# 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.

Save this PDF as:

Size: px
Start display at page:

Download "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."

## Transcription

1 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, and reciprocal functions. Identif and graph step and other piecewise-defined functions. Recognize graphs of parent functions. Wh ou should learn it Step functions can be used to model real-life situations. For instance, in Eercise 6 on page 7, ou will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this tet is to enable ou to recognize the basic shapes of the graphs of different tpes of functions. For instance, ou know that the graph of the linear function f a b is a line with slope m a and -intercept at 0, b. The graph of the linear function has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an -intercept of b m, 0 and a -intercept of 0, b. The graph is increasing if m > 0, decreasing if m < 0, and constant if m 0. Eample Writing a Linear Function Write the linear function f for which f and f 0. Solution To find the equation of the line that passes through,, and,, 0, first find the slope of the line. m 0 Net, use the point-slope form of the equation of a line. m Point-slope form Substitute for,, and m. Simplif. f Function notation The graph of this function is shown in Figure.6. Gett Images f() = + FIGURE.6 Now tr Eercise.

2 0_006.qd /7/0 8:0 AM Page 67 Section.6 A Librar of Parent Functions 67 Additional Eample Write the linear function f for which f 0 and f 8. Solution m m There are two special tpes of linear functions, the constant function and the identit function. A constant function has the form f c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line, as shown in Figure.66. The identit function has the form f. Its domain and range are the set of all real numbers. The identit function has a slope of m and a -intercept 0, 0. The graph of the identit function is a line for which each -coordinate equals the corresponding -coordinate. The graph is alwas increasing, as shown in Figure.67 f() = f() = c FIGURE.66 FIGURE.67 The graph of the squaring function f is a U-shaped curve with the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all nonnegative real numbers. The function is even. The graph has an intercept at 0, 0. The graph is decreasing on the interval, 0 and increasing on the interval 0,. The graph is smmetric with respect to the -ais. The graph has a relative minimum at 0, 0. The graph of the squaring function is shown in Figure.68. f() = (0, 0) FIGURE.68

3 0_006.qd /7/0 8:0 AM Page Chapter Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below.. The graph of the cubic function f has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The function is odd. The graph has an intercept at 0, 0. The graph is increasing on the interval,. The graph is smmetric with respect to the origin. The graph of the cubic function is shown in Figure.69.. The graph of the square root function f has the following characteristics. The domain of the function is the set of all nonnegative real numbers. The range of the function is the set of all nonnegative real numbers. The graph has an intercept at 0, 0. The graph is increasing on the interval 0,. The graph of the square root function is shown in Figure.70.. The graph of the reciprocal function f has the following characteristics. The domain of the function is, 0 0,. The range of the function is, 0 0,. The function is odd. The graph does not have an intercepts. The graph is decreasing on the intervals, 0 and 0,. The graph is smmetric with respect to the origin. The graph of the reciprocal function is shown in Figure.7. f() = f() = f() = (0, 0) (0, 0) Cubic function FIGURE.69 Square root function FIGURE.70 Reciprocal function FIGURE.7

4 0_006.qd /7/0 8:0 AM Page 69 Section.6 A Librar of Parent Functions 69 Demonstrate the real-life nature of step functions b discussing Eercises If writing is a part of our course, this section provides a good opportunit for students to find other eamples of step functions and write brief essas on their applications. FIGURE.7 f () =[[ ]] Technolog When graphing a step function, ou should set our graphing utilit to dot mode. Step and Piecewise-Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted b and defined as f the greatest integer less than or equal to. Some values of the greatest integer function are as follows. greatest integer greatest integer 0 greatest integer 0 0. greatest integer. The graph of the greatest integer function f has the following characteristics, as shown in Figure.7. The domain of the function is the set of all real numbers. The range of the function is the set of all integers. The graph has a -intercept at 0, 0 and -intercepts in the interval 0,. The graph is constant between each pair of consecutive integers. The graph jumps verticall one unit at each integer value. Eample Evaluating a Step Function FIGURE.7 f ()=[[ ]] + Evaluate the function when,, and f Solution For, the greatest integer is, so f 0. For, the greatest integer is, so f. For, the greatest integer is, so f. You can verif our answers b eamining the graph of f shown in Figure.7. Now tr Eercise 9. Recall from Section. that a piecewise-defined function is defined b two or more equations over a specified domain. To graph a piecewise-defined function, graph each equation separatel over the specified domain, as shown in Eample..

5 0_006.qd /7/0 8:0 AM Page Chapter Functions and Their Graphs = + FIGURE = + Eample Sketch the graph of f,, Solution Graphing a Piecewise-Defined Function This piecewise-defined function is composed of two linear functions. At and to the left of the graph is the line, and to the right of the graph is the line, as shown in Figure.7. Notice that the point, is a solid dot and the point, is an open dot. This is because f. Parent Functions >. Now tr Eercise. The eight graphs shown in Figure.7 represent the most commonl used functions in algebra. Familiarit with the basic characteristics of these simple graphs will help ou analze the shapes of more complicated graphs in particular, graphs obtained from these graphs b the rigid and nonrigid transformations studied in the net section. f() = f() = f() = c f() = (a) Constant Function (b) Identit Function (c) Absolute Value Function (d) Square Root Function f() = f() = f() = f () =[[ ]] (e) Quadratic Function FIGURE.7 (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function

6 0_006.qd /7/0 8:0 AM Page 7 Section.6 A Librar of Parent Functions 7.6 Eercises VOCABULARY CHECK: Match each function with its name.. f. f. f. f. f 6. f c f 9. f a b f (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (e) greatest integer function (h) reciprocal function (i) identit function PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 8, (a) write the linear function f such that it has the indicated function values and (b) sketch the graph of the function.. f, f 0 6. f 8, f. f, f 7. f 9, f. 6. f, f f 0, f 6 7. f 6, f 8. f, f In Eercises 9 8, use a graphing utilit to graph the function. Be sure to choose an appropriate viewing window. 9. f 0. f. f 6. f 6. f. f 8. h 6. g f 8. f 8 9. f 0. g. f. f. g. h. f 6. f 7. h 8. k In Eercises 9 6, evaluate the function for the indicated values. 9. f (a) f. (b) f.9 (c) f. (d) 0. g (a) g (b) g 0. (c) g 9. (d) f 7 g. h (a) h (b) h (c) h. (d) h.6. f 7 (a) f 0 (b) f. (c) f 6 (d) f. h (a) h. (b) h. (c) h 7 (d) h. k 6 (a) k (b) k 6. (c) k 0. (d) k. g (a) g.7 (b) g (c) g 0.8 (d) g. 6. g 7 6 (a) (b) g 9 (c) g (d) In Eercises 7, sketch the graph of the function. 7. g 8. g 9. g 0. g. g. g In Eercises 0, graph the function g 8 f,, g 6,, > f,, < 0 0 f,, < f,, > > g

7 0_006.qd /7/0 8:0 AM Page 7 7 Chapter Functions and Their Graphs h,, h,,,, k,, In Eercises and, (a) use a graphing utilit to graph the function, (b) state the domain and range of the function, and (c) describe the pattern of the graph. s.. In Eercises 60, (a) identif the parent function and the transformed parent function shown in the graph, (b) write an equation for the function shown in the graph, and (c) use a graphing utilit to verif our answers in parts (a) and (b) < 0 0 < < 0 0 < > g 6. Communications The cost of a telephone call between Denver and Boise is \$0.60 for the first minute and \$0. for each additional minute or portion of a minute. A model for the total cost C (in dollars) of the phone call is C t, t > 0 where t is the length of the phone call in minutes. (a) Sketch the graph of the model. (b) Determine the cost of a call lasting minutes and 0 seconds. 6. Communications The cost of using a telephone calling card is \$.0 for the first minute and \$0.8 for each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using a calling card for a call lasting t minutes. Which of the following is the appropriate model? Eplain. C t t C t t (b) Graph the appropriate model. Determine the cost of a call lasting 8 minutes and seconds. 6. Deliver Charges The cost of sending an overnight package from Los Angeles to Miami is \$0.7 for a package weighing up to but not including pound and \$.9 for each additional pound or portion of a pound. A model for the total cost C (in dollars) of sending the package is C 0.7.9, > 0 where is the weight in pounds. (a) Sketch a graph of the model. (b) Determine the cost of sending a package that weighs 0. pounds. 6. Deliver Charges The cost of sending an overnight package from New York to Atlanta is \$9.80 for a package weighing up to but not including pound and \$.0 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost C of overnight deliver of a package weighing pounds, > 0. (b) Sketch the graph of the function. 6. Wages A mechanic is paid \$.00 per hour for regular time and time-and-a-half for overtime. The weekl wage function is given b W h h, 8 h 0 80, 0 < h 0 h > 0 where h is the number of hours worked in a week. (a) Evaluate W 0, W 0, W, and W 0. (b) The compan increased the regular work week to hours. What is the new weekl wage function?

8 0_006.qd /7/0 8:0 AM Page 7 Section.6 A Librar of Parent Functions Snowstorm During a nine-hour snowstorm, it snows at a rate of inch per hour for the first hours, at a rate of inches per hour for the net 6 hours, and at a rate of 0. inch per hour for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How man inches of snow accumulated from the storm? Model It 67. Revenue The table shows the monthl revenue (in thousands of dollars) of a landscaping business for each month of the ear 00, with representing Januar. Volume (in gallons) V (60, 00) 00 (0, 7) (0, 7) 7 (, 0) 0 (, 0) (0, 0) (0, ) (0, ) (0, 0) Time (in minutes) FIGURE FOR 68 t Month, Revenue, Snthesis A mathematical model that represents these data is f (a) What is the domain of each part of the piecewisedefined function? How can ou tell? Eplain our reasoning. (b) Sketch a graph of the model. (c) Find f and f, and interpret our results in the contet of the problem. (d) How do the values obtained from the model in part (b) compare with the actual data values? 68. Fluid Flow The intake pipe of a 00-gallon tank has a flow rate of 0 gallons per minute, and two drainpipes have flow rates of gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the hour of time shown on the graph. (There are man correct answers.) True or False? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 69. A piecewise-defined function will alwas have at least one -intercept or at least one -intercept. 70. f,, 6, can be rewritten as f, <. Eploration In Eercises 7 and 7, write equations for the piecewise-defined function shown in the graph (0, 6) (, ) 6 8 Skills Review < < < (8, 0) In Eercises 7 and 7, solve the inequalit and sketch the solution on the real number line > 6 9 L In Eercises 7 and 76, determine whether the lines and L passing through the pairs of points are parallel, perpendicular, or neither. 7. L :,,, L :, 7,, L :,,, 9 L :,,, (, ) (, ) (7, 0) (, ) 6 (0, 0)

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

### 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.,

### 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

### 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

### 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

### 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

### Graphing Linear Equations in Slope-Intercept Form

4.4. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope

### 2.3 Writing Equations of Lines

. Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

### 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

### 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

### 5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

### I 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

### THE POINT-SLOPE FORM

. The Point-Slope Form (-) 67. THE POINT-SLOPE FORM In this section In Section. we wrote the equation of a line given its slope and -intercept. In this section ou will learn to write the equation of a

### 1.6. 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

### Functions 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

### 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.

### 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

### 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

### Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

### FINAL 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) -

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 2.7 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.

### Essential Question How can you solve a system of linear equations? \$15 per night. Cost, C (in dollars) \$75 per Number of. Revenue, R (in dollars)

5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

### 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

### 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.

### Introduction. Introduction

Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

### Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of

### Algebra 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

### 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.

.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

### 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

### LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

### 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 Special Systems of Linear Equations

5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

### 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,

### 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

### 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

### 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

### FINAL 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

### 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

### Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4

Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;

### 135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

### Q (x 1, y 1 ) m = y 1 y 0

. 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 two distinct points in the plane determine

### 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

### Solving Systems of Linear Equations by Graphing

. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. It

### Slope-Intercept Equation. Example

1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

### Translating Points. Subtract 2 from the y-coordinates

CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative

202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and

### Alex and Morgan were asked to graph the equation y = 2x + 1

Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

### Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions

SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life

### SLOPE 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

### 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

### Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes \$25 an hour as an electrician. If he works

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

### Math 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

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Systems of Linear Equations: Solving by Substitution

8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

### 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

### MATH 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

### 6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH

6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions

### 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

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### GASOLINE The graph represents the cost of gasoline at \$3 per gallon.

9-6 Slope-Intercept Form MAIN IDEA Graph linear equations using the slope and -intercept. New Vocabular slope-intercept form -intercept Math Online glencoe.com Etra Eamples Personal Tutor Self-Check Quiz

### 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,

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

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### 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

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated

### 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

### REVIEW 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.

### 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

### 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.

### 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

### The Slope-Intercept Form

7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph

### 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

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### 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

### BookTOC.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

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

### Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

### SECTION 2-2 Straight Lines

- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

### M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant

### Review 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 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

### 3.5 Rational Functions and Asymptotes

7_00.qp 98 /7/06 :0 PM Chapter Page 98 Polynomial and Rational Functions. Rational Functions and Asymptotes What you should learn Introduction to Rational Functions 䊏 A rational function can be written

### 1.6. 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

### Introduction - Algebra I

LIFORNI STNRS TEST lgebra I Introduction - lgebra I The following released test questions are taken from the lgebra I Standards Test. This test is one of the alifornia Standards Tests administered as part

### a > 0 parabola opens a < 0 parabola opens

Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(

### 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

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Section 2.3 Tangent lines, rates of change, and derivatives

Section.3 Tangent lines, rates of change, and derivatives (3/9/08) Overview: The derivative was developed in the seventeenth centur for determining tangent lines to curves and the velocit of moving objects