2.5 Library of Functions; Piecewisedefined Functions


 Barnard Greene
 7 years ago
 Views:
Transcription
1 SECTION.5 Librar of Functions; Piecewisedefined Functions 07.5 Librar of Functions; Piecewisedefined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section., pp. 5 7) Graphs of Ke Equations (Section.: Eample, p. ; Eample, p. 9; Eample, p.0; Eample, p. ) Now work the Are You Prepared? problems on page. OBJECTIVES Graph the Functions Listed in the Librar of Functions Graph Piecewisedefined Functions Figure 6 (, ) (, ) 5 (9, ) 0 We now introduce a few more functions to add to our list of important functions. We begin with the square root function. In Section. we graphed the equation =. Figure shows a graph of f =. Based on the graph, we have the following properties: Properties of f(). The intercept of the graph of f = is 0. The intercept of the graph of f = is also 0.. The function is neither even nor odd.. It is increasing on the interval 0, q.. It has a minimum value of 0 at = 0. EXAMPLE Graphing the Cube Root Function (a) Determine whether f = is even, odd, or neither. State whether the graph of f is smmetric with respect to the ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f =. (c) Graph f =.
2 08 CHAPTER Functions and Their Graphs Solution (a) Because f =  =  = f the function is odd. The graph of f is smmetric with respect to the origin. (b) The intercept is f0 = 0 = 0. The intercept is found b solving the equation f = 0. f = 0 = 0 f() = = 0 Cube both sides of the equation. The intercept is also 0. (c) We use the function to form Table 6 and obtain some points on the graph. Because of the smmetr with respect to the origin, we onl need to find points, for which Ú 0. Figure shows the graph of f =. Table 6 f() (, ) a 8, b (, ) L.6 A, B 8 (8, ) Figure (, ) 8 (, ) (, ) (, ) (, ) (, ) From the results of Eample and Figure, we have the following properties of the cube root function. 8 Properties of f (). The intercept of the graph of f = is 0. The intercept of the graph of f = is also 0.. The function is odd.. It is increasing on the interval  q, q.. It does not have a local minimum or a local maimum. EXAMPLE Graphing the Absolute Value Function (a) Determine whether f = ƒƒ is even, odd, or neither. State whether the graph of f is smmetric with respect to the ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f = ƒƒ. (c) Graph f = ƒƒ. Solution (a) Because f = ƒ ƒ = ƒƒ = f the function is even. The graph of f is smmetric with respect to the ais.
3 SECTION.5 Librar of Functions; Piecewisedefined Functions 09 (b) The intercept is f0 = ƒ0ƒ = 0. The intercept is found b solving the equation f = ƒƒ = 0. So the intercept is 0. (c) We use the function to form Table 7 and obtain some points on the graph. Because of the smmetr with respect to the ais, we onl need to find points, for which Ú 0. Figure shows the graph of f = ƒƒ. Table 7 f() ƒƒ (, ) 0 0 (, ) (, ) (, ) Figure (, ) (, ) (, ) (, ) (, ) (, ) From the results of Eample and Figure, we have the following properties of the absolute value function. Properties of f() ƒƒ. The intercept of the graph of f = ƒƒ is 0. The intercept of the graph of f = ƒƒ is also 0.. The function is even.. It is decreasing on the interval  q, 0. It is increasing on the interval 0, q.. It has a local minimum of 0 at = 0. Seeing the Concept = ƒƒ Graph on a square screen and compare what ou see with Figure. Note that some graphing calculators use abs for absolute value. Graph the Functions Listed in the Librar of Functions We now provide a summar of the ke functions that we have encountered. In going through this list, pa special attention to the properties of each function, particularl to the shape of each graph. Knowing these graphs will la the foundation for later graphing techniques. Figure Linear Function f( ) m b, m 0 Linear Function f = m + b, m and b are real numbers (0, b) See Figure. The domain of a linear function is the set of all real numbers. The graph of this function is a nonvertical line with slope m and intercept b. A linear function is increasing if m 7 0, decreasing if m 6 0, and constant if m = 0.
4 0 CHAPTER Functions and Their Graphs Constant Function Figure 5 Constant Function Figure 6 Identit Function b (0,b) f() = b f() = See Figure 5. A constant function is a special linear function m = 0. Its domain is the set of all real numbers; its range is the set consisting of a single number b. Its graph is a horizontal line whose intercept is b. The constant function is an even function whose graph is constant over its domain. Identit Function f = b, b is a real number f = (, ) Figure 7 Square Function (, ) See Figure 6. The identit function is also a special linear function. Its domain and range are the set of all real numbers. Its graph is a line whose slope is m = and whose intercept is 0. The line consists of all points for which the coordinate equals the coordinate. The identit function is an odd function that is increasing over its domain. Note that the graph bisects quadrants I and III. f() = Square Function (, ) (, ) f = (, ) Figure 8 Cube Function (, ) See Figure 7. The domain of the square function f is the set of all real numbers; its range is the set of nonnegative real numbers. The graph of this function is a parabola whose intercept is at 0, 0. The square function is an even function that is decreasing on the interval  q, 0 and increasing on the interval 0, q. (, ) Figure 9 Square Root Function (, ) f() = f() = (, ) (, ) Cube Function f = See Figure 8. The domain and the range of the cube function is the set of all real numbers.the intercept of the graph is at 0, 0. The cube function is odd and is increasing on the interval  q, q. Square Root Function f = 5 See Figure 9.
5 SECTION.5 Librar of Functions; Piecewisedefined Functions Figure 50 Cube Root Function (, ) 8 (, ) Figure 5 Reciprocal Function (, ) (, ) (, ) (, ) 8 f() = The domain and the range of the square root function is the set of nonnegative real numbers. The intercept of the graph is at 0, 0. The square root function is neither even nor odd and is increasing on the interval 0, q. Cube Root Function f = See Figure 50. The domain and the range of the cube root function is the set of all real numbers.the intercept of the graph is at 0, 0. The cube root function is an odd function that is increasing on the interval  q, q. Reciprocal Function f = (, ) (, ) Refer to Eample, page, for a discussion of the equation = See Figure 5.. The domain and the range of the reciprocal function is the set of all nonzero real numbers. The graph has no intercepts. The reciprocal function is decreasing on the intervals  q, 0 and 0, q and is an odd function. Absolute Value Function Figure 5 Absolute Value Function (, ) (, ) f() = (, ) (, ) f = ƒƒ See Figure 5. The domain of the absolute value function is the set of all real numbers; its range is the set of nonnegative real numbers. The intercept of the graph is at 0, 0. If Ú 0, then f =, and the graph of f is part of the line = ; if 6 0, then f = , and the graph of f is part of the line = . The absolute value function is an even function; it is decreasing on the interval  q, 0 and increasing on the interval 0, q. The notation int stands for the largest integer less than or equal to. For eample, int =, int.5 =, inta b = 0, inta  b = , intp = This tpe of correspondence occurs frequentl enough in mathematics that we give it a name. Greatest Integer Function f = int * = greatest integer less than or equal to * Some books use the notation f = Œœ instead of int.
6 CHAPTER Functions and Their Graphs Table 8 f() int() (, ) (,  ) a  a  a, 0b a, 0b 0 a, 0b,  b,  b NOTE When graphing a function using a graphing utilit, ou can choose either the connected mode, in which points plotted on the screen are connected, making the graph appear without an breaks, or the dot mode, in which onl the points plotted appear. When graphing the greatest integer function with a graphing utilit, it is necessar to be in the dot mode. This is to prevent the utilit from connecting the dots when f changes from one integer value to the net. See Figure 5. We obtain the graph of f = int b plotting several points. See Table 8. For values of,  6 0, the value of f = int is ; for values of, 0 6, the value of f is 0. See Figure 5 for the graph. Figure 5 Greatest Integer Function The domain of the greatest integer function is the set of all real numbers; its range is the set of integers. The intercept of the graph is 0. The intercepts lie in the interval 0,. The greatest integer function is neither even nor odd. It is constant on ever interval of the form k, k +, for k an integer. In Figure 5, we use a solid dot to indicate, for eample, that at = the value of f is f = ; we use an open circle to illustrate that the function does not assume the value of 0 at =. From the graph of the greatest integer function, we can see wh it also called a step function. At = 0, = ;, = ;, and so on, this function ehibits what is called a discontinuit; that is, at integer values, the graph suddenl steps from one value to another without taking on an of the intermediate values. For eample, to the immediate left of =, the coordinates are, and to the immediate right of =, the coordinates are. Figure 5 shows the graph of f = int on a TI8 Plus. Figure 5 f() = int() 6 6 (a) Connected mode 6 6 (b) Dot mode The functions that we have discussed so far are basic. Whenever ou encounter one of them, ou should see a mental picture of its graph. For eample, if ou encounter the function f =, ou should see in our mind s ee a picture like Figure 7. NOW WORK PROBLEMS 9 THROUGH 6. Graph Piecewisedefined Functions Sometimes a function is defined differentl on different parts of its domain. For eample, the absolute value function f = ƒƒ is actuall defined b two equations: f = if Ú 0 and f =  if 6 0. For convenience, we generall combine these equations into one epression as f = ƒƒ = e if Ú 0  if 6 0 When functions are defined b more than one equation, the are called piecewisedefined functions. Let s look at another eample of a piecewisedefined function.
7 SECTION.5 Librar of Functions; Piecewisedefined Functions EXAMPLE Analzing a Piecewisedefined Function The function f is defined as f = c (a) Find f0, f, and f. (b) Determine the domain of f. (c) Graph f b hand. (d) Use the graph to find the range of f.  + if  6 if = if 7 Figure 55 Solution (a) To find f0, we observe that when = 0 the equation for f is given b f =  +. So we have f0 = 0 + = When =, the equation for f is f =. Thus, f = When =, the equation for f is f =. So f = = = 5 (, ) (, ) (, ) (0, ) = + (b) To find the domain of f, we look at its definition. We conclude that the domain of f is 5 ƒ Ú 6, or the interval , q. (c) To graph f b hand, we graph each piece. First we graph the line =  + and keep onl the part for which  6. Then we plot the point, because, when =, f =. Finall, we graph the parabola = and keep onl the part for which 7. See Figure 55. (d) From the graph, we conclude that the range of f is 5 ƒ 7 06, or the interval 0, q. To graph a piecewisedefined function on a graphing calculator, we use the TEST menu to enter inequalities that allow us to restrict the domain function. For eample, to graph the function in Eample using a TI8 Plus graphing calculator, we would enter the function in Y as shown in Figure 56(a). We then graph the function and obtain the result in Figure 56(b). When graphing piecewisedefined functions on a graphing calculator, ou should use dot mode so that the calculator does not attempt to connect the pieces of the function. Figure 56 6 (, ) Y Y (a) (b) NOW WORK PROBLEM 9.
8 CHAPTER Functions and Their Graphs EXAMPLE Cost of Electricit In Ma 00, Commonwealth Edison Compan supplied electricit to residences for a monthl customer charge of $7. plus 8.75 per kilowatthour (kwhr) for the first 00 kwhr supplied in the month and 6.08 per kwhr for all usage over 00 kwhr in the month. (a) What is the charge for using 00 kwhr in a month? (b) What is the charge for using 700 kwhr in a month? (c) If C is the monthl charge for kwhr, epress C as a function of. SOURCE: Commonwealth Edison Co., Chicago, Illinois, 00. Solution (a) For 00 kwhr, the charge is $7. plus 8.75 = $ per kwhr. That is, Charge = $7. + $ = $.96 (b) For 700 kwhr, the charge is $7. plus 8.75 per kwhr for the first 00 kwhr plus 6.08 per kwhr for the 00 kwhr in ecess of 00. That is, Charge = $7. + $ $ = $58.85 Figure 57 Charge (dollars) 80 (700, 58.85) (00, 0.) 0 (00,.96) Usage (kwhr) (c) If 0 00, the monthl charge C (in dollars) can be found b multipling times $ and adding the monthl customer charge of $7.. So, if 0 00, then C = For 7 00, the charge is , since  00 equals the usage in ecess of 00 kwhr, which costs $ per kwhr. That is, if 7 00, then C = The rule for computing C follows two equations: See Figure 57 for the graph. = = if 0 00 C = e if 7 00
5.3 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 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 informationAlgebra II Notes Piecewise Functions Unit 1.5. Piecewise linear functions. Math Background
Piecewise linear functions Math Background Previousl, ou Related a table of values to its graph. Graphed linear functions given a table or an equation. In this unit ou will Determine when a situation requiring
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, 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 information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
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 information1.6. 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 informationFunctions and Their Graphs
3 Functions and Their Graphs On a sales rack of clothes at a department store, ou see a shirt ou like. The original price of the shirt was $00, but it has been discounted 30%. As a preferred shopper, ou
More 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 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 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
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 informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1.6 Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described by piecewise functions. LEARN ABOUT the Math A city parking lot
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 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 informationSection 14 Functions: Graphs and Properties
44 1 FUNCTIONS AND GRAPHS I(r). 2.7r where r represents R & D ependitures. (A) Complete the following table. Round values of I(r) to one decimal place. r (R & D) Net income I(r).66 1.2.7 1..8 1.8.99 2.1
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 informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
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 informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More 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 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 informationPolynomial and Rational Functions
Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
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 informationF.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions
F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions Analyze functions using different representations. 7. Graph functions expressed
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 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 informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More 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 information2.3 TRANSFORMATIONS OF GRAPHS
78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and timeandahalf for the number of hours he works above 0. Let denote the number of hours he
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 informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationMath 152, Intermediate Algebra Practice Problems #1
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
More informationName Date. BreakEven Analysis
Name Date BreakEven Analsis In our business planning so far, have ou ever asked the questions: How much do I have to sell to reach m gross profit goal? What price should I charge to cover m costs and
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationMATH 185 CHAPTER 2 REVIEW
NAME MATH 18 CHAPTER REVIEW Use the slope and intercept to graph the linear function. 1. F() = 4   Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationSlopeIntercept Form and PointSlope Form
SlopeIntercept Form and PointSlope Form In this section we will be discussing SlopeIntercept Form and the PointSlope Form of a line. We will also discuss how to graph using the SlopeIntercept Form.
More informationLinear and Quadratic Functions
Chapter Linear and Quadratic Functions. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
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 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 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 informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
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 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 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 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 informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
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 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 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 informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
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 informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
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 informationSECTION 22 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 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 collegelevel mathematics placement test An answer ke is provided
More informationSECTION 51 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 informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationSection 23 Quadratic Functions
118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationMATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
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 informationAlgebra 2 Unit 10 Tentative Syllabus Cubics & Factoring
Name Algebra Unit 10 Tentative Sllabus Cubics & Factoring DATE CLASS ASSIGNMENT Tuesda Da 1: S.1 Eponent s P: 1, 7 Jan Wednesda Da : S.1 More Eponent s P: 9 Jan Thursda Da : Graphing the cubic parent
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 informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationSection 37. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative
202 Chapter 3 The Derivative Section 37 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and
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 information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
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 reallife
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
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 informationf(x) = g(x), if x A h(x), if x B.
1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piecewise functions. Definition: Piecewisefunction. A piecewisefunction
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationSIMPLIFYING SQUARE ROOTS EXAMPLES
SIMPLIFYING SQUARE ROOTS EXAMPLES 1. Definition of a simplified form for a square root The square root of a positive integer is in simplest form if the radicand has no perfect square factor other than
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationXIV. Mathematics, Grade 8
XIV. Mathematics, Grade 8 Grade 8 Mathematics Test The spring 0 grade 8 Mathematics test was based on standards in the five domains for grade 8 in the Massachusetts Curriculum Framework for Mathematics
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 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 information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.
INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and
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 informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
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 information