Eponential Decay The function y = k a k > 0 is a model for eponential growth if a > l, and a model for eponential decay if 0 < a <. Compound Interest One common application of eponential growth in the financial world is the compounding effect of accrued interest in a savings account. We saw at the beginning of this section that an account paying 5.5% annual interest would multiply by a factor of I.055 every year, so an account starting with principal P would be worth.055 after t years. Some accounts pay interest multiple times per year, which increases the compounding effect. An account earning 6% annual interest compounded monthly would pay one-twelfth of the interest each month, effectively a monthly interest rate of 0.5%. Thus, after t years, the account would be worth P( I.005) t. The general formula is given below. Compound Interest Formula If an account begins with principal P and earns an annual interest rate r compounded n times per year, then the value of the account in t years is n nt EXAMPLE Compound Interest Bernie deposits $500 in an account earning 5.% annual interest. Find the value of the account in 0 years if the interest is compounded (a) annually; (b) quarterly; (c) monthly. (a) 500( , so the account is worth $ (0) (b) , so the account is worth $7.55. O.05 (c) , so the account is worth $8.8. Now Try Eercises 5 and 6. The Number e You may have noticed in Eample that the value of the account becomes greater as the number of interest payments per year increases, but the value does not appear to be increasing without bound. In fact, a bank can even offer to compound the interest continuously without paying that much more per year. This is because the number that serves as the base of the eponential function in the compound interest formula has a finite limit as n approaches infinity: n r n 00
4 6 Chapter Prerequisites for Calculus y = I/X)X You can get an estimate of the number e by looking at values of + increasing values of, as shown graphically and numerically in Figure.. This number e, which is approimately.788, is one of the most important numbers in mathematics. Like 7, it is an irrational number that shows up in many different contets, some of them quite surprising. You will encounter several of them in this course. One place that e appears is in the formula for continuously compounded interest qooo sooo [-0, 0 by [-5, Figure. A graph and table of values for f() = ( + l/) both suggest that as + 00, f() e.78. Continuously Compounded Interest Formula If an account begins with principal P and earns an annual interest rate r compounded continuously, then the value of the account in t years is = Pe rt EXAMPLE 5 Continuously Compounded Interest Bernice deposits $500 in an account earning 5.% annual interest. Find the value of the account in 0 years if the interest is compounded (a) daily; (b) continuously (0) (a) , so the account is worth $ , so the account is worth $90.0. Notice that the difference between "daily" and "continuously" for Bernice is 7 cents! Now Try Eercise 7. Quick Review. (For help, go to Section..) Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises, evaluate the epression. Round your answers to decimal places.. 5/.+6 In Eercises 6, solve the equation. Round your answers to decimal places. = = In Eercises 7 and 8, find the value of investing P dollars for n years with the interest rate r compounded annually. 7. P = $500, r =.75%, n = 5 years 8. P = $000, r = 6.%, n = years In Eercises 9 and 0, simplify the eponential epression. ( Y) ab- ac- - CIO, (y) b Section. Eercises Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises, graph the function. State its domain and range..y=-x+. In Eercises 5 8, rewrite the eponential epression to have the indi - cated base r, base 6. 6, base base base
5 Section. Eponential Functions 7 In Eercises 9, use a graph to find the zeros of the function. 9. f( ) 0. f() = e-. f() = X 05. f() - In Eercises 8, match the function with its graph. Try to do it without using your grapher.. Y = - 6.y = (a) (c) 5, y = (8.» =.5X - (b) (d) 5. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded annually. 6. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded monthly. 7. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded continuously. 8. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded annually. 9. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded daily. 0. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded continuously.. Cholera Bacteria Suppose that a colony of bacteria starts with bacterium and doubles in number every half hour. How many bacteria will the colony contain at the end of h?. Eliminating a Disease Suppose that in any given year, the number of cases of a disease is reduced by 0%. If there are 0,000 cases today, how many years will it take (a) to reduce the number of cases to 000? (b) to eliminate the disease; that is, to reduce the number of cases to less than? (e) In Eercises 9, use an eponential model to solve the problem. 9. Population Growth The population of Knoville is 500,000 and is increasing at the rate of.75% each year. Approimately when will the population reach million? 0. Population Growth The population of Silver Run in the year 890 was 650. Assume the population increased at a rate of.75% per year. (a) Estimate the population in 95 and 90. (b) Approimately when did the population reach 50,000?. Half-life The approimate half-life of titanium- is 6 years. How long will it take a sample to lose (a) 50% of its. titanium-? (b) 75% of its titanium-? c Half-life The amount of silicon- in a sample will decay from 8 grams to 7 grams in approimately 0 years. What is the approimate half-life of silicon-?. Radioactive Decay The half-life of phosphorus- is about days. There are 6.6 grams present initially. (a) Epress the amount of phosphorus- remaining as a function of time t. (b) When will there be gram remaining?. Finding Time If John invests $00 in a savings account with a 6% interest rate compounded annually, how long will it take until John's account has a balance of $50? Group Activity In Eercises 6, copy and complete the table for the function. y=r-. y = + 5. Y' y y 9 Change ( Ay) Change ( Ay) Change ( Ay)
6 8 Chapter Prerequisites for Calculus 6. y = e y Ratio (Yi/Yi-l ) 7. Writing to Learn Eplain how the change Ay is related to the slopes of the lines in Eercises and. If the changes in are constant for a linear function, what would you conclude about the corresponding changes in y? 8. Bacteria Growth The number of bacteria in a petri dish culture after t hours is (a) What was the initial number of bacteria present? (b) How many bacteria are present after 6 hours? (c) Approimately when will the number of bacteria be 00? Estimate the doubling time of the bacteria. 9. The graph of the eponential function y = a b X is shown below. Find a and b. Standardized Test Questions You may use a graphing calculator to solve the following problems.. True or False The number - is negative. Justify your answer.. True or False If = Y, then a = 6. Justify your answer.. Multiple Choice John invests $00 at.5% compounded annually. About how long will it take for John's investment to double in value? (A) 6yr (B) 9yr (C) yr (D) 6yr (E) 0 yr. Multiple Choice Which of the following gives the domain of Y = e - (A) ( 00, 00) (B) [, 00) (C) [, 00) (D) ( 00, ] 5. Multiple Choice Which of the following gives the range of y = K'? (A) ( 00, 00) (B) (-00, ) (C) 00) (D) (-00, ] (E) all reals 6. Multiple Choice Which of the following gives the best approimation for the zero of f() e? = -.86 (B) = 0.86 (C) =.86 (E) There are no zeros. 0. The graph of the eponential function y = a below. Find a and b. (, o. 5) b X is shown Eploration 7. Let = and Y (a) Graph and Y in [ 5, 5] by [, 0]. How many times do you think the two graphs cross? (b) Compare the corresponding changes in and Y as changes from I to, to, and so on. How large must be for the changes in Y to overtake the changes in? (c) Solve for : (d) Solve for : < Etending the Ideas In Eercises 8 and 9, assume that the graph of the eponential function f() = k a passes through the two points. Find the values of a and k. {8. (,.5), (-, 0.5) (9. (,.5), (-, 6) Quick Quiz for AP* Preparation: Sections.. You may use a graphing calculator to solve the following problems.. Multiple Choice Which of the following gives an equation for the line through (, l ) and parallel to the line y=-+? 7 5. Multiple Choice If f() = + I and g() = which of the following gives (f o g )()? (C)9 (D) 0 (E) 5. Multiple Choice The half-life of a certain radioactive substance is 8 hr. There are 5 grams present initially. Which of the following gives the best approimation when there will be gram remaining? (B) 0 (C) 5 (D) 6 (E) 9. Free Response Let f() (a) Find the domain of f. (b) Find the range of f. (c) Find the zeros of f.
1.3 Exponential Functions
22 Chapter 1 Prerequisites for Calculus 1.3 Exponential Functions What you will learn about... Exponential Growth Exponential Decay Applications The Number e and why... Exponential functions model many
More informationMA Lesson 14 Notes Summer 2016 Exponential Functions
Solving Eponential Equations: There are two strategies used for solving an eponential equation. The first strategy, if possible, is to write each side of the equation using the same base. 3 E : Solve:
More informationAssignment #3; Exponential Functions
AP Calculus Assignment #3; Exponential Functions Name: The equation identifies a family of functions called exponential functions. Notice that the ratio of consecutive amounts of outputs always stay the
More informationFunctions and Logarithms
36 Chapter Prerequisites for Calculus.5 Inverse You will be able to find inverses of one-to-one functions and will be able to analyze logarithmic functions algebraically, graphically, and numerically as
More informationMath M111: Lecture Notes For Chapter 10
Math M: Lecture Notes For Chapter 0 Sections 0.: Inverse Function Inverse function (interchange and y): Find the equation of the inverses for: y = + 5 ; y = + 4 3 Function (from section 3.5): (Vertical
More informationUnit 5: Exponential and Logarithmic Functions
71 Rational eponents Unit 5: Eponential and Logarithmic Functions If b is a real number and n and m are positive and have no common factors, then n m m b = b ( b ) m n n Laws of eponents a) b) c) d) e)
More informationChapter 3. Exponential and Logarithmic Functions. Selected Applications
Chapter 3 Eponential and Logarithmic Functions 3. Eponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Eponential and Logarithmic Equations
More informationCHAPTER 6. Exponential Functions
CHAPTER 6 Eponential Functions 6.1 EXPLORING THE CHARACTERISTICS OF EXPONENTIAL FUNCTIONS Chapter 6 EXPONENTIAL FUNCTIONS An eponential function is a function that has an in the eponent. Standard form:
More informationChapter 2 Exponentials and Logarithms
Chapter Eponentials and Logarithms The eponential function is one of the most important functions in the field of mathematics. It is widely used in a variety of applications such as compounded interest,
More information8-1 Exploring Exponential Models
8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =
More informationChapter 8 Prerequisite Skills
Chapter 8 Prerequisite Skills BLM 8. How are 9 and 7 the same? How are they different?. Between which two consecutive whole numbers does the value of each root fall? Which number is it closer to? a) 8
More informationMATH 1431-Precalculus I
MATH 43-Precalculus I Chapter 4- (Composition, Inverse), Eponential, Logarithmic Functions I. Composition of a Function/Composite Function A. Definition: Combining of functions that output of one function
More informationSummer MA Lesson 20 Section 2.7 (part 2), Section 4.1
Summer MA 500 Lesson 0 Section.7 (part ), Section 4. Definition of the Inverse of a Function: Let f and g be two functions such that f ( g ( )) for every in the domain of g and g( f( )) for every in the
More informationHonors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals
Honors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals By the end of this chapter, you should be able to Graph exponential growth functions. (8.1) Graph exponential
More information3.2 Logarithmic Functions and Their Graphs
96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that
More information3.2. Exponential and Logistic Modeling. Finding Growth and Decay Rates. What you ll learn about
290 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.2 Eponential and Logistic Modeling What you ll learn about Constant Percentage Rate and Eponential Functions Eponential Growth and Decay
More informationUnit 8: Exponential & Logarithmic Functions
Date Period Unit 8: Eponential & Logarithmic Functions DAY TOPIC ASSIGNMENT 1 8.1 Eponential Growth Pg 47 48 #1 15 odd; 6, 54, 55 8.1 Eponential Decay Pg 47 48 #16 all; 5 1 odd; 5, 7 4 all; 45 5 all 4
More informationChapter 2 Functions and Graphs
Chapter 2 Functions and Graphs Section 5 Exponential Functions Objectives for Section 2.5 Exponential Functions The student will be able to graph and identify the properties of exponential functions. The
More information1) Now there are 4 bacteria in a dish. Every day we have two more bacteria than on the preceding day.
Math 093 and 117A Linear Functions and Eponential Functions Pages 1, 2, and 3 are due the class after eam 1 Your Name If you need help go to the Math Science Center in MT 02 For each of problems 1-4, do
More informationExponential and Logarithmic Functions
Exponential and Logarithmic Functions Learning Targets 1. I can evaluate, analyze, and graph exponential functions. 2. I can solve problems involving exponential growth & decay. 3. I can evaluate expressions
More informationOBJECTIVE 4 EXPONENTIAL FORM SHAPE OF 5/19/2016. An exponential function is a function of the form. where b > 0 and b 1. Exponential & Log Functions
OBJECTIVE 4 Eponential & Log Functions EXPONENTIAL FORM An eponential function is a function of the form where > 0 and. f ( ) SHAPE OF > increasing 0 < < decreasing PROPERTIES OF THE BASIC EXPONENTIAL
More informationMath 119 Main Points of Discussion
Math 119 Main Points of Discussion 1. Solving equations: When you have an equation like y = 3 + 5, you should see a relationship between two variables, and y. The graph of y = 3 + 5 is the picture of this
More informationPrerequisites for Calculus
CHAPTER Prerequisites for Calculus. Lines. Functions and Graphs.3 Eponential Functions.4 Parametric Equations.5 Functions and Logarithms Eponential functions are used to model situations in which growth
More information( ) ( ) x. The exponential function f(x) with base b is denoted by x
Page of 7 Eponential and Logarithmic Functions Eponential Functions and Their Graphs: Section Objectives: Students will know how to recognize, graph, and evaluate eponential functions. The eponential function
More informationExponential and Logarithmic Functions. Exponential Functions. Example. Example
Eponential and Logarithmic Functions Math 1404 Precalculus Eponential and 1 Eample Eample Suppose you are a salaried employee, that is, you are paid a fied sum each pay period no matter how many hours
More informationExponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
More informationFunctions. Contents. fx ( ) 2 x with the graph of gx ( ) 3 x x 1
Functions Contents WS07.0 Applications of Sequences and Series... Task Investigating Compound Interest... Task Reducing Balance... WS07.0 Eponential Functions... 4 Section A Activity : The Eponential Function,
More informationExponential Growth and Decay Functions (Exponent of t) Read 6.1 Examples 1-3
CC Algebra II HW #42 Name Period Row Date Section 6.1 1. Vocabulary In the eponential growth model Eponential Growth and Decay Functions (Eponent of t) Read 6.1 Eamples 1-3 y = 2.4(1.5), identify the initial
More informationChapter 6: Exponential Functions
Chapter 6: Eponential Functions Section 6.1 Chapter 6: Eponential Functions Section 6.1: Eploring Characteristics of Eponential Functions Terminology: Eponential Functions: A function of the form: y =
More informationMAC 1105 Chapter 6 (6.5 to 6.8) --Sullivan 8th Ed Name: Practice for the Exam Kincade
MAC 05 Chapter 6 (6.5 to 6.8) --Sullivan 8th Ed Name: Practice for the Eam Date: Kincade MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the properties
More informationCOLLEGE ALGEBRA. Practice Problems Exponential and Logarithm Functions. Paul Dawkins
COLLEGE ALGEBRA Practice Problems Eponential and Logarithm Functions Paul Dawkins Table of Contents Preface... ii Eponential and Logarithm Functions... Introduction... Eponential Functions... Logarithm
More information) approaches e
COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural
More informationThe units on the average rate of change in this situation are. change, and we would expect the graph to be. ab where a 0 and b 0.
Lesson 9: Exponential Functions Outline Objectives: I can analyze and interpret the behavior of exponential functions. I can solve exponential equations analytically and graphically. I can determine the
More informationEssential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically
6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one
More informationMaterials: Hw #9-6 answers handout; Do Now and answers overhead; Special note-taking template; Pair Work and answers overhead; hw #9-7
Pre-AP Algebra 2 Unit 9 - Lesson 7 Compound Interest and the Number e Objectives: Students will be able to calculate compounded and continuously compounded interest. Students know that e is an irrational
More informationChapter 8. Exponential and Logarithmic Functions
Chapter 8 Eponential and Logarithmic Functions Lesson 8-1 Eploring Eponential Models Eponential Function The general form of an eponential function is y = ab. Growth Factor When the value of b is greater
More informationDo we have any graphs that behave in this manner that we have already studied?
Boise State, 4 Eponential functions: week 3 Elementary Education As we have seen, eponential functions describe events that grow (or decline) at a constant percent rate, such as placing capitol in a savings
More informationHW#1. Unit 4B Logarithmic Functions HW #1. 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7
HW#1 Name Unit 4B Logarithmic Functions HW #1 Algebra II Mrs. Dailey 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7 2) If the graph of y =6 x is reflected
More informationTwo-Year Algebra 2 A Semester Exam Review
Semester Eam Review Two-Year Algebra A Semester Eam Review 05 06 MCPS Page Semester Eam Review Eam Formulas General Eponential Equation: y ab Eponential Growth: A t A r 0 t Eponential Decay: A t A r Continuous
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More informationPart 4: Exponential and Logarithmic Functions
Part 4: Exponential and Logarithmic Functions Chapter 5 I. Exponential Functions (5.1) II. The Natural Exponential Function (5.2) III. Logarithmic Functions (5.3) IV. Properties of Logarithms (5.4) V.
More information6-1 LESSON MASTER. Name. Skills Objective A In 1 4, evaluate without a calculator In 5 and 6, rewrite each using a radical sign.
6-1 Skills Objective A In 1 4, evaluate without a calculator. 1. 2. 64 1 3 3 3. -343 4. 36 1 2 4 16 In 5 and 6, rewrite each using a radical sign. 5. ( 2 ) 1 4 m 1 6 In 7 10, rewrite without a radical
More informationLogarithmic and Exponential Equations and Inequalities College Costs
Logarithmic and Exponential Equations and Inequalities ACTIVITY 2.6 SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Create Representations Wesley is researching college costs. He is considering
More informationExponential and Logarithmic Functions
Chapter 7 Eponential and Logarithmic Functions Specific Epectations Identify, through investigations, using graphing calculators or graphing software, the key properties of eponential functions of the
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More informationChapter 11 Logarithms
Chapter 11 Logarithms Lesson 1: Introduction to Logs Lesson 2: Graphs of Logs Lesson 3: The Natural Log Lesson 4: Log Laws Lesson 5: Equations of Logs using Log Laws Lesson 6: Exponential Equations using
More information2.6 Solving Inequalities Algebraically and Graphically
7_006.qp //07 8:0 AM Page 9 Section.6 Solving Inequalities Algebraically and Graphically 9.6 Solving Inequalities Algebraically and Graphically Properties of Inequalities Simple inequalities were reviewed
More informationMath 103 Final Exam Review Problems Rockville Campus Fall 2006
Math Final Eam Review Problems Rockville Campus Fall. Define a. relation b. function. For each graph below, eplain why it is or is not a function. a. b. c. d.. Given + y = a. Find the -intercept. b. Find
More informationSection Exponential Functions
121 Section 4.1 - Exponential Functions Exponential functions are extremely important in both economics and science. It allows us to discuss the growth of money in a money market account as well as the
More informationLog1 Contest Round 2 Theta Logarithms & Exponents. 4 points each
5 Log Contest Round Theta Logarithms & Eponents Name: points each Simplify: log log65 log6 log6log9 log5 Evaluate: log Find the sum:... A square has a diagonal whose length is feet, enclosed by the square.
More informationLogarithmic, Exponential, and Other Transcendental Functions. Copyright Cengage Learning. All rights reserved.
5 Logarithmic, Exponential, and Other Transcendental Functions Copyright Cengage Learning. All rights reserved. 5.5 Bases Other Than e and Applications Copyright Cengage Learning. All rights reserved.
More informationDo you know how to find the distance between two points?
Some notation to understand: is the line through points A and B is the ray starting at point A and extending (infinitely) through B is the line segment connecting points A and B is the length of the line
More informationFirst Order Differential Equations
First Order Differential Equations CHAPTER 7 7.1 7.2 SEPARABLE DIFFERENTIAL 7.3 DIRECTION FIELDS AND EULER S METHOD 7.4 SYSTEMS OF FIRST ORDER DIFFERENTIAL Slide 1 Exponential Growth The table indicates
More informationEXPONENTIAL FUNCTIONS REVIEW PACKET FOR UNIT TEST TOPICS OF STUDY: MEMORIZE: General Form of an Exponential Function y = a b x-h + k
EXPONENTIAL FUNCTIONS REVIEW PACKET FOR UNIT TEST TOPICS OF STUDY: o Recognizing Eponential Functions from Equations, Graphs, and Tables o Graphing Eponential Functions Using a Table of Values o Identifying
More informationwhere is a constant other than ( and ) and
Section 12.1: EXPONENTIAL FUNCTIONS When you are done with your homework you should be able to Evaluate eponential functions Graph eponential functions Evaluate functions with base e Use compound interest
More informationChapter 6: Exponential and Logarithmic Functions
Section 6.1: Algebra and Composition of Functions #1-9: Let f(x) = 2x + 3 and g(x) = 3 x. Find each function. 1) (f + g)(x) 2) (g f)(x) 3) (f/g)(x) 4) ( )( ) 5) ( g/f)(x) 6) ( )( ) 7) ( )( ) 8) (g+f)(x)
More informationLESSON #25 - EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II
1 LESSON #5 - EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II Eponential functions are very important in modeling a variety of real world phenomena because certain things either
More informationSolving Exponential Equations (Applied Problems) Class Work
Solving Exponential Equations (Applied Problems) Class Work Objective: You will be able to solve problems involving exponential situations. Quick Review: Solve each equation for the variable. A. 2 = 4e
More information17 Exponential and Logarithmic Functions
17 Exponential and Logarithmic Functions Concepts: Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions Exponential Growth
More informationHomework 2 Solution Section 2.2 and 2.3.
Homework Solution Section. and.3...4. Write each of the following autonomous equations in the form x n+1 = ax n + b and identify the constants a and b. a) x n+1 = 4 x n )/3. x n+1 = 4 x n) 3 = 8 x n 3
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More informationLogarithms. Professor Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University
Logarithms Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math211 1. Definition of Logarithm For a > 0, a 1,
More informationDo you know how to find the distance between two points?
Some notation to understand: is the line through points A and B is the ray starting at point A and extending (infinitely) through B is the line segment connecting points A and B is the length of the line
More informationHonors Pre Calculus Worksheet 3.1. A. Find the exponential equation for the given points, and then sketch an accurate graph (no calculator). 2.
Honors Pre Calculus Worksheet 3.1 A. Find the eponential equation for the given points, and then sketch an accurate graph (no calculator). 1., 3, 9 1,. ( 1, ),, 9 1 1 1 8 8 B. Sketch a graph the following
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Write the equation of the line that goes through the points ( 3, 7) and (4, 5)
More informationEvaluate the expression using the values given in the table. 1) (f g)(6) x f(x) x g(x)
M60 (Precalculus) ch5 practice test Evaluate the expression using the values given in the table. 1) (f g)(6) 1) x 1 4 8 1 f(x) -4 8 0 15 x -5-4 1 6 g(x) 1-5 4 8 For the given functions f and g, find the
More informationMath 11A Graphing Exponents and Logs CLASSWORK Day 1 Logarithms Applications
Log Apps Packet Revised: 3/26/2012 Math 11A Graphing Eponents and Logs CLASSWORK Day 1 Logarithms Applications Eponential Function: Eponential Growth: Asymptote: Eponential Decay: Parent function for Eponential
More informationevery hour 8760 A every minute 525,000 A continuously n A
In the previous lesson we introduced Eponential Functions and their graphs, and covered an application of Eponential Functions (Compound Interest). We saw that when interest is compounded n times per year
More informationComplete Week 18 Package
Complete Week 18 Package Jeanette Stein Table of Contents Unit 4 Pacing Chart -------------------------------------------------------------------------------------------- 1 Day 86 Bellringer --------------------------------------------------------------------------------------------
More informationMath 1 Exponential Functions Unit 2018
1 Math 1 Exponential Functions Unit 2018 Points: /10 Name: Graphing Exponential Functions/Domain and Range Exponential Functions (Growth and Decay) Tables/Word Problems Linear vs Exponential Functions
More informationSample Questions. Please be aware that the worked solutions shown are possible strategies; there may be other strategies that could be used.
Sample Questions Students who achieve the acceptable standard should be able to answer all the following questions, ecept for any part of a question labelled SE. Parts labelled SE are appropriate eamples
More informationFunctions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f.
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing
More informationExponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
3 Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 3.1 Exponential Functions and Their Graphs Copyright Cengage Learning. All rights reserved. What You Should Learn
More informationMATH 1113 Exam 2 Review. Spring 2018
MATH 1113 Exam 2 Review Spring 2018 Section 3.1: Inverse Functions Topics Covered Section 3.2: Exponential Functions Section 3.3: Logarithmic Functions Section 3.4: Properties of Logarithms Section 3.5:
More information, identify what the letters P, r, n and t stand for.
1.In the formula At p 1 r n nt, identify what the letters P, r, n and t stand for. 2. Find the exponential function whose graph is given f(x) = a x 3. State the domain and range of the function (Enter
More informationName: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions
Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following:
More informationAlgebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents
Slide 1 / 261 Algebra II Slide 2 / 261 Linear, Exponential and 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 261 Linear Functions Exponential Functions Properties
More informationPolynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial
Polynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial Shapes of Polynomials Look at the shape of the odd degree polynomials
More information3.4 Solving Exponential and Logarithmic Equations
214 Chapter 3 Exponential and Logarithmic Functions 3.4 Solving Exponential and Logarithmic Equations Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential
More informationMath 180 Chapter 4 Lecture Notes. Professor Miguel Ornelas
Math 80 Chapter 4 Lecture Notes Professor Miguel Ornelas M. Ornelas Math 80 Lecture Notes Section 4. Section 4. Inverse Functions Definition of One-to-One Function A function f with domain D and range
More informationSection II: Exponential and Logarithmic Functions. Module 6: Solving Exponential Equations and More
Haberman MTH 111c Section II: Eponential and Logarithmic Functions Module 6: Solving Eponential Equations and More EXAMPLE: Solve the equation 10 = 100 for. Obtain an eact solution. This equation is so
More informationSection 6.3. x m+1 x m = i n x m, (6.3.1)
Difference Equations to Differential Equations Section 6.3 Models of Growth and Decay In this section we will look at several applications of the exponential and logarithm functions to problems involving
More informationChapter 10 Resource Masters
Chapter 0 Resource Masters DATE PERIOD 0 Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 0. As you study the chapter,
More informationEXPONENTS AND LOGS (CHAPTER 10)
EXPONENTS AND LOGS (CHAPTER 0) POINT SLOPE FORMULA The point slope formula is: y y m( ) where, y are the coordinates of a point on the line and m is the slope of the line. ) Write the equation of a line
More informationMath 120 Handouts. Functions Worksheet I (will be provided in class) Point Slope Equation of the Line 5. Functions Worksheet III 17
Math 0 Handouts HW # (will be provided to class) Lines: Concepts from Previous Classes (emailed to the class) Parabola Plots # (will be provided in class) Functions Worksheet I (will be provided in class)
More informationPage Points Score Total: 100
Math 1130 Spring 2019 Sample Midterm 2a 2/28/19 Name (Print): Username.#: Lecturer: Rec. Instructor: Rec. Time: This exam contains 10 pages (including this cover page) and 9 problems. Check to see if any
More information4.6 (Part A) Exponential and Logarithmic Equations
4.6 (Part A) Eponential and Logarithmic Equations In this section you will learn to: solve eponential equations using like ases solve eponential equations using logarithms solve logarithmic equations using
More informationWarm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2
4-5 Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Solve. 1. log 16 x = 3 2 64 2. log x 1.331 = 3 1.1 3. log10,000 = x 4 Objectives Solve exponential and logarithmic equations and equalities.
More informationUNIT #6 EXPONENTS, EXPONENTS, AND MORE EXPONENTS REVIEW QUESTIONS
Name: Date: UNIT #6 EXPONENTS, EXPONENTS, AND MORE EXPONENTS REVIEW QUESTIONS Part I Questions 1. The epression 9 5 10 can be simplified to (1) 6 () () 1 1 6 (4). Which of the following is equivalent to
More information5.1. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS
5.1. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS 1 What You Should Learn Recognize and evaluate exponential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate,
More information10 Exponential and Logarithmic Functions
10 Exponential and Logarithmic Functions Concepts: Rules of Exponents Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions
More informationAlgebra II Non-Calculator Spring Semester Exam Review
Algebra II Non-Calculator Spring Semester Eam Review Name: Date: Block: Simplify the epression. Leave only positive eponents.. ( a ). ( p s ). mn 9cd cd. mn. ( w )( w ). 7. 7 7 Write the answer in scientific
More information3.1 Exponential Functions and Their Graphs
.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
More informationMATH Section 4.1
MATH 1311 Section 4.1 Exponential Growth and Decay As we saw in the previous chapter, functions are linear if adding or subtracting the same value will get you to different coordinate points. Exponential
More informationIntroduction to Exponential Functions (plus Exponential Models)
Haberman MTH Introduction to Eponential Functions (plus Eponential Models) Eponential functions are functions in which the variable appears in the eponent. For eample, f( ) 80 (0.35) is an eponential function
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University February 3-5, 2016 Outline 1 growth (doubling time) Suppose a single bacterium
More information2015 2nd Semester Exam Review
Algebra 2 2015 2nd Semester Exam Review 1. Write a function whose graph is a translation of the graph of the function in two directions. Describe the translation. 2. What are the solutions to the equation?
More informationExponential Growth. b.) What will the population be in 3 years?
0 Eponential Growth y = a b a b Suppose your school has 4512 students this year. The student population is growing 2.5% each year. a.) Write an equation to model the student population. b.) What will the
More information4.1 Exponential Functions
Chapter 4 Exponential and Logarithmic Functions 531 4.1 Exponential Functions In this section, you will: Learning Objectives 4.1.1 Evaluate exponential functions. 4.1.2 Find the equation of an exponential
More informationAlgebra 1B Assignments Exponential Functions (All graphs must be drawn on graph paper!)
Name Score Algebra 1B Assignments Eponential Functions (All graphs must be drawn on graph paper!) 8-6 Pages 463-465: #1-17 odd, 35, 37-40, 43, 45-47, 50, 51, 54, 55-61 odd 8-7 Pages 470-473: #1-11 odd,
More information