I think that starting


 Charles Hodges
 1 years ago
 Views:
Transcription
1 . 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 of mathematical endeavor are enough to achieve such enormousl impressive results in phsics. P.W.C. Davies Graph of a Function A function f assigns a range element to each domain element, so it is often useful to think of the function as pairing numbers (if the domain and range are sets of numbers). The function is completel determined b these ordered pairs. In mathematical notation, f, f D, where D is the domain of f. When the function is defined b an equation, f, then f, f, D. The ordered pairs that define f look like coordinates of points in the plane, and we can use this feature to define the graph of a function. Definition: graph of a function If f is a function with domain D, thenthegraphoffis the set of points with coordinates, such that D and f. An accurate graph of a function makes both the domain and the range of the function apparent. The domain is the set of values of points on the graph, and the range is the set of values. Function Properties and Graphs I think that starting mathematics earl had given me a certain selfreliance. I felt ou didn t learn anthing in class, ou just figured it out ourself. Paul Cohen It is natural to think of drawing a graph b plotting points and connecting them in an appropriate wa to get a curve. This is precisel how a computer or graphing calculator shows graphs on a screen. Without the capabilit to compute hundreds of function valuesquickl, however, pencil and paper techniques are time consuming and tedious. With or without access to the tools of technolog, some additional tools can help us draw graphs and understand the properties of functions. In this section, we eamine smmetr properties of graphs and introduce the notion of even and odd functions. Also, certain core graphs are given. Knowing a single core graph and how it is affected b simple changes, we can draw graphs of a whole famil of related functions. Core Graphs There are a few graphs that should be familiar to ever precalculus student. We will do lots of graphing with calculators and the aid of technolog, but ou should be able to sketch each of the following core graphs without an help. Knowing the properties of these simple functions and some of their ke features will make discussions of all kinds of functional behavior more meaningful. Use our graphing calculator as needed to help absorb the ideas, but make ourself confident that ou know the graphs of the functions in Figure.
2 70 Chapter Functions (a) f()= (b) f()= (c) f() = (d) f()= (e) f()= (f) f()= FIGURE Catalog of core graphs Intercepts, Smmetr, Even and Odd Functions We are alwas interested in the points where a graph meets the coordinate aes. If 0 is in the domain of f, thenf 0 is called the intercept and the point 0, f 0 is the intercept point. A graph need not meet the ais, but if it does, an points where it does are called intercept points, and if f c 0, then the number c is called an intercept. Given a point A a, b, three smmetricall located points can help in graphing. We can reflect A in the ais to the point C a, b, intheais to the point D a, b or in the origin to the point E a, b. Figure a shows a firstquadrant point A and its reflections. The graph of a function ma have all sorts of smmetr properties (or none). Suppose that for ever arbitrar point a, b on the graph of a function, the graph also contains the reflection of a, b in the ais. That is, whenever a, b is on the graph, a, b isalsoonthegraph.thenwesathatthegraphissmmetric about the ais and that the function is an even function. In functional notation, since to sa that a, b is on the graph means that f a b, a function is even when f f. See Figure b. If, whenever a, b is on the graph of f, a, b is also on the graph, then the graph is smmetric about the origin and the function is an odd function. In functional notation, a function is odd if f f for ever in D. See Figure c.
3 . Graphs of Functions 7 Reflection in ais A(a, b) C( a, b) ( a, b) (a, b) (a, b) E( a, b) 0 Reflection in ais Reflection in origin D(a, b) ( a, b) (a) Point Reflections (b) Even; Smmetric about ais (c) Odd; Smmetric about origin FIGURE The catalog of core graphs in Figure has eamples that should help ou remember the distinction between even and odd functions. The parabola and the absolute value function are even; the graphs are clearl smmetric about the ais. The line, the cubic, and the reciprocal are all smmetric about the origin; the functions are odd. The square root function is neither odd nor even. We sum up in the following. Definition: even and odd functions, smmetr properties Suppose f is a function with domain D. If, for ever in D, wehave f f,thenfis an even function; f f,thenfis an odd function. The graph of an even function is smmetric about the ais. The graph of an odd function is smmetric about the origin. In addition to just looking at a single graph, graphing calculators can be used to see whether or not f f or f f ; compare the graph of f with the graph of f ( ), where we replace each in f b. EXAMPLE Even and odd functions Sketch the graph and determine, both graphicall and algebraicall, whether the function is odd or even. (a) f ( (b) g( (a) Algebraic f, so f f, and b definition, f is an even function. Graphical The graph, shown in Figure (a), is smmetric about the ais and is clearl the graph of an even function. (b) Algebraic For the function g, wehaveg g, and so from the definition, g is an odd function. Graphical The calculator confirms that the graph is smmetric about the origin, and hence the graph of an odd function. See Figure b.
4 7 Chapter Functions = + = (a) Even Function (b) Odd Function FIGURE A function ma be neither even nor odd, as the net eample shows. EXAMPLE Neither even nor odd Show that the function is neither even nor odd, and sketch its graph (a) F (b) G Appl the definitions for even and odd functions. F F G G Since F F and F F, F is neither even nor odd, and similarl, G is neither even nor odd. To graph the two functions, plot points as in Figure 5. (, 0) = 5 6 = (, ) (, ) (, ) (0, ) (a) F is neither even nor odd. (b) G is neither even nor odd. FIGURE 5
5 . Graphs of Functions 7 Piecewise, Step Functions, and Calculator Graphs We have alread seen some eamples of an important class of functions called piecewise functions because the are defined in pieces (with different rules for different portions of their domains). The first eample we encountered is the absolute value function, even though we did not recognize it when we first introduced absolute values. Among the properties of absolute values in Section. we listed the equalit, which can be justified b considering values, but it ma also be helpful to get calculator reinforcement. Graph both Y ABS(X) and Y X and see that the graphs are identical. The function can also be written in pieces: if 0 if 0 The graph consists of part of the line and part of the line. You ma want to refer again to the Technolog Tip in Section. to make sure that ou know how to graph piecewisedefined functions on our calculator. (, ) [, ] b [.5,.5] FIGURE 6 EXAMPLE Calculator graphs of piecewise functions Draw a calculator graph of the piecewise function from Eample of Section., f if if The graph is shown in Figure 6. You ma wish to eperiment with different windows to see how changing the view alters the shape of the pieces we see. Whatever the window, however, the graph of f consists of two pieces that meet at the point (, ). Greatest Integer Function Another piecewisedefined function that is useful in man different contets is the greatest integer function, which is programmed into most graphing calculators, defined as the largest integer that is less than or equal to. An real number is either an integer or it lies between two integers. If n n, the largest integerlessthanorequaltois n, soforsuchan,int n. The function is denoted b Int on graphing calculators, or sometimes Floor, or, in older books, b. Definition: The greatest integer function The greatest integer function of, denoted b Int, isthe largest integer that is less than or equal to. As eamples, Int, and for an integer n, Int n n, Int 0. 0, and for an number between 0 and, Int 0, Int because is the largest integer less than, Int because is between and. Postal rates are eamples of functions defined piecewise, where the definition can make use of Int. Mailing cost is a function of the weight of a letter. It remains constant for a while and then suddenl jumps to a new value.
6 7 Chapter Functions (, 78) (, 55) (0, ) (, 0) 5 FIGURE 7 C W Int W EXAMPLE The postage function In 995, postal rates for first class letters delivered within the United States were set as follows: the cost is cents for anthing less than ounce; foreach additional ounce (up to ounces) the cost increases in increments of cents. Epress the cost C (in cents) of firstclass postage as a function of the weight W (ounces) and draw a graph. In mathematical language, we can epress the cost C as a function of the weight W either piecewise, or b using the greatest integer function. C Int W, 0 W. The graph of the cost function is shown in Figure 7. TECHNOLOGY TIP Graphing in dot mode In connected mode, a calculator graph of a piecewisedefined function can make it appear as if there is a vertical piece of the graph connecting pieces that should not be connected. Don t be fooled. There can be a jump from one piece to another between piels. When values differ in adjacent piel columns, the calculator connects piels in vertical columns joining separated points. It is often helpful to change to a nonconnected format to see jumps in graphs of piecewise functions. In calculator graphs, remember that unless the calculator is in dot mode, the calculator connects separated piels in adjacent columns. Not all equations define functions, and some familiar graphs are not graphs of functions. A function assigns to each domain element eactl one element of the range. This implies that for an function f and an given domain number c, there is eactl one point, c, f c, onthegraphoff. A vertical line can meet the graph of f in at most one point, which is the basis for the following hand test. Vertical line test For a given graph, if at each number c of the domain, the vertical line c intersects the graph in eactl one point, then the graph represents the graph of a function. If some vertical line meets a graph in more than a single point, then the graph is not the graph of a function. [, ] b [, ] FIGURE 8 Vertical line meets graph in two places. EXAMPLE 5 Vertical line test Useacalculatortosketchagraphofall points that satisf the equation. Use the vertical line test to verif that the graph is not that of a function. In function mode, we cannot enter equations ecept in the form of Y...,sowhen we solve the given equation for,weget. On the same screen we graph Y X and Y X. Thetwographstogetherformaparabolaopeningtotheright, as shown in Figure 8. Since an vertical line through the positive ais meets the graph twice, the vertical line test tells us that the graph is not the graph of a function.
7 . Graphs of Functions 75 EXERCISES. Check Your Understanding Eercises 6 True or False. Give reasons.. The graph of a function cannot have more than one  intercept point.. The graph of isidenticaltothegraphof.. For the greatest integer function f Int, (a) f.5 f.5 (b) f f.. The distance between the  andintercept points of the graph of is. 5. For an function f, the function g f is an even function. 6. For an even function f, if(, ) is on the graph of f, then (, ) must also be on the graph of f. Eercises 7 0 Fill in the blank so that the following statement is true. If ou draw a graph of function f using [ 0, 0] [ 0, 0], then the number of intercept points shown in the displa is. 7. f f f 0. f 6 Develop Master Eercises Isolated Points A function is given along with its domain. Draw a graph of the function. The graph consists of isolated points. State the range of the function.. f ; D,,. f ; D, 0,,. f ; D,, 0,,. f ; D,, Eercises 5 8 Make a table of several, ordered pairs that satisf the equation. Plot the points in our table and drawagraph Eercises 9 Find the value of or so that point P is on the graph of f. 9. f ; P, 0. f ; P,. f ; P,. f 8; P, 5 Eercises 6 Odd, Even Determine whether function f is odd, even, or neither. Do the same for g. First draw a graph then use algebra to support our conclusion.. f, g. f, g 5. f, g 6. f, g Eercises 7 8 Graphs Draw a graph of f. Give the coordinates of the  and intercept points. 7. f 8. f 9. g 0. g. f. g 6. g. g 5. f 6. g 7. g 8. f 8 Eercises 9 0 Calculator Graph Suppose ou are interested in using a graph to help ou get information about the zeros of f. Which of the given windows would ou use? 9. f 7 75 (i) 0, 0 0, 0 (ii) 0, 0 00, 00 (iii) 0, 0 00, f 5 00 (i) 0, 0 0, 0 (ii) 6, 0 5, 0 (iii) 0, 0 5, 00 Eercises Window Dimensions The graph of f has two intercept points and either a highest or lowest point. Give the dimensions of a window for which the graph of f will show this information. (Answers ma var.). f f 0 8. f 5. f 8 0 Eercises 5 8 Limited Domain (a) Draw a graph of f, with the indicated domain. (b) Find the range of f. 5. D 0 ; f 6. D 0 ; f 7. D ; f 8. D ; f
8 76 Chapter Functions Eercises 9 Piecewise Graph Draw a graph of the given function and determine the range. 9. f,, 0. f. f if 0 if 0 if 0 if 0 if if 9. Use the vertical line test to determine which of these graphs are graphs of functions that have as the independent variable. Eercises 8 Refer to the function f whose graph is shown in the diagram with domain, 6. = f () 5 (a) (b). From the graph, give f, f 0, andf.. Order the following numbers from smallest to largest: f, f, f, f 9.. (a) What is the maimum value (the largest value) of f? (b) What is the minimum value (the smallest value) of f? (c) What is the range of f? 5. Give the coordinates of the highest and the lowest point on the graph. 6. Give the coordinates of the intercept point and the intercept points. 7. (a) Forwhatvaluesofis f negative? (b) Forwhatvaluesofis f positive? 8. True or false. Eplain. (a) f is less than f. (b) f. isanegativenumber. (c) f f isanegativenumber. (d) There are three intercept points for the graph of f. (c) Eercises 50 5 Interesting Functions (a) Give the domain for f and for g. Evaluate f at 0,,0, 0, 0, 56. (b) Draw graphs of f and g on separate screens. (c) Look at the graphs and describe an interesting features. (d) Are functions f and g identical? Eplain. (e) What is the solution set for f? f? 50. f 6 6 g 8 5. f g Eercises 5 5 Functions Involving Abs (a) Draw a graph of f. Use a decimal window. (b) Use the graph to find the solution set for f f 5. f Eercises 5 55 (a) Evaluate f at 5,, 0,.5. (b) Give a formula for f in piecewise form. (c) Usepart(b) to draw a graph of f. 5. f min, f ma, 7
9 . Graphs of Functions 77 Eercises Graphs Involving Int Draw a graph of f. Use dot mode and a decimal window. (a) What is the rangeoff?(b) What is the solution set for f 5? 56. f Int 57. f Int 58. f Int f Int Int Eercises 60 6 Equation Involving [] Draw a graph of f. Use dot mode and a decimal window. Find the solution set for the given equation. Note: [] is the same as Int (). 60. f ; f 6. f ; f 6. f ; f 6. f ; f 0 Eercises 6 67 Set Find the solution set for the open sentence. 6. Int 5Int 0. (Hint: Factor.) Int p where p is a prime. 67. Int Postage Costs This section discussed postage charges. When a parcel or letter eceeds ounces, a different rule applies for determining mailing cost as a function of weight. The rule depends upon mailing zones as well as weight and is given in tabular form. For eample, when mailing from Zone to Zone 8, the table below lists charges where w is the weight (not eceeding the number of pounds) and c is the cost in dollars. (a) Draw a graph of c as a function of w. (b) What is the cost of mailing a package that weighs pounds 5 ounces? pounds ounces? (c) If the cost of mailing a package is $6.00, what do we know about its weight? w c Pounds Dollars Bug on a Ladder A bug starts at point (, 0) and travels along the line segment AB toward point (0, ) as shown in the diagram. If P, denotes the location of the bug when it has traveled a distance d from (, 0), epress the coordinates and as functions of the distance d. Eercises 70 7 B(0, ) d Parking Costs Bug has crawled a distance d to point P(, ) A(, 0) 70. A parking garage charges $.00 for parking up to one hour and $0.50 for each additional hour (or fraction thereof ), with a maimum of $8.00 if ou park hours or longer. Suppose denotes the number of hours ou park and (dollars) the corresponding cost. Then is a function of given in piecewise form: 0.50 Int 8 if if (a) Draw a graph of this function. Use dot mode. (b) If ou have onl $5.00, how long can ou park? 7. A parking garage charges $.00 for parking up to one hour and $0.0 for each additional fifteen minutes (or fraction thereof ), with a maimum of $0. Suppose denotes the number of hours ou park and (dollars) the corresponding cost. Then is a function of given in piecewise form: Int 0 if if 6.75 if 6.75 (a) Useseveralvaluesofto check this formula. (b) What is the cost for hours and 0 minutes? (c) Forwhatvaluesofis the cost $.50? Check using thegraph.usedotmodeandadecimalwindow.
The Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationSTRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs
6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous
More information2 Analysis of Graphs of
ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
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 information1.6 Graphs of Functions
.6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each coordinate was matched with onl one coordinate. We spent most of our time
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 information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More information2.5 Library of Functions; Piecewisedefined Functions
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.,
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
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 informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationThe only idea of real
80 Chapter Polnomial and Rational Functions. RATIONAL FUNCTIONS What is mathematics about? I think it s reall summed up in what I frequentl tell m classes. That is that proofs reall aren t there to convince
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
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 informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
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 informationA Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0.
0_006.qd 66 /7/0 Chapter.6 8:0 AM Page 66 Functions and Their Graphs A Librar of Parent Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root,
More information3 Functions and Graphs
54617_CH03_155224.QXP 9/14/10 1:04 PM Page 155 3 Functions and Graphs In This Chapter 3.1 Functions and Graphs 3.2 Smmetr and Transformations 3.3 Linear and Quadratic Functions 3.4 PiecewiseDefined Functions
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 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 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 informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More informationGraphing and transforming functions
Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
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 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 informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationINTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane.
P ( 1, 1 ) INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing Distance Between Two Points On The Coordinate Plane Q (, ) R (, 1 ) The straight line distance between two points P ( 1, 1
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 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 informationRational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza
Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia
More information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More informationModifying Functions  Families of Graphs
Worksheet 47 Modifing Functions  Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
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 information4.1 PiecewiseDefined Functions
Section 4.1 PiecewiseDefined Functions 335 4.1 PiecewiseDefined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More information8. Bilateral symmetry
. Bilateral smmetr Our purpose here is to investigate the notion of bilateral smmetr both geometricall and algebraicall. Actuall there's another absolutel huge idea that makes an appearance here and that's
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 informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
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 information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
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 informationGraphing Quadratic Functions
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
More informationRational Functions, Equations, and Inequalities
Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions
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 information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More information7.3 Graphing Rational Functions
Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
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 information17.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More informationFamilies of Quadratics
Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More informationQuadratic Functions. MathsStart. Topic 3
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
More informationTransformations of Function Graphs
   0        Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),
More information8 Graphs of Quadratic Expressions: The Parabola
8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be
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 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 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 information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationYears t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More information14.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes. Learning Style
The Area Bounded b a Curve 14.3 Introduction One of the important applications of integration is to find the area bounded b a curve. Often such an area can have a phsical significance like the work done
More informationREVIEW, pages
REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ  + ƒ b) = ƒ ( + )(  ) ƒ 8 ( )( ) Draw the graph of. It has intercept.. Reflect, in
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
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 APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationSolving x < a. Section 4.4 Absolute Value Inequalities 391
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
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 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 information3.1 Quadratic Functions
Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain
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 information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
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 informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
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 informationGraphs of Functions of Two Variables and Contour Diagrams
Graphs of Functions of Two Variables and Contour Diagrams In the previous section we introduced functions of two variables. We presented those functions primaril as tables. We eamined the differences between
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
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 informationSection 0.2 Set notation and solving inequalities
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
More informationMath 1400/1650 (Cherry): Quadratic Functions
Math 100/1650 (Cherr): Quadratic Functions A quadratic function is a function Q() of the form Q() = a + b + c with a 0. For eample, Q() = 3 7 + 5 is a quadratic function, and here a = 3, b = 7 and c =
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
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 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 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 information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular
More informationAttributes and Transformations of Reciprocal Functions VOCABULARY
TEKS FOCUS  Attributes and Transformations of Reciprocal Functions VOCABULARY TEKS (6)(G) Analze the effect on the graphs of f () = when f () is replaced b af (), f (b), f (  c), and f () + d for specific
More informationFunctions. Chapter A B C D E F G H I J. The reciprocal function x
Chapter 1 Functions Contents: A B C D E F G H I J Relations and functions Function notation, domain and range Composite functions, f± g Sign diagrams Inequalities (inequations) The modulus function 1 The
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
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 informationCALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES
6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits
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 informationChapter 3. Curve Sketching. By the end of this chapter, you will
Chapter 3 Curve Sketching How much metal would be required to make a ml soup can? What is the least amount of cardboard needed to build a bo that holds 3 cm 3 of cereal? The answers to questions like
More information