3.4. Properties of Exponential Functions. Investigate
|
|
- Alaina Nicholson
- 7 years ago
- Views:
Transcription
1 3. Properties of Eponential Functions Capacitors are used to store electric potential energ. When a capacitor in a resistor-capacitor (RC) circuit is discharged, the electric potential across the capacitor decas eponentiall over time. This sort of circuit is used in a variet of electronic devices, such as televisions, computers, and MP3 plaers. Engineers and technicians who design and build such devices must have a solid understanding of eponential functions. Man situations can be modelled using functions of the form f () 5 ab, where a and b, b 1. How do the values of a and b affect the properties of this tpe of function? Tools computer with The Geometer s Sketchpad or grid paper or graphing calculator interval an unbroken part of the real number line is either all of or has one of the following forms: < a, > a, a, a, a < < b, a b, a < b, a < b, where a, b, and a < b Connections It is important to be careful around discarded electrical equipment, such as television sets. Even if the device is not connected to a power source, stored electrical energ ma be present in the capacitors. Investigate How can ou discover the characteristics of the graph of an eponential function? A: The Effect of b on the Graph of = ab Start with the function f () 5. In this case, a a) Graph the function. b) Describe the shape of the graph.. Use algebraic and/or graphical reasoning to justif our answers to the following. a) What are the domain and the range of the function? b) What is the -intercept? c) Is there an -intercept? d) Over what interval is the function i) increasing? ii) decreasing? 3. Change the value of b. Use values greater than. a) Compare each graph to the graph of 5. Describe how the graphs are alike. How do the differ? b) Describe how the value of b has affected the characteristics listed in step. c) Eplain wh a value of b greater than has this effect on the graph. 178 MHR Functions 11 Chapter 3
2 If ou are using The Geometer s Sketchpad, ou can set b as a parameter whose value ou can change dnamicall: From the Graph menu, choose New Parameter. Set the name as b and its initial value to. Click OK. From the Graph menu, choose Plot New Function. Click on the parameter b, and then click on ^ and OK. You can change the value of b in three was: Click on parameter b and press and on the keboard to increase or decrease the value of b in 1-unit increments. Right-click on parameter b and choose Edit Parameter to enter a specific value. Right-click on parameter b and choose Animate Parameter. Use the buttons on the Motion Controller to see the effects of changing b continuousl.. Change the value of b again. This time, use values between and 1. a) How has the graph changed? b) Describe how the values of b affect the characteristics listed in step. c) Eplain wh a value of b between and 1 has this effect on the graph. Connections In Section 3.3, ou saw how to deal with rational eponents. The definition of an eponent can be etended to include real numbers, and so the domain of a function like f () = is all real numbers. Tr to evaluate π using a calculator. 5. What happens to the graph when ou set b 5 1? Eplain this result. 6. Reflect Summarize how the values of b affect the shape and characteristics of the graph of f () 5 b. B: The Effect of a on the Graph of = ab Use the function f () 5 a. In this part of the Investigate, keep b 5, and eplore what happens as ou change the value of a. 1. Set a 5 1. This gives the original graph of f () 5. Eplore what happens when a) a 1 b) a 1 c) a. Reflect Write a summar of the effects of various values of a on the graph of the function f () 5 a. Include the following characteristics: domain, range, - and -intercepts, and intervals of increase and decrease. Eplain wh changing the value of a has these effects. Sketch diagrams to support our eplanations. Technolog Tip With a graphing calculator, ou can var the line stle to see multiple graphs simultaneousl. Refer to the Technolog Appendi on pages 96 to Properties of Eponential Functions MHR 179
3 Connections You encountered horizontal and vertical asmptotes in Chapter 1 Functions. One of the interesting features of an eponential function is its asmptotic behaviour. Consider the function f () 5. If ou keep looking left at decreasing values of, ou will see that the corresponding -value of the function gets closer and closer to, but never reaches, the -ais. In this case, the -ais is an asmptote. Eample 1 Analse the Graph of an Eponential Function Graph each eponential function. Identif the domain range - and -intercepts, if the eist intervals of increase/decrease asmptote a) 5 ( 1_ ) b) 5 3 Solution a) 5 ( 1_ ) Method 1: Use a Table of Values Select negative and positive values of that will make it eas to compute corresponding values of _ 1_ ( 1_ ) = ( _ 1 ) = 16 Use the table of values to graph the function = ( ) 1 18 MHR Functions 11 Chapter 3
4 Method : Use a Graphing Calculator Use a graphing calculator to eplore the graph of this function. The function is defined for all values of. Therefore, the domain is { }. The function has positive values for, but never reaches zero. Therefore, the range is {, }. The graph never crosses the -ais, which means there is no -intercept. The graph crosses the -ais at. Therefore, the -intercept is. The graph falls to the right throughout its domain, so the -values decrease as the -values increase. Therefore, the function is decreasing over its domain. As the -values increase, the -values get closer and closer to, but never reach, the -ais. Therefore, the -ais, or the line 5, is an asmptote. b) 5 3 The domain is { }. All function values are negative. Therefore, the range is {, }. There is no -intercept. The -intercept is 1. The graph rises throughout its domain. Therefore, the function is increasing for all values of. The -ais, whose equation is 5, is an asmptote. Connections Could ou use transformations to quickl sketch the graph of = 3? You will eplore this option in Section Properties of Eponential Functions MHR 181
5 Eample Write an Eponential Equation Given Its Graph Write the equation in the form 5 ab that describes the graph shown. Solution Read some ordered pairs from the graph (3, 5) (, 18) 1 6 (1, 6) (, ) Note that as changes b 1 unit, increases b a factor of 3, confirming that this function is an eponential function Change in Since each successive value increases b a factor of 3, this function must have b 5 3. Since all points on this graph must satisf the equation 5 ab, substitute the coordinates of one of the points, and the value of b, to find the value of a. Pick a point that is eas to work with, such as (1, 6). Substitute 5 1, 5 6, and b 5 3: 5 ab 6 5 a a 3 a 5 Therefore, the equation that describes this curve is MHR Functions 11 Chapter 3
6 Eample 3 Write an Eponential Function Given Its Properties A radioactive sample has a half-life of 3 das. The initial sample is mg. a) Write a function to relate the amount remaining, in milligrams, to the time, in das. b) Restrict the domain of the function so that the mathematical model fits the situation it is describing. Solution a) This eponential deca can be modelled using a function of the form A() 5 A ( _ 1 ), where is the time, in half-life periods; A is the initial amount, in milligrams; and A is the amount remaining, in milligrams, after time. Start with mg. After ever half-life, the amount is reduced b half. Substituting A 5 into this equation gives A() 5 ( 1_ ). This epresses A as a function of, the number of half-lives. To epress A as a function of t, measured in das, replace with t_ 3. The half-life of this material is 3 das. Therefore, the number of elapsed half-lives at an given point is the number of das divided b 3. t_ A(t) 5 ( 1_ ) 3 This equation relates the amount, A, in milligrams, of radioactive material remaining to time, t, in das. b) A graph of this function reveals a limitation of the mathematical model. The initial sample size, at t 5, was mg. It is not clear that the function has an meaning before this time. Since it is onl certain that the mathematical model fits this situation for non-negative values of t, it makes sense to restrict its domain: A(t) 5 ( 1_ t_ 3 ) for {t, t }. A t A(t) = ( 1 ) 3 t A t 3. Properties of Eponential Functions MHR 183
7 Ke Concepts The graph of an eponential function of the form 5 ab is i ncreasing if a and b 1 d ecreasing if a and b 1 d ecreasing if a and b 1 i ncreasing if a and < b 1 The graph of an eponential function of the form 5 ab, where a and b, has domain { } range {, } a horizontal asmptote at 5 a -intercept of a The graph of an eponential function of the form 5 ab, where a and b, has domain { } range {, } a horizontal asmptote at 5 a -intercept of a You can write an equation to model an eponential function if ou are given enough information about its graph or properties. Sometimes it makes sense to restrict the domain of an eponential model based on the situation it represents. Communicate Your Understanding C1 a) Is an eponential function either alwas increasing or alwas decreasing? Eplain. b) Is it possible for an eponential function of the form 5 ab to have an -intercept? If es, given an eample. If no, eplain wh not. ( _1 ) C Consider the eponential functions f () 5 1 and g() 5 1(). a) Which function has a graph with range i) {, }? ii) {, }? Eplain how ou can tell b inspecting the equations. b) Which function is i) increasing? ii) decreasing? Eplain how ou can tell b inspecting the equations. C3 Describe what is meant b asmptotic behaviour. Support our eplanation with one or more sketches. 18 MHR Functions 11 Chapter 3 Functions 11 CH3.indd 18 6/1/9 :5:5 PM
8 A Practise For help with questions 1 to 3, refer to Eample Match each graph with its corresponding equation. a) b) c) A 5 B 5 ( 1_ ) C 5 1_ D 5. a) Sketch the graph of an eponential function that satisfies all of these conditions: domain { } range {, } -intercept 5 function increasing b) Is this the onl possible graph? Eplain. 3. a) Sketch the graph of an eponential function that satisfies all of these conditions: domain { } range {, } -intercept function decreasing b) Is this the onl possible graph? Eplain. For help with questions and 5, refer to Eample.. Write an eponential equation to match the graph shown (, 16) d) 8 (1, 8) (, ) Properties of Eponential Functions MHR 185
9 5. Write an eponential equation to match the graph shown. For help with question 6, refer to Eample A radioactive sample with an initial mass of 5 mg has a half-life of das. B a) Which equation models this eponential deca, where t is the time, in das, and A is the amount of the substance that remains? A A 5 5 t_ B A 5 5 ( 1_ ) t C A 5 5 ( 1_ ) D A 5 5 t_ b) What is the amount of radioactive material remaining after 7 das? t_ Connect and Appl 7. Graph each function and identif the i) domain ii) range iii) - and -intercepts, if the eist iv) intervals of increase/decrease v) asmptote a) f () 5 ( 1_ ) b) c) 5 ( 1_ 3 ) (, ) (1, 1) (, 6) 8. a) Graph each function. i) f () 5 ii) r() 5 _ Connecting b) Describe how the graphs are alike. How do the differ? c) Compare the asmptotes of these functions. What do ou observe? 9. a) Graph each function. i) g() 5 ( 1_ ) ii) r() 5 _ b) Describe how the graphs are alike. How do the differ? c) Compare the asmptotes of these functions. What do ou observe? 1. a) Predict how the graphs of the following functions are related. i) f () 5 3 ii) g() 5 ( 1_ 3 ) b) Graph both functions and check our prediction from part a). c) Use algebraic reasoning to eplain this relationship. 11. The graph shows the voltage drop across a capacitor over time while discharging an RC circuit. At t 5 s, the circuit begins to discharge. a) What is the domain of this function? b) What is the range? Representing Voltage Drop (V) Reasoning and Proving Problem Solving Communicating Selecting Tools Reflecting c) What is the initial voltage drop across the capacitor? d) What value does the voltage drop across the capacitor approach as more time passes? e) Approimatel how long will it take the voltage drop to reach 5% of the initial value? V t Time (milliseconds) 186 MHR Functions 11 Chapter 3
10 1. A flwheel is Reasoning and Proving rotating under Representing friction. The Problem Solving number, R, of Connecting revolutions per minute after Communicating t minutes can be determined using the function R(t) 5 (.75) t. a) Eplain the roles of the numbers.75 and in the equation. b) Graph the function. c) Which value in the equation indicates that the flwheel is slowing? d) Determine the number of revolutions per minute after i) 1 min ii) 3 min C Etend 13. Use Technolog Refer to question 11. The equation that models this situation is given b V 5 V b t_ RC, where V is the voltage drop, in volts; V is the initial voltage drop; t is the time, in seconds; R is the resistance, in ohms (Ω); and C is the capacitance, in farads (F). For this circuit, R 5 Ω and C 5 1 µf. Note that 1 µf 5. 1 F. a) Determine the value of the base, b. b) Eplain our method. Selecting Tools Reflecting c) Graph the function using a graphing calculator or graphing software. Use the window settings shown. d) What are the domain and range of this function? e) Eplain how and wh the domain and range are restricted, as illustrated in the graph of question Suppose a square-based pramid has a fied height of 5 m. a) Write an equation, using rational eponents where appropriate, to epress the side length of the base of a squarebased pramid in terms of its volume. b) How should ou limit the domain of this function so that the mathematical model fits the situation? c) What impact does doubling the volume have on the side length of the base? Eplain. 15. Suppose that a shelf can hold clindrical drums with a fied height of 1 m. a) Write a simplified equation, using rational eponents where appropriate, to epress the surface area in terms of the volume for drums that will fit on the shelf. b) Find the surface area and diameter of a drum with a volume of.8 m 3. c) What are the restrictions on the domain of the function used in this model? d) Graph the function for the restricted domain. 16. Math Contest Find all solutions to Math Contest Consider the function 5 1 ( 1_ ) 3. The -intercept is b and the -intercept is a. The sum of a and b is A 11 B 6 C 7 D Math Contest A number is between and 3. When this number is subtracted from its cube, the result is When the same number is added to its cube, the answer is A B C D Properties of Eponential Functions MHR 187
LESSON 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 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 informationSECTION 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 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 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 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 informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
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 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 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 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 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 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 informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
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 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 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 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 informationGraphing Piecewise Functions
Graphing Piecewise Functions Course: Algebra II, Advanced Functions and Modeling Materials: student computers with Geometer s Sketchpad, Smart Board, worksheets (p. -7 of this document), colored pencils
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 informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
- NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
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 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 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 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 informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
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 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 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 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 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 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 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 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 informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationExponential Functions
Eponential Functions Deinition: An Eponential Function is an unction that has the orm ( a, where a > 0. The number a is called the base. Eample:Let For eample (0, (, ( It is clear what the unction means
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 informationName Class Date. Additional Vocabulary Support
- Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the
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 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 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 informationExponent Law Review 3 + 3 0. 12 13 b. 1 d. 0. x 5 d. x 11. a 5 b. b 8 a 8. b 2 a 2 d. 81u 8 v 10 81. u 8 v 20 81. Name: Class: Date:
Name: Class: Date: Eponent Law Review Multiple Choice Identify the choice that best completes the statement or answers the question The epression + 0 is equal to 0 Simplify 6 6 8 6 6 6 0 Simplify ( ) (
More informationSome Tools for Teaching Mathematical Literacy
Some Tools for Teaching Mathematical Literac Julie Learned, Universit of Michigan Januar 200. Reading Mathematical Word Problems 2. Fraer Model of Concept Development 3. Building Mathematical Vocabular
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 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 informationChapter 13 Introduction to Linear Regression and Correlation Analysis
Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
More informationSlope-Intercept Form and Point-Slope Form
Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.
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 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 information6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
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 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 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 informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationImagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x
OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations
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 informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
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 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 informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
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 informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationAP Calculus AB 2005 Scoring Guidelines Form B
AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college
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 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 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 informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More informationClassifying Solutions to Systems of Equations
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham
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 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 informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationThe Big Picture. Correlation. Scatter Plots. Data
The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered
More informationBy Clicking on the Worksheet you are in an active Math Region. In order to insert a text region either go to INSERT -TEXT REGION or simply
Introduction and Basics Tet Regions By Clicking on the Worksheet you are in an active Math Region In order to insert a tet region either go to INSERT -TEXT REGION or simply start typing --the first time
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 informationPearson s Correlation Coefficient
Pearson s Correlation Coefficient In this lesson, we will find a quantitative measure to describe the strength of a linear relationship (instead of using the terms strong or weak). A quantitative measure
More information