every hour 8760 A every minute 525,000 A continuously n A
|
|
- Charla Cannon
- 5 years ago
- Views:
Transcription
1 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 for some number of years, the accumulated value of that investment can be found by using the formula A = P (1 + r n )nt. Using that formula we found the accumulated value of investments when interest is compounded bi-annually (n = 2), quarterly (n = 4), monthly (n = 12), weekly (n = 52), etc. But what would happen if the number of compounding periods per year (n) continued to increase? For instance what if interest were compounded every day (n = 365), or every hour (n = 8760), or every minute (n = 525,000), or every second (n = 31,536, 000)? For a simple eample, imagine that one dollar is invested for one year at a rate of 100% (P, t, and r = 1); what would that investment accumulate to if the number of compounding periods continued to increase? How often is interest compounded? n A = P (1 + r n ) nt monthly 12 A = 1 ( ) weekly 52 A = 1 ( ) daily 365 A = 1 ( ) every hour 8760 A every minute 525,000 A every second 31,536,000 A continuously n A
2 What we see is that as the number of compounding periods per year increase (n ), the accumulated value A continues to get closer and closer to the value This value that we get closer and closer to is known as the natural number e, and it is used etensively in finance (as we ll see later with our second compound interest formula), as well as in other disciplines such as statistics and engineering. The Natural Number e: - a non-terminating constant o to find the value on your calculator press the number 1, then the 2 nd function button, then the LN button - it is NOT a variable like or y, but rather a number like π The way we will use the natural number e is either as the base of an eponential function, which is known as the natural eponential function, or as the base of a logarithmic function, which is known as the natural logarithmic function g() = ln(). We will work with the Natural Eponential Function in this lesson, and we ll cover the Natural Logarithmic Function g() = ln() in Lesson 32. Natural Eponential Functions: -, where the eponent is a variable and base e is the natural number Just like any other eponential function, the domain of the natural eponential function is unrestricted (, ) and the range is only positive numbers (0, ). And just like any other eponential function, the natural eponential function can be transformed, which in the case of a vertical transformation would alter the range (more on this later).
3 Eample 1: Given the input/output table for the function, as well as its graph, find its domain, range, zeros, positive/negative intervals, increasing/decreasing intervals, and intercepts. f() 0 5 f( 5) = e 5 = 1 e f( 4) = e 4 = 1 e f( 3) = e 3 = 1 e f( 2) = e 2 = 1 e Notice that the graph of is increasing throughout its domain. When a function is always increasing or always decreasing that function is one-to-one, and it will have an inverse function. All eponential functions are one-to-one, therefore all eponential functions have an inverse (we ll discuss this further in Lessons 31 & 32). f() 1 f( 1) = e 1 = 1 e f(0) = e 0 = 1 1 f(1) = e f(2) = e f(3) = e f(4) = e f(5) = e f() Domain: (, ) Range: (0, ) Zeros: f() = 0 when = NONE Positive intervals: f() > 0 when is (, ) Negative intervals: f() < 0 when is NONE Increasing intervals: f() is rising when is (, ) Decreasing intervals: f() is falling when is NONE Intercepts: intercept: NONE y intercept: (0, 1)
4 Notice that just like the graph of 2, the graph of e has a horizontal asymptote at y = 0 (the -ais). That is because the graphs of both (2 and e ) approach the -ais, but they never touch it or cross it. Eample 2: Re-write the following function in terms of. Then find its y-intercept and sketch its graph using transformations and the y-intercept. Enter eact answers only, no approimations. a. g() = e f() Re write g() in terms of f(): g() = y intercept: g(0) = (0, )
5 Eample 3: Re-write each of the following functions in terms of, then match the transformation with the appropriate graph. Also, find the y-intercept for each function. Enter eact answers only, no approimations. f() f() e e e e f() a. h() = e b. j() = e +2 c. k() = e + 2 b. c. f() = F( ) h() = j() = k() = d. y int: y int: y int: e. f. l() = 2e e. m() = e 2 g.f() = F( ) l() = m() = h. y int: y int:
6 A. B. C. D. E. F.
7 Answers to Eamples: 3a. A, (0, 1) ; 3b. C, (0, e 2 ) ; 3c. F, (0, 3) ; 3d. B, (0, 2) ; 3e. E, (0, 1) ;
( ) ( ) 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 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 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 informationWe want to determine what the graph of an exponential function. y = a x looks like for all values of a such that 0 > a > 1
Section 5 B: Graphs of Decreasing Eponential Functions We want to determine what the graph of an eponential function y = a looks like for all values of a such that 0 > a > We will select a value of a such
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 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 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 informationExponential and Logarithmic Functions
Lesson 6 Eponential and Logarithmic Fu tions Lesson 6 Eponential and Logarithmic Functions Eponential functions are of the form y = a where a is a constant greater than zero and not equal to one and is
More informationChapter 8 Notes SN AA U2C8
Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of
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 informationExponential Functions
Exponential Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: recognize and evaluate exponential functions with base a,
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 informationSection 4.5 Graphs of Logarithmic Functions
6 Chapter 4 Section 4. Graphs of Logarithmic Functions Recall that the eponential function f ( ) would produce this table of values -3 - -1 0 1 3 f() 1/8 ¼ ½ 1 4 8 Since the arithmic function is an inverse
More information8 f(8) = 0 (8,0) 4 f(4) = 4 (4, 4) 2 f(2) = 3 (2, 3) 6 f(6) = 3 (6, 3) Outputs. Inputs
In the previous set of notes we covered how to transform a graph by stretching or compressing it vertically. In this lesson we will focus on stretching or compressing a graph horizontally, which like the
More informationExample. Determine the inverse of the given function (if it exists). f(x) = 3
Example. Determine the inverse of the given function (if it exists). f(x) = g(x) = p x + x We know want to look at two di erent types of functions, called logarithmic functions and exponential functions.
More informationGoal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation
Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is
More informationNONLINEAR FUNCTIONS A. Absolute Value Exercises: 2. We need to scale the graph of Qx ( )
NONLINEAR FUNCTIONS A. Absolute Value Eercises:. We need to scale the graph of Q ( ) f ( ) =. The graph is given below. = by the factor of to get the graph of 9 - - - - -. We need to scale the graph of
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 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 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 informationMath 1314 Lesson 1: Prerequisites
Math 131 Lesson 1: Prerequisites Prerequisites are topics you should have mastered before you enter this class. Because of the emphasis on technology in this course, there are few skills which you will
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 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 information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationGraphing Exponential Functions
MHF UI Unit Da Graphing Eponential Functions. Using a table of values (no decimals), graph the function.. For the function, state: a) domain b) range c) equation of the asmptote d) -intercept e) -intercept
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 informationPure Math 30: Explained!
Pure Math 30: Eplained! www.puremath30.com 9 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first type of eponential graph
More information4 Exponential and Logarithmic Functions
4 Exponential and Logarithmic Functions 4.1 Exponential Functions Definition 4.1 If a > 0 and a 1, then the exponential function with base a is given by fx) = a x. Examples: fx) = x, gx) = 10 x, hx) =
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 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 informationMath 1160 Final Review (Sponsored by The Learning Center) cos xcsc tan. 2 x. . Make the trigonometric substitution into
Math 60 Final Review (Sponsored by The Learning Center). Simplify cot csc csc. Prove the following identities: cos csc csc sin. Let 7sin simplify.. Prove: tan y csc y cos y sec y cos y cos sin y cos csc
More informationf 2a.) f 4a.) increasing:
MSA Pre-Calculus Midterm Eam Review December 06. Evaluate the function at each specified value of the independent variable and simplify. f ( ) f ( ) c. ( b ) f ) ). Evaluate the function at each specified
More informationWrite each expression as a sum or difference of logarithms. All variables are positive. 4) log ( ) 843 6) Solve for x: 8 2x+3 = 467
Write each expression as a single logarithm: 10 Name Period 1) 2 log 6 - ½ log 9 + log 5 2) 4 ln 2 - ¾ ln 16 Write each expression as a sum or difference of logarithms. All variables are positive. 3) ln
More informationUnit 3 Exam Review Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Unit Eam Review Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Some Useful Formulas: Compound interest formula: A=P + r nt n Continuously
More information1. Evaluate the function at each specified value of the independent variable and simplify. f 2a.)
Honors Pre-Calculus Midterm Eam Review Name: January 04 Chapter : Functions and Their Graphs. Evaluate the function at each specified value of the independent variable and simplify. f ( ) f () b. f ( )
More informationUnit 7 Study Guide (2,25/16)
Unit 7 Study Guide 1) The point (-3, n) eists on the eponential graph shown. What is the value of n? (2,25/16) (-3,n) (3,125/64) a)y = 1 2 b)y = 4 5 c)y = 64 125 d)y = 64 125 2) The point (-2, n) eists
More information1.1 Checkpoint GCF Checkpoint GCF 2 1. Circle the smaller number in each pair. Name the GCF of the following:
39 0 . Checkpoint GCF Name the GCF of the following:.. 3.. + 9 + 0 + 0 6 y + 5ab + 8 5. 3 3 y 5y + 7 y 6. 3 3 y 8 y + y.. Checkpoint GCF. Circle the smaller number in each pair. 5, 0 8, 0,,,, 3 0 3 5,,,
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 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 informationL43-Mon-12-Dec-2016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45 Page 27. L43-Mon-12-Dec-2016-Rev-Cpt-4-HW44-and-Rev-Cpt-5-for-Final-HW45
L43-Mon-1-Dec-016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45 Page 7 L43-Mon-1-Dec-016-Rev-Cpt-4-HW44-and-Rev-Cpt-5-for-Final-HW45 L43-Mon-1-Dec-016-Rev-Cpt-4-for-Final-HW44-and-Rev-Cpt-5-for-Final-HW45
More informationHonors Calculus Summer Preparation 2018
Honors Calculus Summer Preparation 08 Name: ARCHBISHOP CURLEY HIGH SCHOOL Honors Calculus Summer Preparation 08 Honors Calculus Summer Work and List of Topical Understandings In order to be a successful
More informationName Date Period. Pre-Calculus Midterm Review Packet (Chapters 1, 2, 3)
Name Date Period Sections and Scoring Pre-Calculus Midterm Review Packet (Chapters,, ) Your midterm eam will test your knowledge of the topics we have studied in the first half of the school year There
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 informationExponential functions: week 13 STEM
Boise State, 4 Eponential functions: week 3 STEM As we have seen, eponential functions describe events that grow (or decline) at a constant percent rate, such as placing finances in a savings account.
More informationName. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
REVIEW Eam #3 : 3.2-3.6, 4.1-4.5, 5.1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Leading Coefficient Test to determine the end behavior
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 informationOne-to-One Functions YouTube Video
Section 9. One-to-One Functions YouTue Video A function in which each element in the range corresponds to one and only one element in the domain. Determine if the following are One-to-one functions:,,
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
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 informationMath 137 Exam #3 Review Guide
Math 7 Exam # Review Guide The third exam will cover Sections.-.6, 4.-4.7. The problems on this review guide are representative of the type of problems worked on homework and during class time. Do not
More informationMATH 175: Final Exam Review for Pre-calculus
MATH 75: Final Eam Review for Pre-calculus In order to prepare for the final eam, you need to be able to work problems involving the following topics:. Can you find and simplify the composition of two
More informationIntermediate Algebra Chapter 12 Review
Intermediate Algebra Chapter 1 Review Set up a Table of Coordinates and graph the given functions. Find the y-intercept. Label at least three points on the graph. Your graph must have the correct shape.
More informationSec 3.1. lim and lim e 0. Exponential Functions. f x 9, write the equation of the graph that results from: A. Limit Rules
Sec 3. Eponential Functions A. Limit Rules. r lim a a r. I a, then lim a and lim a 0 3. I 0 a, then lim a 0 and lim a 4. lim e 0 5. e lim and lim e 0 Eamples:. Starting with the graph o a.) Shiting 9 units
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions 6.3 Logarithmic Functions. 9 = 3 is equivalent to = log 3 9. 6 = 4 is equivalent to = log 4 6 3. a =.6 is equivalent to = log a.6 4. a 3 =. is equivalent
More informationMATH 175: Final Exam Review for Pre-calculus
MATH 75: Final Eam Review for Pre-calculus In order to prepare for the final eam, you need too be able to work problems involving the following topics:. Can you graph rational functions by hand after algebraically
More informationHonors Accelerated Pre-Calculus Midterm Exam Review Name: January 2010 Chapter 1: Functions and Their Graphs
Honors Accelerated Pre-Calculus Midterm Eam Review Name: January 010 Chapter 1: Functions and Their Graphs 1. Evaluate the function at each specified value of the independent variable and simplify. 1 f
More informationnotes.notebook April 08, 2014
Chapter 7: Exponential Functions graphs solving equations word problems Graphs (Section 7.1 & 7.2): c is the common ratio (can not be 0,1 or a negative) if c > 1, growth curve (graph will be increasing)
More informationChapter 6 Logarithmic and Exponential Functions
Chapter 6 Logarithmic and Eponential Functions 6.1 The Definition of e Eample 1 Consider the eponential function f 1 ( ) 1 What happens to f() as gets very large? Type the function into Y=. Let Y 1 1 1/.
More informationIntermediate Algebra Section 9.3 Logarithmic Functions
Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where
More informationLesson Goals. Unit 5 Exponential/Logarithmic Functions Exponential Functions (Unit 5.1) Exponential Functions. Exponential Growth: f (x) = ab x, b > 1
Unit 5 Eponential/Logarithmic Functions Eponential Functions Unit 5.1) William Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Recognize and
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 informationx x x 2. Use your graphing calculator to graph each of the functions below over the interval 2,2
MSLC Math 48 Final Eam Review Disclaimer: This should NOT be used as your only guide for what to study.. Use the piece-wise defined function 4 4 if 0 f( ) to answer the following: if a) Compute f(-), f(-),
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 informationSummary sheet: Exponentials and logarithms
F Know and use the function a and its graph, where a is positive Know and use the function e and its graph F2 Know that the gradient of e k is equal to ke k and hence understand why the eponential model
More informationMath 0210 Common Final Review Questions (2 5 i)(2 5 i )
Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )
More informationSTUDENT NAME CLASS DAYS/TIME MATH 102, COLLEGE ALGEBRA UNIT 3 LECTURE NOTES JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY
STUDENT NAME CLASS DAYS/TIME MATH 10, COLLEGE ALGEBRA UNIT 3 LECTURE NOTES JILL TRIMBLE, BLACK HILLS STATE UNIVERSITY Math10 College Algebra Unit 3 Outcome/Homework 1 Students will be able to add, subtract,
More informationComplete your Parent Function Packet!!!!
PARENT FUNCTIONS Pre-Ap Algebra 2 Complete your Parent Function Packet!!!! There are two slides per Parent Function. The Parent Functions are numbered in the bottom right corner of each slide. The Function
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More information2. Use your graphing calculator to graph each of the functions below over the interval 2, 2
MSLC Math 0 Final Eam Review Disclaimer: This should NOT be used as your only guide for what to study. if 0. Use the piece-wise defined function f( ) to answer the following: if a. Compute f(0), f(), f(-),
More informationLesson 5.6 Exercises, pages
Lesson 5.6 Eercises, pages 05 0 A. Approimate the value of each logarithm, to the nearest thousanth. a) log 9 b) log 00 Use the change of base formula to change the base of the logarithms to base 0. log
More information3. (12 points) Find an equation for the line tangent to the graph of f(x) =
April 8, 2015 Name The total number of points available is 168 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions
More informationChapter 12 and 13 Math 125 Practice set Note: the actual test differs. Given f(x) and g(x), find the indicated composition and
Chapter 1 and 13 Math 1 Practice set Note: the actual test differs. Given f() and g(), find the indicated composition. 1) f() = - ; g() = 3 + Find (f g)(). Determine whether the function is one-to-one.
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 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 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 informationObjectives. Use the number e to write and graph exponential functions representing realworld
Objectives Use the number e to write and graph exponential functions representing realworld situations. Solve equations and problems involving e or natural logarithms. natural logarithm Vocabulary natural
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 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 information9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically
9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically Use this blank page to compile the most important things you want to remember for cycle 9.5: 181 Even and Odd Functions Even Functions:
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 informationSummer Mathematics Prep
Summer Mathematics Prep Entering Calculus Chesterfield County Public Schools Department of Mathematics SOLUTIONS Domain and Range Domain: All Real Numbers Range: {y: y } Domain: { : } Range:{ y : y 0}
More informationfor every x in the gomain of g
Section.7 Definition of Inverse Function Let f and g be two functions such that f(g(x)) = x for every x in the gomain of g and g(f(x)) = x for every x in the gomain of f Under these conditions, the function
More informationThis problem set is a good representation of some of the key skills you should have when entering this course.
Math 4 Review of Previous Material: This problem set is a good representation of some of the key skills you should have when entering this course. Based on the course work leading up to Math 4, you should
More information5.1 Exponential and Logarithmic Functions
Math 0 Student Notes. Eponential and Logarithmic Functions Eponential Function: the equation f() = > 0, defines an eponential function for each different constant, called the ase. The independent variale
More information2018 Pre-Cal Spring Semester Review Name: Per:
08 Pre-Cal Spring Semester Review Name: Per: For # 4, find the domain of each function. USE INTERVAL NOTATION!!. 4 f ( ) 5. f ( ) 6 5. f( ) 5 4. f( ) 4 For #5-6, find the domain and range of each graph.
More informationReview of Exponential Relations
Review of Exponential Relations Integrated Math 2 1 Concepts to Know From Video Notes/ HW & Lesson Notes Zero and Integer Exponents Exponent Laws Scientific Notation Analyzing Data Sets (M&M Lab & HW/video
More informationSection 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.
Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph
More informationPRECAL REVIEW DAY 11/14/17
PRECAL REVIEW DAY 11/14/17 COPY THE FOLLOWING INTO JOURNAL 1 of 3 Transformations of logs Vertical Transformation Horizontal Transformation g x = log b x + c g x = log b x c g x = log b (x + c) g x = log
More informationConcept Category 2. Exponential and Log Functions
Concept Category 2 Exponential and Log Functions Concept Category 2 Check List *Find the inverse and composition of functions *Identify an exponential from a table, graph and equation *Identify the difference
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationComposition of and the Transformation of Functions
1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of
More information* Circle these problems: 23-27, 37, 40-44, 48, No Calculator!
AdvPreCal 1 st Semester Final Eam Review Name 1. Solve using interval notation: 7 8 * Circle these problems: -7, 7, 0-, 8, 6-66 No Calculator!. Solve and graph: 0. Solve using a number line and leave answer
More informationMATH 121 Precalculus Practice problems for Exam 1
MATH 11 Precalculus Practice problems for Eam 1 1. Analze the function and then sketch its graph. Find - and -intercepts of the graph. Determine the behavior of the graph near -intercepts. Find the vertical
More informationMATH 135 Sample Review for the Final Exam
MATH 5 Sample Review for the Final Eam This review is a collection of sample questions used by instructors of this course at Missouri State University. It contains a sampling of problems representing the
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 informationTest # 33 QUESTIONS MATH131 091700 COLLEGE ALGEBRA Name atfm131bli www.alvarezmathhelp.com website MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationTopic 33: One-to-One Functions. Are the following functions one-to-one over their domains?
Topic 33: One-to-One Functions Definition: A function f is said to be one-to-one if for every value f(x) in the range of f there is exactly one corresponding x-value in the domain of f. Ex. Are the following
More information1 st Semester Final Review Date No
CHAPTER 1 REVIEW 1. Simplify the epression and eliminate any negative eponents. Assume that all letters denote positive numbers. r s 6r s. Perform the division and simplify. 6 8 9 1 10. Simplify the epression.
More information