# Lesson 7.1 Polynomial Degree and Finite Differences

Save this PDF as:

Size: px
Start display at page:

Download "Lesson 7.1 Polynomial Degree and Finite Differences"

## Transcription

1 Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b c Determine which of the epressions are polnomials. For each polnomial, state its degree and write it in general form. If it is not a polnomial, eplain wh not. a b. 1 c. 3. The figures below show wh the numbers in the sequence 1, 3, 6, 10,... are called triangular numbers. a. Complete the table. n nth triangular number b. Calculate the finite differences for the completed table. c. What is the degree of the polnomial function that ou would use to model this data set? d. Write a polnomial function t that gives the nth triangular number as a function of n. (Hint: Create and solve a sstem of equations to find the coefficients.) Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

2 Lesson 7. Equivalent Quadratic Forms 1. Identif each quadratic function as being in general form, verte form, factored form, or none of these forms. a. 3 4 b. (.) 7. c. 1.( ). Convert each quadratic function to general form. a. ( 3) b. ( 3)( ) 30 c. 3( 1.) Find the verte of the graph of each quadratic function. a. b. ( 1) 6 c ( 4) 4. Find the zeros of each quadratic function. a. ( 1)( 6) b. 0.( ) c. ( 7.). Consider this table of values generated b a quadratic function a. What is the line of smmetr for the graph of the quadratic function? b. Identif the verte of the graph of this quadratic function, and determine whether it is a maimum or a minimum. c. Use the table of values to write the quadratic function in verte form. 4 CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

3 Lesson 7.3 Completing the Square 1. Factor each quadratic epression. a. 10 b. 1 4 c What value is required to complete the square? a. 18 b. c Convert each quadratic function to verte form b completing the square. a b c. 4. Find the verte of the graph of each quadratic function, and state whether the verte is a maimum or a minimum. a. ( )( 6) b c Rewrite each epression in the form a b c, and then identif the coefficients a, b, and c. a b. ( 8) c. ( 3)( ) 6. A ball is thrown up and off the roof of a 7 m tall building with an initial velocit of 14.7 m/s. a. Let t represent the time in seconds and h represent the height of the ball in meters. Write an equation that models the height of the ball. b. At what time does the ball reach maimum height? What is the ball s maimum height? c. At what time does the ball hit the ground? Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

4 Lesson 7.4 The Quadratic Formula 1. Evaluate each epression. Round our answers to the nearest thousandth. a (1)( ) b. 4 ( 4) 4()(1) (1) () c. ( ) 4(4)( 3) (4) d ()() (). Solve b an method. Give our answers in eact form. a b. 1 c d e..8 f Use the quadratic formula to find the zeros of each function. Then, write each equation in factored form, a r 1 r, where r 1 and r are the zeros of the function. a. 4 b. 8 6 c Write a quadratic function in general form that satisfies the given conditions. a. a 1; -intercepts of graph are 4 and b. -intercepts of graph are 0 and 13; graph contains point (, ) c. -intercept of graph is 4.8; -intercept is CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

5 Lesson 7. Comple Numbers 1. Add, subtract, or multipl. a. ( 6i ) (1 i ) b. (.4.6i) (.9 1.8i) c. 4i( 6 i ) d. (. 1.i)( i). Find the conjugate of each comple number. a. 4i b. 7i c i 3. Rewrite each quotient in the form a bi. a. b. 1 i 3 i 1 i c. 3 i 6i 4. Solve each equation. Use substitution to check our solutions. Label each solution as real, imaginar, and/or comple. a. 0 b. 7 0 c. ( ) 1 d. 1 0 e f. ( 7)( 3). Write a quadratic function in general form that has the given zeros and leading coefficient of 1. a. 4, 7 b. 11i, 11i c. 3i, 3i 6. Name the comple number associated with each point, A through C, on the comple plane shown. C Imaginar A Real B Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

6 Lesson 7.6 Factoring Polnomials 1. Without graphing, find the -intercepts and the -intercept for the graph of each equation. a. ( 8) b. 3( 4)( ) c. 0.7( )( 6). Write the factored form of the quadratic function for each graph. Don t forget the vertical scale factor. a. b. (0, 4) ( 3, 0) (4, 0) ( 4, 0) (, 0) 0 (0, 4) 3. Convert each polnomial function to general form. a. (.)(.) b. 0.( 3) c. ( 1)( 1) 4. Write each polnomial as a product of factors. Some factors ma include irrational numbers. a b. 3 3 c. 169 d. 1 e f Sketch a graph for each situation if possible. a. A quadratic function with two real zeros, whose graph has the line as its ais of smmetr b. A cubic function with three real zeros, whose graph has a positive -intercept 46 CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

7 Lesson 7.7 Higher-Degree Polnomials 1. Refer to these two graphs of polnomial functions. i. ii ( 3, 0) (, 0) (0, 0) (0, 4) (, 0) (, 0) 10 0 a. Identif the zeros of each function. b. Find the -intercept of each graph. c. Identif the lowest possible degree of each polnomial function. d. Write the factored form for each polnomial function. Check our work b graphing on our calculator.. Write a polnomial function with the given features. a. A quadratic function whose graph has verte (3, 8), which is a minimum, and two -intercepts, one of which is b. A fourth-degree polnomial function with two double roots, 0 and, and whose graph contains the point (1, 1) 3. Write the lowest-degree polnomial function that has the given set of zeros and whose graph has the given -intercept. Write each polnomial function in factored form. Give the degree of each function. a. Zeros: 3, ; -intercept: 30 b. Zeros: i, (double root), ; -intercept: 80 Discovering Advanced Algebra More Practice Your Skills CHAPTER Ke Curriculum Press

8 Lesson 7.8 More About Finding Solutions 1. Divide. a. ) b. 4 ) Varsha started out dividing two polnomials b snthetic division this wa: a. Identif the dividend and divisor. b. Write the numbers that will appear in the second line of the snthetic division. c. Write the numbers that will appear in the last line of the snthetic division. d. Write the quotient and remainder for this division. 3. In each division problem, use the polnomial that defines P as the dividend and the binomial that defines D as the divisor. Write the result of the division in the form P() D() Q() R, where the polnomial that defines Q is the quotient and R is an integer remainder. (It is not necessar to write the remainder if R 0.) a. P() 9 ; D() b. P() 3 8 ; D() 1 4. Make a list of the possible rational roots of each equation. a b Find all the zeros of each polnomial function. Then write the function in factored form. a b CHAPTER 7 Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

9 d. Vertices: 14 3, 0, (6, 0), (0, 4.), (0, 3.) e. 14 3, 0 : \$140; (6, 0): \$180; (0, 4.): \$147; (0, 3.): \$1.0 f. 6 batches of full sheet cakes, 0 batches of halfsheet cakes; \$180 LESSON 7.1 Polnomial Degree and Finite Differences 1. a. b. 3 c.. a. Polnomial; 4; a. b. Not a polnomial; 1 has a negative eponent c. Polnomial; 0; alread in general form n nth triangular number b. D 1 {, 3, 4,, 6, 7}, D {1, 1, 1, 1, 1} c. d. t (n) 1 n 1 n, or t (n) 0.n 0.n LESSON 7. Equivalent Quadratic Forms 1. a. General form b. Verte form c. Factored form. a. 6 9 b. c a. (0, 0) b. (1, 6) c. ( 4, 6.) 4. a. 1 and 6 b. 0 and c. 7.. a. 1. b. ( 1., ); minimum c. ( 1.) LESSON 7.3 Completing the Square 1. a. ( ) b. 1 c. (3 4). a. 81 b., or 6. c a. ( 7) 1 b. ( 1) 8 c. ( 1.) a. (, 16); minimum b. ( 1, 3.); maimum c. ( 4., 30.); minimum. a. 3 6 ; a 3, b 6, c b. 16; a, b 16, c 0 c. 7 1; a, b 7, c 1 6. a. h 4.9t 14.7t 7 b. 1. s; 86.0 m c..69 s LESSON 7.4 The Quadratic Formula 1. a b c d a. or b. 3 or 4 c. 7 d e. 0 or.8 f a. ( 8)( 3) b. ( 3)( 1) c. 4( 1)( 0.) 4. a. 6 8 b. 13 c LESSON 7. Comple Numbers 1. a. 6 7i b i c. 4 4i d i. a. 4i b. 7i c i 3. a. 3 1 i, or i b. i, or 0 i c. 6 1 i 4. a. 1 i; comple b. i 7 ; imaginar and comple c. 9 ; real and comple 1 i d. 3 ; comple e. 3_ i, or 1.i; imaginar and comple 6 f. 140, or 3 3 ; real and comple. a b c A: 3i, or 0 3i ; B: 3i ; C: 4 i Discovering Advanced Algebra More Practice Your Skills ANSWERS Ke Curriculum Press

10 LESSON 7.6 Factoring Polnomials 1. a. -intercept: 8; -intercept: 64 b. -intercepts: 4, ; -intercept: 4 c. -intercepts: 0,, 6; -intercept: 0. a. ( 3)( 4) b. 0.( 4)( ) 3. a. 1. b c a. ( 7) b. ( 1)( ) c. ( 13i)( 13i) d. 1 1 e. ( 1)( 1)( 3)( 3) f. 3( 1)( 4)( ). Possible graphs: a. b. 3. a. 9 ( )( 1) 7 b. 3 8 ( 1) 3 4. a. 1,, 4, 8 b. 1,, 3,, 6, 10, 10, 1, 30, 1 3,,, 1. a., 3i, and 3i; ( )( 3i)( 3i), or ( ) 9 b., 4_ 3, and 1_ ; 6( ) 4_ 3 1_, or ( )(3 4)( 1) LESSON 8.1 Using the Distance Formula 1. a. 13 units b. units c. 0 units, or units d. 80 units, or 4 units e. a 9 units f. units. a. or 14 b. 4 7 LESSON 7.7 Higher-Degree Polnomials 1. a. i. 3, 0, ; ii., b. i. 0; ii. 4 c. i. 3; ii. 4 d. i. ( 3)( ); ii. 0.( ) ( ). a. ( 1)( ), or 1 10 b. ( ), or a. ( 3)( ); degree b. ( i )( i )( ) ( ), or 4 ( ) ( ); degree a. ( ) ( 3) LESSON 7.8 More About Finding Solutions 1. a. 3 1 b a. Dividend: ; divisor: 3 b c d. Quotient: ; remainder: 4 b ANSWERS Discovering Advanced Algebra More Practice Your Skills 010 Ke Curriculum Press

### 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,

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

### Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.

Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the

### Lesson 9.1 Using the Distance Formula

Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

### Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and

### Lesson 10.1 Solving Quadratic Equations

Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One -intercept and all nonnegative y-values b. The verte in the third quadrant and no

### 2.1 Evaluate and Graph Polynomial

2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

### Unit 2 Notes Packet on Quadratic Functions and Factoring

Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

### N x. You should know how to decompose a rational function into partial fractions.

Section 7. Partial Fractions 0. 0 7 0 0 0 0 Solution:, 0 Equation Equation Eq. Eq. 07. nswers will var. Section 7. Partial Fractions N You should know how to decompose a rational function into partial

### C)not a function. B) function domain: {-3, 2, 4, 6} range: {-7, 4, 2, -1}

Name Spring Semester Final Review (Dual) Precalculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the relation represents a function.

### Name Please print your name as it appears on the class roster.

Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes

### MATH College Algebra Review for Test 2

MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of

### MAT 1033C -- Martin-Gay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam

MAT 33C -- Martin-Ga Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.

Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression

### Polynomial and Rational Functions

Polnomial and Rational Functions 5 Figure 1 35-mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia

### Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities

### Solving Quadratic Equations

9 Solving Quadratic Equations 9. Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square

### Polynomial Functions of Higher Degree

SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona

### Shape and Structure. Forms of Quadratic Functions. Lesson 2.1 Assignment

Lesson.1 Assignment Name Date Shape and Structure Forms of Quadratic Functions 1. Analze the graph of the quadratic function. a. The standard form of a quadratic function is f() 5 a 1 b 1 c. What possible

### 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

### Lesson 3.1 Linear Equations and Arithmetic Sequences

Lesson 3.1 Linear Equations and Arithmetic Sequences 1. Find an eplicit formula for each recursivel defined arithmetic sequence. a. u 0 18.25 b. t 0 0 u n u n 1 4.75 where n 1 t n t n 1 100 where n 1 2.

### Polynomial and Synthetic Division

Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Final Eam Review MAC 1 Fall 011 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve and check the linear equation. 1) (- + ) - = -( - 7) A)

### Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

### 5. Determine the discriminant for each and describe the nature of the roots.

4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.

Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions -: Points and Lines Sstem of Linear Equations: - two or more linear equations on the same coordinate grid. Solution of

### Factoring Polynomials

5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with

### INTRODUCTION GOOD LUCK!

INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills

### Mth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula

Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b

### Dividing Polynomials: Remainder and Factor Theorems

Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.

### Instructor: Imelda Valencia Course: A3 Honors Pre Calculus

Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression

### 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

### Using Intercept Form

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

### Graphing Calculator Computations 2

Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)

### Lesson Master 9-1B. REPRESENTATIONS Objective G. Questions on SPUR Objectives. 1. Let f(x) = 1. a. What are the coordinates of the vertex?

Back to Lesson 9-9-B REPRESENTATIONS Objective G. Let f() =. a. What are the coordinates of the verte? b. Is the verte a minimum or a maimum? c. Complete the table of values below. 3 0 3 f() d. Graph the

### ALGEBRA 1 CP FINAL EXAM REVIEW

ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8

### Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed

### f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.

Test Review Section.. Given the following function: f ( ) = + 5 - Determine the implied domain of the given function. Epress your answer in interval notation.. Find the verte of the following quadratic

### A. Incorrect! Apply the rational root test to determine if any rational roots exist.

College Algebra - Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root

### Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

### Solving Quadratic Equations by Graphs and Factoring

Solving Quadratic Equations b Graphs and Factoring Algebra Unit: 05 Lesson: 0 Consider the equation - 8 + 15 = 0. Show (numericall) that = 5 is a solution. There is also another solution to the equation.

### 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

### k y = where k is the constant of variation and

Syllabus Objectives: 9. The student will solve a problem by applying inverse and joint variation. 9.6 The student will develop mathematical models involving rational epressions to solve realworld problems.

### Law of Sines, Law of Cosines, Heron s Formula:

PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,

### Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry

Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function = - -1 0 1 1 Eample 1: Make a table,

### = 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

### Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For

### Nonlinear Systems. No solution One solution Two solutions. Solve the system by graphing. Check your answer.

8-10 Nonlinear Sstems CC.9-1.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two

### Section 2.5: Graphs of Functions

Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree

### LESSON #17 - FACTORING COMMON CORE ALGEBRA II FACTOR TWO IMPORTANT MEANINGS

1 LESSON #17 - FACTORING COMMON CORE ALGEBRA II In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more comple

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.

### 3.3 Real Zeros of Polynomial Functions

71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2

### Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

### MATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)

MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the

### + = + + = x = + = + = 36x

Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

### Lesson 2.1: Quadratic Functions

Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

### 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

### 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

### 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

### Solve Quadratic Equations by Graphing

0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6.

### 4Cubic. polynomials UNCORRECTED PAGE PROOFS

4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review

### SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.

Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of

### KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

### 17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96

Section.3 Zeros of the Quadratic 473.3 Eercises In Eercises 1-8, factor the given quadratic polnomial. 1. 2 + 9 + 14 2. 2 + 6 + 3. 2 + + 9 4. 2 + 4 21. 2 4 6. 2 + 7 8 7. 2 7 + 12 8. 2 + 24 In Eercises

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Spring 0 Math 08 Eam Preparation Ch Dressler Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the quadratic equation b the square root propert.

### Math 1050 REVIEW for Exam 1. Use synthetic division to find the quotient and the remainder. 1) x3 - x2 + 6 is divided by x + 2

Math 0 REVIEW for Eam 1 Use snthetic division to find the quotient and the remainder. 1) 3-2 + 6 is divided b + 2 Use snthetic division to determine whether - c is a factor of the given polnomial. 2) 3-32

### Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review

Name: Class: Date: Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review 1 Write 3x 2 ( 2x 2 5x 3 ) in standard form State whether the function is even, odd, or neither Show your work

### 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

### 3.1 Graph Quadratic Functions

3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

### Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

### Honors Algebra 2 ~ Spring 2014 Name 1 Unit 3: Quadratic Functions and Equations

Honors Algebra ~ Spring Name Unit : Quadratic Functions and Equations NC Objectives Covered:. Define and compute with comple numbers. Operate with algebraic epressions (polnomial, rational, comple fractions)

### Lesson 5.1 Exponential Functions

Lesson.1 Eponential Functions 1. Evaluate each function at the given value. Round to four decimal places if necessar. a. r (t) 2(1 0.0) t, t 8 b. j() 9.(1 0.09), 10 2. Record the net three terms for each

### 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

### 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

### C) x m A) 260 sq. m B) 26 sq. m C) 40 sq. m D) 364 sq. m. 7) x x - (6x + 24) = -4 A) 0 B) all real numbers C) 4 D) no solution

Sample Departmental Final - Math 46 Perform the indicated operation. Simplif if possible. 1) 7 - - 2-2 + 3 2 - A) + - 2 B) - + 4-2 C) + 4-2 D) - + - 2 Solve the problem. 2) The sum of a number and its

### Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.

Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +

### Lesson 3.1 Linear Equations and Arithmetic Sequences

Lesson 3.1 Linear Equations and Arithmetic Sequences 1. Find an eplicit formula for each recursivel defined arithmetic sequence. a. u 0 18.25 b. t 0 0 u n u n 1 4.75 where n 1 t n t n 1 100 where n 1 2.

### 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

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Practice for the Final Eam MAC 1 Sullivan Version 1 (2007) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the distance d(p1, P2) between the points

### 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

### NAME DATE PERIOD. Study Guide and Intervention. Transformations of Quadratic Graphs

NAME DATE PERID Stud Guide and Intervention Write Quadratic Equations in Verte Form A quadratic function is easier to graph when it is in verte form. You can write a quadratic function of the form = a

### Name: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown.

SM Name: Period: 7.5 Starter on Reading Quadratic Graph This graph and equation represent the path of an object being thrown. 1. What is the -ais measuring?. What is the y-ais measuring? 3. What are the

### Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions

Name Class Date -1 Finding Real Roots of Polnomial Equations Etension: Graphing Factorable Polnomial Functions Essential question: How do ou use zeros to graph polnomial functions? Video Tutor prep for

### Polynomial Functions and Models

1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

### Fair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal

Name Date Chapter Graph the linear equation. Fair Game Review. =. = +. =. =. = +. = + Copright Big Ideas Learning, LLC Big Ideas Math Algebra Name Date Chapter Fair Game Review (continued) Evaluate the

### Review Topics for MATH 1400 Elements of Calculus Table of Contents

Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

### INTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE:

INTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE: You decide to open a small business making gluten-free cakes. Your start-up costs were \$, 000. In addition, it costs \$ 0 to produce each cake. What is the

### Why? _ v a There are different ways to simplify the expression. one fraction. term by 2a. = _ b 2

Dividing Polynomials Then You divided rational expressions. (Lesson 11-5) Now 1Divide a polynomial by a monomial. 2Divide a polynomial by a binomial. Why? The equation below describes the distance d a

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### Chapter 6 Resource Masters

Chapter 6 Resource Masters Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook 0-07-8809-X

### 3.1. Shape and Structure Forms of Quadratic Functions ESSENTIAL IDEAS TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS 169A

Shape and Structure Forms of Quadratic Functions.1 LEARNING GOALS KEY TERMS In this lesson, ou will: Match a quadratic function with its corresponding graph. Identif ke characteristics of quadratic functions

### 25) x x + 30 x2 + 15x ) x Graph the equation. 30) y = - x - 1

Pre-AP Algebra Final Eam Review Solve. ) A stone is dropped from a tower that is feet high. The formula h = - t describes the stoneʹs height above the ground, h, in feet, t seconds after it was dropped.

### Find the distance between the pair of points. 2) (7, -7) and (3, -5) A) 12 3 units B) 2 5 units C) 6 units D) 12 units B) 8 C) 63 2

Sample Departmental Final - Math 9 Write the first five terms of the sequence whose general term is given. 1) a n = n 2 - n 0, 2,, 12, 20 B) 2,, 12, 20, 30 C) 0, 3, 8, 1, 2 D) 1,, 9, 1, 2 Find the distance

### 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

### Properties of Rational Exponents PROPERTIES OF RATIONAL EXPONENTS AND RADICALS. =, a 0 25 º1/ =, b /3 2. b m

Page of 8. Properties of Rational Eponents What ou should learn GOAL Use properties of rational eponents to evaluate and simplif epressions. GOAL Use properties of rational eponents to solve real-life

### One of the most common applications of Calculus involves determining maximum or minimum values.

8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..

### Zeros of Polynomial Functions

OpenStax-CNX module: m49349 1 Zeros of Polynomial Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:

### Chapter 9 Prerequisite Skills

Name: Date: Chapter 9 Prerequisite Skills BLM 9. Consider the function f() 3. a) Show that 3 is a factor of f(). If f() ( 3)g(), what is g()?. Factor each epression fully. a) 30g 4g 6fg 8g c) 6 5 d) 5