Using Intercept Form

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Using Intercept Form"

Transcription

1 8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of the form f () = ( p)( q) or f () = ( p)( q). Write the function represented b each graph. Eplain our reasoning. a. b. c. d. e. f. g. h. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. Communicate Your Answer. What are some of the characteristics of the graph of f () = a( p)( q)? 3. Consider the graph of f () = a( p)( q). a. Does changing the sign of a change the -intercepts? Does changing the sign of a change the -intercept? Eplain our reasoning. b. Does changing the value of p change the -intercepts? Does changing the value of p change the -intercept? Eplain our reasoning. Section 8.5 Using Intercept Form 9

2 8.5 Lesson What You Will Learn Core Vocabular intercept form, p. 50 Graph quadratic functions of the form f () = a( p)( q). Use intercept form to find zeros of functions. Use characteristics to graph and write quadratic functions. Use characteristics to graph and write cubic functions. Graphing f ( ) = a( p)( q) You have alread graphed quadratic functions written in several different forms, such as f () = a + b + c (standard form) and g() = a( h) + k (verte form). Quadratic functions can also be written in intercept form, f () = a( p)( q), where a 0. In this form, the polnomial that defines a function is in factored form and the -intercepts of the graph can be easil determined. Core Concept Graphing f ( ) = a( p)( q) The -intercepts are p and q. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is = p + q. The graph opens up when a > 0, and the graph opens down when a < 0. (p, 0) = p + q (q, 0) Graphing f () = a( p)( q) Graph f () = ( + 1)( 5). Describe the domain and range. Step 1 Identif the -intercepts. Because the -intercepts are p = 1 and q = 5, plot ( 1, 0) and (5, 0). Step Find and graph the ais of smmetr. f() = ( + 1)( 5) = p + q = = 10 (, 9) Step 3 Find and plot the verte. 8 The -coordinate of the verte is. To find the -coordinate of the verte, substitute for and simplif. f () = ( + 1)( 5) = 9 So, the verte is (, 9). ( 1, 0) (5, 0) Step Draw a parabola through the verte and the points where the -intercepts occur. The domain is all real numbers. The range is Chapter 8 Graphing Quadratic Functions

3 Graphing a Quadratic Function Graph f () = 8. Describe the domain and range. Step 1 Rewrite the quadratic function in intercept form. f () = 8 Write the function. = ( ) Factor out common factor. = ( + )( ) Difference of two squares pattern Step Identif the -intercepts. Because the -intercepts are p = and q =, plot (, 0) and (, 0). Step 3 Find and graph the ais of smmetr. (, 0) (, 0) = p + q + = 0 Step Find and plot the verte. The -coordinate of the verte is 0. The -coordinate of the verte is f (0) = (0) 8 = 8. So, the verte is (0, 8). Step 5 Draw a parabola through the verte and the points where the -intercepts occur. (0, 8) f() = 8 The domain is all real numbers. The range is 8. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the quadratic function. Label the verte, ais of smmetr, and -intercepts. Describe the domain and range of the function. REMEMBER Functions have zeros, and graphs have -intercepts. 1. f () = ( + )( 3). g() = ( )( + 1) 3. h() = 3 Using Intercept Form to Find Zeros of Functions In Section 8., ou learned that a zero of a function is an -value for which f () = 0. You can use the intercept form of a function to find the zeros of the function. 5 Check Finding Zeros of a Function Find the zeros of f () = ( 1)( + ). To find the zeros, determine the -values for which f () is 0. f () = ( 1)( + ) Write the function. 0 = ( 1)( + ) Substitute 0 for f(). 1 = 0 or + = 0 Zero-Product Propert = 1 or = Solve for. So, the zeros of the function are and 1. Section 8.5 Using Intercept Form 51

4 Core Concept Factors and Zeros For an factor n of a polnomial, n is a zero of the function defined b the polnomial. Finding Zeros of Functions Find the zeros of each function. a. f () = 10 1 b. h() = ( 1)( 1) LOOKING FOR STRUCTURE The function in Eample (b) is called a cubic function. You can etend the concept of intercept form to cubic functions. You will graph a cubic function in Eample 7. Write each function in intercept form to identif the zeros. a. f () = 10 1 Write the function. = ( ) Factor out common factor. = ( + 3)( + ) Factor the trinomial. So, the zeros of the function are 3 and. b. h() = ( 1)( 1) Write the function. = ( 1)( + )( ) Difference of two squares pattern So, the zeros of the function are, 1, and. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the zero(s) of the function.. f () = ( )( 1) 5. g() = h() = ( 1) Using Characteristics to Graph and Write Quadratic Functions Use zeros to graph h() = 3. Graphing a Quadratic Function Using Zeros The function is in standard form. You know that the parabola opens up (a > 0) and the -intercept is 3. So, begin b plotting (0, 3). ATTENDING TO PRECISION To sketch a more precise graph, make a table of values and plot other points on the graph. Notice that the polnomial that defines the function is factorable. So, write the function in intercept form and identif the zeros. h() = 3 Write the function. = ( + 1)( 3) Factor the trinomial. The zeros of the function are 1 and 3. So, plot ( 1, 0) and (3, 0). Draw a parabola through the points. h() = 3 5 Chapter 8 Graphing Quadratic Functions

5 Writing Quadratic Functions STUDY TIP In part (a), man possible functions satisf the given condition. The value a can be an nonzero number. To allow easier calculations, let a = 1. B letting a =, the resulting function would be f () = Write a quadratic function in standard form whose graph satisfies the given condition(s). a. verte: ( 3, ) b. passes through ( 9, 0), (, 0), and (, 0) a. Because ou know the verte, use verte form to write a function. f () = a( h) + k Verte form = 1( + 3) + Substitute for a, h, and k. = Find the product ( + 3). = Combine like terms. b. The given points indicate that the -intercepts are 9 and. So, use intercept form to write a function. f () = a( p)( q) Intercept form = a( + 9)( + ) Substitute for p and q. Use the other given point, (, 0), to find the value of a. 0 = a( + 9)( + ) Substitute for and 0 for f(). 0 = a(5)( ) Simplif. = a Solve for a. Use the value of a to write the function. f () = ( + 9)( + ) Substitute for a. = 3 Simplif. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use zeros to graph the function. 7. f () = ( 1)( ) 8. g() = + 1 Write a quadratic function in standard form whose graph satisfies the given condition(s). 9. -intercepts: 1 and verte: (8, 8) 11. passes through (0, 0), (10, 0), and (, 1) 1. passes through ( 5, 0), (, 0), and (3, 1) Using Characteristics to Graph and Write Cubic Functions In Eample, ou etended the concept of intercept form to cubic functions. f () = a( p)( q)( r), a 0 Intercept form of a cubic function The -intercepts of the graph of f are p, q, and r. Section 8.5 Using Intercept Form 53

6 Use zeros to graph f () = 3. Graphing a Cubic Function Using Zeros Notice that the polnomial that defines the function is factorable. So, write the function in intercept form and identif the zeros. f () = 3 Write the function. = ( ) Factor out. = ( + )( ) Difference of two squares pattern The zeros of the function are, 0, and. So, plot (, 0), (0, 0), and (, 0). To help determine the shape of the graph, find points between the zeros f() 3 3 f() = 3 Plot ( 1, 3) and (1, 3). Draw a smooth curve through the points. Writing a Cubic Function The graph represents a cubic function. Write the function. (0, 0) 8 (3, 1) (, 0) (5, 0) From the graph, ou can see that the -intercepts are 0,, and 5. Use intercept form to write a function. f () = a( p)( q)( r) Intercept form = a( 0)( )( 5) Substitute for p, q, and r. = a()( )( 5) Simplif. Use the other given point, (3, 1), to find the value of a. 1 = a(3)(3 )(3 5) Substitute 3 for and 1 for f(). = a Solve for a. Use the value of a to write the function. f () = ()( )( 5) Substitute for a. = Simplif. The function represented b the graph is f () = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use zeros to graph the function. 13. g() = ( 1)( 3)( + 3) 1. h() = The zeros of a cubic function are 3, 1, and 1. The graph of the function passes through the point (0, 3). Write the function. 5 Chapter 8 Graphing Quadratic Functions

7 8.5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The values p and q are of the graph of the function f () = a( p)( q).. WRITING Eplain how to find the maimum value or minimum value of a quadratic function when the function is given in intercept form. Monitoring Progress and Modeling with Mathematics In Eercises 3, find the -intercepts and ais of smmetr of the graph of the function. 3. = ( 1)( + 3) 1. = ( )( 5) 5. = g() = 8 7. f () = ( + 5)( ) 8. h() = ( 3)( 11) 9. = = In Eercises 31 3, match the function with its graph. 31. = ( + 5)( + 3) 3. = ( + 5)( 3) 33. = ( 5)( + 3) 3. = ( 5)( 3) 5. f () = 5( + 7)( 5). g() = ( + 8) 3 In Eercises 7 1, graph the quadratic function. Label the verte, ais of smmetr, and -intercepts. Describe the domain and range of the function. (See Eample 1.) 7. f () = ( + )( + 1) 8. = ( )( + ) 35. = ( + 5)( 5) 3. = ( + 3)( 3) A. 8 B = ( + )( ) 10. h() = ( 7)( 3) g() = 5( + 1)( + ) 1. = ( 3)( + ) In Eercises 13 0, graph the quadratic function. Label the verte, ais of smmetr, and -intercepts. Describe the domain and range of the function. (See Eample.) 13. = 9 1. f () = 8 C. 8 D h() = = q() = p() = + 7 E. F. 19. = = In Eercises 1 30, find the zero(s) of the function. (See Eamples 3 and.) 1. = ( )( 10). f () = 1 ( + 5)( 1) g() = + 5. = Section 8.5 Using Intercept Form 55

8 In Eercises 37, use zeros to graph the function (See Eample 5.) Dnamic Solutions available (, at 3) BigIdeasMath.com (, 0) (10, 0) 37. f () = ( + )( ) 38. g() = 3( + 1)( + 7) 39. = = = h() = (, 0) (, 0) 8 (8, ) ERROR ANALYSIS In Eercises 3 and, describe and correct the error in finding the zeros of the function. 3.. = 5( + 3)( ) The zeros of the function are 3 and. = ( + )( 9) The zeros of the function are and 9. In Eercises 5 5, write a quadratic function in standard form whose graph satisfies the given condition(s). (See Eample.) 5. verte: (7, 3). verte: (, 8) 7. -intercepts: 1 and intercepts: and 5 9. passes through (, 0), (3, 0), and (, 18) 50. passes through ( 5, 0), ( 1, 0), and (, 3) 51. passes through (7, 0) 5. passes through (0, 0) and (, 0) 53. ais of smmetr: = 5 5. increases as increases when < ; decreases as increases when >. 55. range: 3 5. range: 10 In Eercises 57 0, write the quadratic function represented b the graph ( 3, 0) 8 (, 0) (1, 8) (1, 0) (7, 0) (, 5) In Eercises 1 8, use zeros to graph the function. (See Eample 7.) 1. = 5( + )( ). f () = ( + 9)( + 3) 3. h() = ( )( + )( + 7). = ( + 1)( 5)( ) 5. f () = = = g() = In Eercises 9 7, write the cubic function represented b the graph. (See Eample 8.) ( 1, 0) (0, 0) (1, ) (, 0) (, 0) 0 ( 7, 0) (0, 0) (, 0) ( 3, 0) ( 1, 3) (0, 0) (, 0) (1, 0) (, 0) (3, 0) (5, 0) In Eercises 73 7, write a cubic function whose graph satisfies the given condition(s) intercepts:, 3, and intercepts: 7, 5, and passes through (1, 0) and (7, 0) 7. passes through (0, ) 5 Chapter 8 Graphing Quadratic Functions

9 In Eercises 77 80, all the zeros of a function are given. Use the zeros and the other point given to write a quadratic or cubic function represented b the table. 8. MODELING WITH MATHEMATICS A professional basketball plaer s shot is modeled b the function shown, where and are measured in feet = ( ) (3, 0) a. Does the plaer make the shot? Eplain. b. The basketball plaer releases another shot from the point (13, 0) and makes the shot. The shot also passes through the point (10, 1.). Write a quadratic function in standard form that models the path of the shot. In Eercises 81 8, sketch a parabola that satisfies the given conditions intercepts: and ; range: 3 8. ais of smmetr: = ; passes through (, 15) 83. range: 5; passes through (0, ) 8. -intercept: ; -intercept: 1; range: USING STRUCTURE In Eercises 87 90, match the function with its graph. 87. = = = = MODELING WITH MATHEMATICS Satellite dishes are shaped like parabolas to optimall receive signals. The cross section of a satellite dish can be modeled b the function shown, where and are measured in feet. The -ais represents the top of the opening of the dish. 1 = ( ) 8 A. B. C. D. 1 a. How wide is the satellite dish? b. How deep is the satellite dish? c. Write a quadratic function in standard form that models the cross section of a satellite dish that is feet wide and 1.5 feet deep. 91. CRITICAL THINKING Write a quadratic function represented b the table, if possible. If not, eplain wh Section 8.5 Using Intercept Form 57

10 9. HOW DO YOU SEE IT? The graph shows the parabolic arch that supports the roof of a convention center, where and are measured in feet a. The arch can be represented b a function of the form f () = a( p)( q). Estimate the values of p and q. b. Estimate the width and height of the arch. Eplain how ou can use our height estimate to calculate a. ANALYZING EQUATIONS In Eercises 93 and 9, (a) rewrite the quadratic function in intercept form and (b) graph the function using an method. Eplain the method ou used. 93. f () = 3( + 1) g() = ( 1) 95. WRITING Can a quadratic function with eactl one real zero be written in intercept form? Eplain. 9. MAKING AN ARGUMENT Your friend claims that an quadratic function can be written in standard form and in verte form. Is our friend correct? Eplain. 97. PROBLEM SOLVING Write the function represented b the graph in intercept form. 80 (1, 0) (, 0) ( 5, 0) (8, 0) 10 (, 0) 10 (0, 3) 98. THOUGHT PROVOKING Sketch the graph of each function. Eplain our procedure. a. f () = ( 1)( ) b. g() = ( 1)( ) 99. REASONING Let k be a constant. Find the zeros of the function f () = k k k 3 in terms of k. PROBLEM SOLVING In Eercises 100 and 101, write a sstem of two quadratic equations whose graphs intersect at the given points. Eplain our reasoning (, 0) and (, 0) 101. (3, ) and (7, ) Maintaining Mathematical Proficienc The scatter plot shows the amounts (in grams) of fat and the numbers of calories in 1 burgers at a fast-food restaurant. (Section.) 10. How man calories are in the burger that contains 1 grams of fat? 103. How man grams of fat are in the burger that contains 00 calories? 10. What tends to happen to the number of calories as the number of grams of fat increases? Reviewing what ou learned in previous grades and lessons Calories Burgers Determine whether the sequence is arithmetic, geometric, or neither. Eplain our reasoning. (Section.) , 11, 1, 33, 7, ,, 18, 5, , 18, 10,,, , 5, 9, 1, 3, Fat (grams) 58 Chapter 8 Graphing Quadratic Functions

4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? 3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the

More information

Writing Equations in Point-Slope Form

Writing Equations in Point-Slope Form . Writing Equations in Point-Slope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch

More information

Linear Functions. Essential Question How can you determine whether a function is linear or nonlinear?

Linear Functions. Essential Question How can you determine whether a function is linear or nonlinear? . Linear Functions Essential Question How can ou determine whether a function is linear or nonlinear? Finding Patterns for Similar Figures Work with a partner. Cop and complete each table for the sequence

More information

Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a

Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a 0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value

More information

Essential Question How can you determine whether a polynomial equation has imaginary solutions? 2 B. 4 D. 4 F.

Essential Question How can you determine whether a polynomial equation has imaginary solutions? 2 B. 4 D. 4 F. 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.A The Fundamental Theorem of Algebra Essential Question How can ou determine whether a polnomial equation has imaginar solutions? Cubic Equations and Imaginar

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

More information

How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines

How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines .7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the

More information

Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson 1.1 Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s

More information

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

3.2. Parallel Lines and Transversals

3.2. Parallel Lines and Transversals . Parallel Lines and Transversals Essential Question When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? Exploring Parallel Lines Work with a partner.

More information

Definition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs Applications

Definition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs Applications 616 9 Additional Topics in Analtic Geometr 53. Space Science. A designer of a 00-foot-diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 100 feet above the verte

More information

Algebra Notes Quadratic Functions and Equations Unit 08

Algebra Notes Quadratic Functions and Equations Unit 08 Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic

More information

Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks).

Graphs of Polynomials: Polynomial functions of degree 2 or higher are smooth and continuous. (No sharp corners or breaks). Graphs of Polynomials: Polynomial functions of degree or higher are smooth and continuous. (No sharp corners or breaks). These are graphs of polynomials. These are NOT graphs of polynomials There is a

More information

How can you use multiplication or division to solve an inequality? ACTIVITY: Using a Table to Solve an Inequality

How can you use multiplication or division to solve an inequality? ACTIVITY: Using a Table to Solve an Inequality . Solving Inequalities Using Multiplication or Division How can you use multiplication or division to solve an inequality? 1 ACTIVITY: Using a Table to Solve an Inequality Work with a partner. Copy and

More information

Attributes of Polynomial Functions VOCABULARY

Attributes of Polynomial Functions VOCABULARY 8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze

More information

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE

Answer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

More information

Answers. Investigation 2. ACE Assignment Choices. Applications. Problem 2.5. Problem 2.1. Problem 2.2. Problem 2.3. Problem 2.4

Answers. Investigation 2. ACE Assignment Choices. Applications. Problem 2.5. Problem 2.1. Problem 2.2. Problem 2.3. Problem 2.4 Answers Investigation ACE Assignment Choices Problem. Core, Problem. Core, Other Applications ; Connections, 3; unassigned choices from previous problems Problem.3 Core Other Connections, ; unassigned

More information

The Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set

The Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using

More information

290 Chapter 1 Functions and Graphs

290 Chapter 1 Functions and Graphs 90 Chapter Functions and Graphs 7. Studies show that teting while driving is as risk as driving with a 0.08 blood alcohol level, the standard for drunk driving. The bar graph shows the number of fatalities

More information

APPENDIX D Rotation and the General Second-Degree Equation

APPENDIX D Rotation and the General Second-Degree Equation APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the

More information

Table of Contents. Module 1

Table of Contents. Module 1 Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra

More information

1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY

1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Quickwrite. Graph both = - for < 3 and = - + 7 for

More information

CHAPTER 3 Graphs and Functions

CHAPTER 3 Graphs and Functions CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.

More information

2.1 The Rectangular Coordinate System

2.1 The Rectangular Coordinate System . The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table

More information

where a =, and k =. Example 1: Determine if the function is a power function. For those that are not, explain why not.

where a =, and k =. Example 1: Determine if the function is a power function. For those that are not, explain why not. . Power Functions with Modeling PreCalculus. POWER FUNCTIONS WITH MODELING Learning Targets: 1. Identify a power functions.. Model power functions using the regression capabilities of your calculator.

More information

Chapter 9 BUILD YOUR VOCABULARY

Chapter 9 BUILD YOUR VOCABULARY C H A P T E R 9 BUILD YUR VCABULARY Chapter 9 This is an alphabetical list of new vocabular terms ou will learn in Chapter 9. As ou complete the stud notes for the chapter, ou will see Build Your Vocabular

More information

SECOND-DEGREE INEQUALITIES

SECOND-DEGREE INEQUALITIES 60 (-40) Chapter Nonlinear Sstems and the Conic Sections 0 0 4 FIGURE FOR EXERCISE GETTING MORE INVOLVED. Cooperative learning. Let (, ) be an arbitrar point on an ellipse with foci (c, 0) and ( c, 0)

More information

Last modified Spring 2016

Last modified Spring 2016 Math 00 Final Review Questions In problems 6, perform the indicated operations and simplif if necessar.. 8 6 8. 7 6. ( i) ( 4 i) 4. (8 i). ( 9 i)( 7 i) 6. ( i)( i) In problems 7-, solve the following applications.

More information

Mathematics 2201 Common Mathematics Assessment June 12, 2013

Mathematics 2201 Common Mathematics Assessment June 12, 2013 Common Mathematics Assessment June 1, 013 Name: Mathematics Teacher: 8 Selected Response 8 marks 13 Constructed Response marks FINAL 70 Marks TIME: HOURS NOTE Diagrams are not necessaril drawn to scale.

More information

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS

MATH 021 UNIT 1 HOMEWORK ASSIGNMENTS MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,

More information

Final Exam Review Part 2 #4

Final Exam Review Part 2 #4 Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + = 8. Solve for, where is a real number. 9 1 = 3. Solve

More information

Chapter 4E - Combinations of Functions

Chapter 4E - Combinations of Functions Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?

More information

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line. MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for

More information

MODELING WITH FUNCTIONS

MODELING WITH FUNCTIONS MATH HIGH SCHOOL MODELING WITH FUNCTIONS Copright 05 b Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected b copright,

More information

Quadratic Functions. Graphing Quadratic Functions

Quadratic Functions. Graphing Quadratic Functions UNIT 3 Quadratic Functions CONTENTS COMMON CORE F-BF.B.3 F-BF.B.3 F-IF.B. MODULE Graphing Quadratic Functions Lesson.1 Understanding Quadratic Functions................... 11 Lesson. Transforming Quadratic

More information

New Jersey Quality Single Accountability Continuum (NJQSAC) A-SSE 1-2; A-CED 1,4; A-REI 1-3, F-IF 1-5, 7a

New Jersey Quality Single Accountability Continuum (NJQSAC) A-SSE 1-2; A-CED 1,4; A-REI 1-3, F-IF 1-5, 7a ALGEBRA 2 HONORS Date: Unit 1, September 4-30 How do we use functions to solve real world problems? What is the meaning of the domain and range of a function? What is the difference between dependent variable

More information

mentoringminds.com MATH LEVEL 6 Student Edition Sample Page Unit 33 Introduction Use the coordinate grid to answer questions 1 9.

mentoringminds.com MATH LEVEL 6 Student Edition Sample Page Unit 33 Introduction Use the coordinate grid to answer questions 1 9. Student Edition Sample Page Name Standard 6.11(A) Readiness Unit 33 Introduction Use the coordinate grid to answer questions 1 9. A 6 F 5 L E 4 3 I 1 B K 6 5 4 3 1 1 3 4 5 6 1 H C D 3 G 4 5 J 6 1 Which

More information

Systems of Linear Equations

Systems of Linear Equations Sstems of Linear Equations Monetar Sstems Overload Lesson 3-1 Learning Targets: Use graphing, substitution, and elimination to solve sstems of linear equations in two variables. Formulate sstems of linear

More information

Lesson 24: Modeling with Quadratic Functions

Lesson 24: Modeling with Quadratic Functions Student Outcomes Students create a quadratic function from a data set based on a contextual situation, sketch its graph, and interpret both the function and the graph in context. They answer questions

More information

16x y 8x. 16x 81. U n i t 3 P t 1 H o n o r s P a g e 1. Math 3 Unit 3 Day 1 - Factoring Review. I. Greatest Common Factor GCF.

16x y 8x. 16x 81. U n i t 3 P t 1 H o n o r s P a g e 1. Math 3 Unit 3 Day 1 - Factoring Review. I. Greatest Common Factor GCF. P a g e 1 Math 3 Unit 3 Day 1 - Factoring Review I. Greatest Common Factor GCF Eamples: A. 3 6 B. 4 8 4 C. 16 y 8 II. Difference of Two Squares Draw ( - ) ( + ) Square Root 1 st and Last Term Eamples:

More information

Parametric Equations for Circles and Ellipses

Parametric Equations for Circles and Ellipses Lesson 5-8 Parametric Equations for Circles and Ellipses BIG IDEA Parametric equations use separate functions to defi ne coordinates and and to produce graphs Vocabular parameter parametric equations equation

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) x y =

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) x y = Santa Monica College Practicing College Algebra MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the standard equation for the circle. 1) Center

More information

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

More information

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

More information

Original site. translation. transformation. Decide whether the red figure is a translation of the blue figure. Compare a Figure and Its Image

Original site. translation. transformation. Decide whether the red figure is a translation of the blue figure. Compare a Figure and Its Image Page of 8 3.7 Translations Goal Identif and use translations. Ke Words translation image transformation In 996, New York Cit s Empire Theater was slid 70 feet up 2nd Street to a new location. Original

More information

Syllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.

Syllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function. Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in

More information

Chapter 1 Coordinates, points and lines

Chapter 1 Coordinates, points and lines Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and

More information

Chapter 4. Chapter 4 Opener. Section 4.1. Big Ideas Math Blue Worked-Out Solutions. x 2. Try It Yourself (p. 147) x 0 1. y ( ) x 2

Chapter 4. Chapter 4 Opener. Section 4.1. Big Ideas Math Blue Worked-Out Solutions. x 2. Try It Yourself (p. 147) x 0 1. y ( ) x 2 Chapter Chapter Opener Tr It Yourself (p. 7). As the input decreases b, the output increases b.. Input As the input increases b, the output increases b.. As the input decreases b, the output decreases

More information

Systems of Linear Equations Monetary Systems Overload

Systems of Linear Equations Monetary Systems Overload Sstems of Linear Equations SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Interactive Word Wall Have ou ever noticed that when an item is popular and man people want to bu it, the price

More information

Solving Systems of Linear Equations and Inequalities

Solving Systems of Linear Equations and Inequalities Solving Sstems of Linear Equations and Inequalities Lesson 7-1 Solve sstems of linear equations b graphing. Lessons 7-2 through 7-4 Solve sstems of linear equations algebraicall. Lesson 7-5 Solve sstems

More information

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1. 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

More information

California Common Core State Standards for Mathematics Standards Map Algebra I

California Common Core State Standards for Mathematics Standards Map Algebra I A Correlation of Pearson CME Project Algebra 1 Common Core 2013 to the California Common Core State s for Mathematics s Map Algebra I California Common Core State s for Mathematics s Map Algebra I Indicates

More information

Intermediate Algebra 100A Final Exam Review Fall 2007

Intermediate Algebra 100A Final Exam Review Fall 2007 1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,

More information

Diagnostic Tests Study Guide

Diagnostic Tests Study Guide California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra

More information

Bridge to Algebra II Standards for Mathematical Practice

Bridge to Algebra II Standards for Mathematical Practice Bridge to Algebra II Standards for Mathematical Practice The Standards for Mathematical Practices are to be interwoven and should be addressed throughout the year in as many different units and tasks as

More information

Solving Equations Quick Reference

Solving Equations Quick Reference Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First

More information

Functions. Academic Vocabulary causation. Math Terms

Functions. Academic Vocabulary causation. Math Terms Functions Unit Overview In this unit, ou will build linear models and use them to stud functions, domain, and range. Linear models are the foundation for studing slope as a rate of change, intercepts,

More information

UNIT 5 CONGRUENCE, PROOF, AND CONSTRUCTIONS Lesson 2: Defining and Applying Rotations, Reflections, and Translations Instruction

UNIT 5 CONGRUENCE, PROOF, AND CONSTRUCTIONS Lesson 2: Defining and Applying Rotations, Reflections, and Translations Instruction UNIT ONGRUENE, PROOF, ND ONSTRUTIONS Lesson : Defining and ppling Rotations, Reflections, and Translations Prerequisite Skills This lesson requires the use of the following skills: understanding the coordinate

More information

(x a) (a, b, c) P. (z c) E (y b)

(x a) (a, b, c) P. (z c) E (y b) ( a). FUNCTIONS OF TWO VARIABLES 67 G (,, ) ( c) (a, b, c) P E ( b) Figure.: The diagonal PGgives the distance between the points (,, ) and (a, b, c) F Using Pthagoras theorem twice gives (PG) =(PF) +(FG)

More information

Secondary 1 Vocabulary Cards and Word Walls Revised: March 16, 2012

Secondary 1 Vocabulary Cards and Word Walls Revised: March 16, 2012 Secondary 1 Vocabulary Cards and Word Walls Revised: March 16, 2012 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State

More information

3.1 Quadratic Functions and Their Models. Quadratic Functions. Graphing a Quadratic Function Using Transformations

3.1 Quadratic Functions and Their Models. Quadratic Functions. Graphing a Quadratic Function Using Transformations 3.1 Quadratic Functions and Their Models Quadratic Functions Graphing a Quadratic Function Using Transformations Graphing a Quadratic Function Using Transformations from Basic Parent Function, ) ( f Equations

More information

y x+ 2. A rectangular frame for a painting has a perimeter of 82 inches. If the length of the frame is 25 inches, find the width of the frame.

y x+ 2. A rectangular frame for a painting has a perimeter of 82 inches. If the length of the frame is 25 inches, find the width of the frame. . Simplif the complex fraction x 4 4 x. x 4 x x 4 x x 4 x x 4 x x E) 4 x. A rectangular frame for a painting has a perimeter of 8 inches. If the length of the frame is inches, find the width of the frame.

More information

Physics Gravitational force. 2. Strong or color force. 3. Electroweak force

Physics Gravitational force. 2. Strong or color force. 3. Electroweak force Phsics 360 Notes on Griffths - pluses and minuses No tetbook is perfect, and Griffithsisnoeception. Themajorplusisthat it is prett readable. For minuses, see below. Much of what G sas about the del operator

More information

Chapter 1: Linear Equations and Functions

Chapter 1: Linear Equations and Functions Chapter : Linear Equations and Functions Eercise.. 7 8+ 7+ 7 8 8+ + 7 8. + 8 8( + ) + 8 8+ 8 8 8 8 7 0 0. 8( ) ( ) 8 8 8 8 + 0 8 8 0 7. ( ) 8. 8. ( 7) ( + ) + + +. 7 7 7 7 7 7 ( ). 8 8 0 0 7. + + + 8 +

More information

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2 6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x

More information

Georgia Standards of Excellence Frameworks. Mathematics

Georgia Standards of Excellence Frameworks. Mathematics Georgia Standards of Ecellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 2: Geometric and Algebraic Connections These materials are for nonprofit educational purposes

More information

Lesson 3: Free fall, Vectors, Motion in a plane (sections )

Lesson 3: Free fall, Vectors, Motion in a plane (sections ) Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)

More information

Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.

Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website

More information

Appendices. Appendix A.1: Factoring Polynomials. Techniques for Factoring Trinomials Factorability Test for Trinomials:

Appendices. Appendix A.1: Factoring Polynomials. Techniques for Factoring Trinomials Factorability Test for Trinomials: APPENDICES Appendices Appendi A.1: Factoring Polynomials Techniques for Factoring Trinomials Techniques for Factoring Trinomials Factorability Test for Trinomials: Eample: Solution: 696 APPENDIX A.1 Factoring

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

1. m = 3, P (3, 1) 2. m = 2, P ( 5, 8) 3. m = 1, P ( 7, 1) 4. m = m = 0, P (3, 117) 8. m = 2, P (0, 3)

1. m = 3, P (3, 1) 2. m = 2, P ( 5, 8) 3. m = 1, P ( 7, 1) 4. m = m = 0, P (3, 117) 8. m = 2, P (0, 3) . Linear Functions 69.. Eercises To see all of the help resources associated with this section, click OSttS Chapter. In Eercises - 0, find both the point-slope form and the slope-intercept form of the

More information

Lesson 5.6 Exercises, pages

Lesson 5.6 Exercises, pages Lesson 5.6 Eercises, pages 05 0 A. Approimate the value of each logarithm, to the nearest thousanth. a) log 9 b) log 00 Use the change of base formula to change the base of the logarithms to base 0. log

More information

Algebra SEMESTER ONE. K12.com { Pg. 1 } Course Overview. Unit 1: Algebra Basics. Unit 2: Properties of Real Numbers

Algebra SEMESTER ONE. K12.com { Pg. 1 } Course Overview. Unit 1: Algebra Basics. Unit 2: Properties of Real Numbers Algebra Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform manipulations with numbers, variables, equations, and inequalities. They also learn

More information

University of Bergen. Solutions to Exam in MAT111 - Calculus 1

University of Bergen. Solutions to Exam in MAT111 - Calculus 1 Universit of Bergen The Facult of Mathematics and Natural Sciences English Solutions to Eam in MAT - Calculus Wednesda Ma 0, 07, 09.00-4.00 Eercise. a) Find the real numbers and such that the comple variable

More information

Polynomial, Power, and Rational Functions

Polynomial, Power, and Rational Functions Section 157 Chapter 2 2.1 Linear and Quadratic Functions and Modeling 2.2 Power Functions with Modeling 2.3 Polynomial Functions of Higher Degree with Modeling 2.4 Real Zeros of Polynomial Functions 2.5

More information

College Algebra with Current Interesting Applications and Facts by Acosta & Karwowski, Kendall Hunt, Textbook Supplement (Revised 8/2013)

College Algebra with Current Interesting Applications and Facts by Acosta & Karwowski, Kendall Hunt, Textbook Supplement (Revised 8/2013) College Algebra with Current Interesting Applications and Facts b Acosta & Karwowski, Kendall Hunt, 01 Tetbook Supplement (Revised 8/013) Contents A. Comple Numbers B. Distance Formula, Midpoint Formula,

More information

Zero and Negative Exponents

Zero and Negative Exponents 0.4 Zero and Negative Exponents How can you evaluate a nonzero number with an exponent of zero? How can you evaluate a nonzero number with a negative integer exponent? ACTIVITY: Using the Quotient of Powers

More information

Rewriting Absolute Value Functions as Piece-wise Defined Functions

Rewriting Absolute Value Functions as Piece-wise Defined Functions Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the

More information

Lesson 8 Solving Quadratic Equations

Lesson 8 Solving Quadratic Equations Lesson 8 Solving Quadratic Equations Lesson 8 Solving Quadratic Equations We will continue our work with quadratic equations in this lesson and will learn the classic method to solve them the Quadratic

More information

Introduction to Inequalities

Introduction to Inequalities Algebra Module A7 Introduction to Inequalities Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED Nov., 007 Introduction to Inequalities Statement

More information

Marina s Fish Shop Student Worksheet Name

Marina s Fish Shop Student Worksheet Name Marina s Fish Shop Student Worksheet Name Marina owns a fish shop, and wants to create a new sign above the shop. She likes geometric ideas, and thinks a square with a triangle looks like a fish. Marina

More information

B.3 Solving Equations Algebraically and Graphically

B.3 Solving Equations Algebraically and Graphically B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find

More information

Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x

Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x SECTION 9 Differentials 5 Section 9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph f In the same viewing window, graph the tangent line to the graph of f at the point, Zoom in twice

More information

2. In Exercise 1, suppose the two pricing plans changed as follows. Complete parts (a) (d) based on these two plans.

2. In Exercise 1, suppose the two pricing plans changed as follows. Complete parts (a) (d) based on these two plans. A C E Applications Connections Extensions Applications 1. A school is planning a Saturday Back-to-School Festival to raise funds for the school art and music programs. Some of the planned activities are

More information

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y. BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n +n ( 9 )( ) + + 6 + 9ab a+b

More information

3) Find the distance for each set of ordered pairs (remember to provide EXACT answers): 5) x 2 + y 2 + 6x 6y + 9 = 0 A) Ellipse (x 1) 2

3) Find the distance for each set of ordered pairs (remember to provide EXACT answers): 5) x 2 + y 2 + 6x 6y + 9 = 0 A) Ellipse (x 1) 2 Algebra Chapter Review 1) State the Midpoint Formula: State the Distance Formula: ID: 1 Name Date ) Find the midpoint for each set of ordered pairs: a) (1, ), (, ) b) (-, 0), (-, 3) Period c) (, ), (-,

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Methods of Solving Ordinary Differential Equations (Online)

Methods of Solving Ordinary Differential Equations (Online) 7in 0in Felder c0_online.te V3 - Januar, 05 0:5 A.M. Page CHAPTER 0 Methods of Solving Ordinar Differential Equations (Online) 0.3 Phase Portraits Just as a slope field (Section.4) gives us a wa to visualize

More information

Exponents, Radicals, and Polynomials

Exponents, Radicals, and Polynomials Eponents, Radicals, and Polynomials 01 College Board. All rights reserved. Unit Overview In this unit you will eplore multiplicative patterns and representations of nonlinear data. Eponential growth and

More information

Adv Algebra S1 Final Exam Practice Test Calculator

Adv Algebra S1 Final Exam Practice Test Calculator Adv Algebra S1 Final Exam Practice Test Calculator Name Period NOTE: ANY TALKING OR SUSPICIOUS COMMUNICATION DURING OR AFTER THE IN- CLASS FINAL EXAM TEST WILL RESULT IN AN AUTOMATIC GRADE OF 0%. Multiple

More information

Grade 12- PreCalculus

Grade 12- PreCalculus Albuquerque School of Excellence Math Curriculum Overview Grade 12- PreCalculus Module Complex Numbers and Transformations Module Vectors and Matrices Module Rational and Exponential Functions Module Trigonometry

More information

Polynomial, Power, and Rational Functions

Polynomial, Power, and Rational Functions 5144_Demana_Ch0pp169-74 1/13/06 11:4 AM Page 169 CHAPTER Polynomial, Power, and Rational Functions.1 Linear and Quadratic Functions and Modeling. Power Functions with Modeling.3 Polynomial Functions of

More information

Common Core State Standards for Mathematics - High School PARRC Model Content Frameworks Mathematics Algebra 2

Common Core State Standards for Mathematics - High School PARRC Model Content Frameworks Mathematics Algebra 2 A Correlation of CME Project Algebra 2 Common Core 2013 to the Common Core State Standards for , Common Core Correlated to the Number and Quantity The Real Number System N RN Extend the properties of exponents

More information

Topics Covered in Math 115

Topics Covered in Math 115 Topics Covered in Math 115 Basic Concepts Integer Exponents Use bases and exponents. Evaluate exponential expressions. Apply the product, quotient, and power rules. Polynomial Expressions Perform addition

More information

Antiderivatives and Indefinite Integration

Antiderivatives and Indefinite Integration 60_00.q //0 : PM Page 8 8 CHAPTER Integration Section. EXPLORATION Fining Antierivatives For each erivative, escribe the original function F. a. F b. F c. F. F e. F f. F cos What strateg i ou use to fin

More information

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N. Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,

More information

!!! 1.! 4x 5 8x 4 32x 3 = 0. Algebra II 3-6. Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation.

!!! 1.! 4x 5 8x 4 32x 3 = 0. Algebra II 3-6. Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation. Page 1 of 15 Fundamental Theorem of Algebra Attendance Problems. Identify all the real roots of each equation. 1. 4x 5 8x 4 32x 3 = 0 2. x 3 x 2 + 9 = 9x 3. x 4 +16 = 17x 2 Page 2 of 15 4. 3x 3 + 75x =

More information

1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y.

1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y. 1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = -2x 2 4x + 2 x y y = x 2 + 6x -3 x y domain= range= -4-3 -2-1 0 1 2 3 4 domain= range=

More information