# I think that starting

Save this PDF as:

Size: px
Start display at page:

## 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 self-reliance. 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, inthe-ais to the point D a, b or in the origin to the point E a, b. Figure a shows a first-quadrant 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 piecewise-defined 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 piecewise-defined 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 first-class 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 piecewise-defined 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 non-connected 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 - and-intercept 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

### STRETCHING, 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

### 2 Analysis of Graphs of

ch.pgs1-16 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,

### 2.3 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 time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he

### 1.6 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

### INVESTIGATIONS 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.

### 2.4 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

### HI-RES STILL TO BE SUPPLIED

1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

### 2.5 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.,

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

### When 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

### Section 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.

### The 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

### Solving 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

### EQUATIONS 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,

### Simplification 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

### C3: 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

### A 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,

### 3 Functions and Graphs

54617_CH03_155-224.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 Piecewise-Defined Functions

### 1.6. 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

### Solving 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

### Section 1-4 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

### GRAPHS 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

### Graphing 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

### Polynomial 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

### SAMPLE. 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

### Downloaded 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

### P1. 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

### INTRODUCTION 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

.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

### 6. 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

### Rational 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

### 4 Non-Linear relationships

NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

### Modifying 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

### Let (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. Slope-Intercept Form

8 (-) Chapter Linear Equations in Two Variables and Their Graphs In this section Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications

### Higher. 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

### Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

### More 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

### 4.1 Piecewise-Defined Functions

Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.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

### 8. 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

### 5.2 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,

### If (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

### Higher. 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

### 13 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,

### 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

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

### Rational 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

### Zeros 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

. 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

### 7.3 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

### SECTION 2-5 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

### Exponential 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,

### 17.1 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,

### Lesson 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

### Graphing 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

### Analyzing 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

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

### A 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

### 8.7 Systems of Non-Linear Equations and Inequalities

8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

### Quadratic 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

### Transformations 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),

### 8 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

### Graphing 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

### LINEAR 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

### Solving 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

### 1.2 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

### Filling 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

### Years 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 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

### 14.3. 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

### REVIEW, 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

### Objectives. 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

### D.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

### Solving 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

### Linear 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.

### Linear 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

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

### Functions 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

### 10.2 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

### THE 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

### Systems 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.

### D.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

### Graphs 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

### Polynomial 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

### 1. 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,

### Section 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.

### Math 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 =

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

### In 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

### Polynomials. 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

### POLYNOMIAL 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

### 2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math

Chapter 2 Lesson 2-5 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

### Attributes 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

### Functions. 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

### 3 Rectangular Coordinate System and Graphs

060_CH03_13-154.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

### 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

### CALCULUS 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