171S4.1 Polynomial Functions and Models. March 20, 2012
|
|
- Leon Simon
- 6 years ago
- Views:
Transcription
1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial Division; The Remainder and Factor Theorems 4.4 Theorems about Zeros of Polynomial Functions 4.5 Rational Functions 4.6 Polynomial and Rational Inequalities Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Polynomial Functions. User: able7oxygen Quick Launch: 1022 (Polynomial patterns), 1441 (Exploring Graphs of Polynomial Functions) Pearson Interactive Figures Polynomial Leading Term Test 4.1 Polynomial Functions and Models Determine the behavior of the graph of a polynomial function using the leading term test. Factor polynomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polynomial function and find its real number zeros. See the following lesson in Course Documents of CourseCompass: 171Session4 171Session4 ( Package file ) This lesson is a brief discussion of and suggestions relative to studying Chapter 4. You may use the "Polynomial Roots" program to graph polynomial functions and find the real roots (zeros). Explanations and Exploratory Exercises Polynomial Function Quadratic Function A polynomial function P is given by where the coefficients a n, a n 1,, a 1, a 0 are real numbers and the exponents are whole numbers. 1
2 Cubic Function Examples of Polynomial Functions Examples of Nonpolynomial Functions Polynomial Functions The graph of a polynomial function is continuous and smooth. The domain of a polynomial function is the set of all real numbers 2
3 The Leading Term Test Example Using the leading term test, match each of the following functions with one of the graphs A D, which follow. a) b) c) d) Solution Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f (c) = 0), then (c, 0) is an x intercept of the graph of the function. Graphs Example: Find the zeros of 3
4 Finding Zeros of Factored Polynomial Functions continued Solution: To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0. The zeros of f(x) are 1 and 2. Finding Real Zeros on a Calculator Find the zeros of f (x) = 0.2x 3 1.5x 2 0.3x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x axis. We use the ZERO feature to find them. See graph on right. The zeros are approximately 1.164, 1,142, and Even and Odd Multiplicity If (x c) k, k 1, is a factor of a polynomial function P(x) and (x c) k + 1 is not a factor and: k is odd, then the graph crosses the x axis at (c, 0); k is even, then the graph is tangent to the x axis at (c, 0). Example Find the zeros of f (x) = x 4 + 8x Solution We factor as follows: f (x) = x 4 + 8x 2 33 = (x )(x 2 3). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products. 4
5 307/2. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 15x x 4 7x 3 307/12. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = (1/4)x 4 + (1/2)x 3 6x 2 + x 5 308/18. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = 2x + x 3 5x 5 308/20. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = 2x 4 x
6 308/24. Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = 2x 3 3x 2 + x /42. Find the zeros of the polynomial function and state the multiplicity of each f(x) = 3x 3 + x 2 48x /28. Find the zeros of the polynomial function and state the multiplicity of each f(x) = (x + 5) 3 (x 4)(x + 1) 2 6
7 308/46. Using a graphing calculator, find the real zeros of the function f(x) = x 4 2x /56. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function f(x) = 2x 4 5.6x /58. Determine whether true or false: If P(x) = (x + 2) 2 (x 1/4) 5, then the graph of the polynomial function y = P(x) crosses the x axis at (1/4, 0). 309/62. Projectile Motion. A stone thrown downward with an initial velocity of 34.3 m/sec will travel a distance of s meters, where s(t) = 4.9t t and t is in seconds. If a stone is thrown downward at 34.3 m/sec from a height of 294 m, how long will it take the stone to hit the ground? 7
8 310/66. Windmill Power. Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by P(v) = 0.015v 3. (a) Find the power generated by 15 mph winds. (b) How fast must the wind blow in order to generate 120 watts of power in 1 hr? 310/70. Threshold Weight. In a study performed by Alvin Shemesh, it was found that the threshold weight W, defined as the weight above which the risk of death rises dramatically, is given by W(h) = (h/12.3) 3, where W is in pounds and h is a persons height, in inches. Find the threshold weight of a person who is 5 ft 7 in. tall. 310/74. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not quadratic or linear 311/79. Unemployed. The table below shows the number of unemployed in the United States from 1996 through a) Use a graphing calculator to fit cubic and quartic functions to the data. Let x represent the number of years since b) Use the functions found in part (a) to estimate the number of unemployed in Compare the estimates and determine which model gives the more realistic estimate. R 2 = See TI tutorial at R 2 = See TI tutorial at See TI tutorial at 8
March 20, S4.1q Polynomial Functions and Models
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationOctober 28, S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S3.4 Solving Rational Equations & Radical Equations. February 23, Some Media for this Section
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More information171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationFebruary 27, S3.4q Solving Rational Equations and Radical Equations TRUE. The solution is 5.
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More information171S4.6q Polynomial and Rational Inequalities. April 02, Polynomial and Rational Inequalities. Polynomial Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information171S3.2 Quadratic Equations, Functions, Zeros, and Models September 30, Quadratic Equations, Functions, Zeros, and Models
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationCHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
171S MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros,
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationCHAPTER 3: Quadratic Functions and Equations; Inequalities
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and
More informationSecondary 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
More informationwhere 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 information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationFinal Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
More information3 What is the degree of the polynomial function that generates the data shown below?
hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior
More informationLesson 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
More informationMaintaining 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
More informationCumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).
More informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,
More informationPolynomial 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
More informationPrecalculus. How to do with no calculator 1a)
Precalculus UNIT 2 Review NAME PERIOD This assessment covers many concepts which you must be able to understand without the use of your calculator to view the graph. Please complete the following table
More informationPolynomial Functions
Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1 A cross-section of a honeycomb has a pattern with one
More information2 If ax + bx + c = 0, then x = b) What are the x-intercepts of the graph or the real roots of f(x)? Round to 4 decimal places.
Quadratic Formula - Key Background: So far in this course we have solved quadratic equations by the square root method and the factoring method. Each of these methods has its strengths and limitations.
More informationFall 09/MAT 140/Worksheet 1 Name: Show all your work. 1. (6pts) Simplify and write the answer so all exponents are positive:
Fall 09/MAT 140/Worksheet 1 Name: Show all your work. 1. (6pts) Simplify and write the answer so all exponents are positive: a) (x 3 y 6 ) 3 x 4 y 5 = b) 4x 2 (3y) 2 (6x 3 y 4 ) 2 = 2. (2pts) Convert to
More informationIdentify polynomial functions
EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x 4 1 x 2
More informationChapter 2 Prerequisite Skills BLM Evaluate Functions 1. Given P(x) = x 4 3x 2 + 5x 11, evaluate.
Chapter Prerequisite Skills BLM 1.. Evaluate Functions 1. Given P(x) = x 4 x + 5x 11, evaluate. a) P( ) b) P() c) P( 1) 1 d) P 4 Simplify Expressions. Expand and simplify. a) (x x x + 4)(x 1) + b) (x +
More information2-2: Evaluate and Graph Polynomial Functions
2-2: Evaluate and Graph Polynomial Functions What is a polynomial? -A monomial or sum of monomials with whole number exponents. Degree of a polynomial: - The highest exponent of the polynomial How do we
More information171S1.1 Graphs, Functions, Models August 23, 2012
MAT 171 D10 8:00 9:15 MAT 171 D11 9:30 10:45 Aug 21 7:11 AM Aug 21 7:14 AM MAT 171 D14 2:00 3:15 MAT 171 D15 3:30 4:45 Aug 21 7:14 AM Aug 21 7:14 AM 1 MAT 171 Dr. Claude Moore, CFCC CHAPTER 1: Graphs,
More information9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.
Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1
More informationFinal Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
More informationThe highest degree term is x $, therefore the function is degree 4 (quartic) c) What are the x-intercepts?
L3 1.3 Factored Form Polynomial Functions Lesson MHF4U Jensen In this section, you will investigate the relationship between the factored form of a polynomial function and the x-intercepts of the corresponding
More informationCollege Algebra and College Algebra with Review Final Review
The final exam comprises 30 questions. Each of the 20 multiple choice questions is worth 3 points and each of the 10 open-ended questions is worth 4 points. Instructions for the Actual Final Exam: Work
More informationSection 4.1: Polynomial Functions and Models
Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial
More informationLet's look at some higher order equations (cubic and quartic) that can also be solved by factoring.
GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More informationFinal Exam Review for DMAT 0310
Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x
More informationMATH College Algebra Review for Test 2
MATH 34 - College Algebra Review for Test 2 Sections 3. and 3.2. For f (x) = x 2 + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the
More informationMATH 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
More informationMAT 171. August 22, S1.4 Equations of Lines and Modeling. Section 1.4 Equations of Lines and Modeling
MAT 171 WebAdvisor: http://reg.cfcc.edu Dr. Claude Moore, CFCC Session 1 introduces the Course, CourseCompass, and Chapter 1: Graphs, Functions, and Models. This session is available in CourseCompass.
More informationMath 95 Practice Final Exam
Part 1: No Calculator Evaluating Functions and Determining their Domain and Range The domain of a function is the set of all possible inputs, which are typically x-values. The range of a function is the
More informationChapter 8 ~ Quadratic Functions and Equations In this chapter you will study... You can use these skills...
Chapter 8 ~ Quadratic Functions and Equations In this chapter you will study... identifying and graphing quadratic functions transforming quadratic equations solving quadratic equations using factoring
More informationExample. 171S Introduction to Graphing. January 06, Introduction to Graphing. Cartesian Coordinate System
MAT 171 Dr. Claude Moore, CFCC CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling
More informationPolynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1
Polynomial Functions A polynomial function is a sum of multiples of an independent variable raised to various integer powers. The general form of a polynomial function is f x = a o x n a 1 x n 1 a 2 x
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More informationPrecalculus Chapter 7 Page 1
Section 7.1 Polynomial Functions 1. To evaluate polynomial functions.. To identify general shapes of the graphs of polynomial functions. I. Terminology A. Polynomials in one variable B. Examples: Determine
More informationPolynomial functions right- and left-hand behavior (end behavior):
Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify
More informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More informationPolynomial and Rational Functions
Polnomial and Rational Functions. Polnomial Functions and Modeling. Graphing Polnomial Functions. Polnomial Division; The Remainder and Factor Theorems. Theorems about Zeros of Polnomial Functions. Rational
More informationNAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.
2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
6-7 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Identify all the real roots of each equation. 1. x 3 7x 2 + 8x + 16 = 0 1, 4 2. 2x 3 14x 12 = 0 1, 2, 3 3. x 4 + x 3 25x 2 27x = 0 4. x 4 26x 2 + 25
More informationPrecalculus Lesson 4.1 Polynomial Functions and Models Mrs. Snow, Instructor
Precalculus Lesson 4.1 Polynomial Functions and Models Mrs. Snow, Instructor Let s review the definition of a polynomial. A polynomial function of degree n is a function of the form P(x) = a n x n + a
More informationLesson 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 informationFormative Assignment PART A
MHF4U_2011: Advanced Functions, Grade 12, University Preparation Unit 2: Advanced Polynomial and Rational Functions Activity 2: Families of polynomial functions Formative Assignment PART A For each of
More informationMidterm. Multiple Choice Identify the choice that best completes the statement or answers the question.
Midterm Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor completely. If the polynomial cannot be factored, say it is prime. 10x 2-95x + 225 2. Solve
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationS56 (5.1) Polynomials.notebook August 25, 2016
Q1. Simplify Daily Practice 28.6.2016 Q2. Evaluate Today we will be learning about Polynomials. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line joining (0, 3) and (4,
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More information171S2.2q The Algebra of Functions. February 13, MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
More information1. The graph of a quadratic function is shown. Each square is one unit.
1. The graph of a quadratic function is shown. Each square is one unit. a. What is the vertex of the function? b. If the lead coefficient (the value of a) is 1, write the formula for the function in vertex
More informationWhen using interval notation use instead of open circles, and use instead of solid dots.
P.1 Real Numbers PreCalculus P.1 REAL NUMBERS Learning Targets for P1 1. Describe an interval on the number line using inequalities. Describe an interval on the number line using interval notation (closed
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationA repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows
More informationCollege Algebra Joysheet 1 MAT 140, Fall 2015 D. Ivanšić. Name: Simplify and write the answer so all exponents are positive:
College Algebra Joysheet 1 MAT 140, Fall 2015 D. Ivanšić Name: Covers: R.1 R.4 Show all your work! Simplify and write the answer so all exponents are positive: 1. (5pts) (3x 4 y 2 ) 2 (5x 2 y 6 ) 3 = 2.
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More informationTable of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials
Table of contents Quadratic Functions Graphs of Polynomial Division Finding Roots of Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall 2010 1 / 65 Concepts Quadratic Functions The Definition of
More informationYou analyzed parent functions and their families of graphs. (Lesson 1-5)
You analyzed parent functions and their families of graphs. (Lesson 1-5) Graph and analyze power functions. Graph and analyze radical functions, and solve radical equations. power function monomial function
More informationCHAPTER 5: Exponential and Logarithmic Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More informationChapter 3 Polynomial Functions
Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1.
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More information171S5.6o Applications and Models: Growth and Decay; and Compound Interest November 21, 2011
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More informationUMUC MATH-107 Final Exam Information
UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
More information5.1 Polynomial Functions
5.1 Polynomial Functions In this section, we will study the following topics: Identifying polynomial functions and their degree Determining end behavior of polynomial graphs Finding real zeros of polynomial
More informationUNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS
UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
More informationEvaluate and Graph Polynomial Functions
Evaluate and Graph Polynomial Functions Section 2.2 How do you identify and evaluate polynomial functions? What is synthetic substitution? How do you graph polynomial functions? Polynomial Function f(x)
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More informationChapter 3 Page 1 of 23. Lecture Guide. Math College Algebra Chapter 3. to accompany. College Algebra by Julie Miller
Chapter 3 Page 1 of 23 Lecture Guide Math 105 - College Algebra Chapter 3 to accompany College Algebra by Julie Miller Corresponding Lecture Videos can be found at Prepared by Stephen Toner & Nichole DuBal
More informationMath 1311 Section 5.5 Polynomials and Rational Functions
Math 1311 Section 5.5 Polynomials and Rational Functions In addition to linear, exponential, logarithmic, and power functions, many other types of functions occur in mathematics and its applications. In
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY
More informationTopic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx
More informationUNIT 3 MATHEMATICAL METHODS ALGEBRA
UNIT 3 MATHEMATICAL METHODS ALGEBRA Substitution of Values Rearrangement and Substitution Polynomial Expressions Expanding Expressions Expanding Expressions by Rule Perfect Squares The Difference of Two
More informationWarm 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 informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More information6A The language of polynomials. A Polynomial function follows the rule. Degree of a polynomial is the highest power of x with a non-zero coefficient.
Unit Mathematical Methods Chapter 6: Polynomials Objectives To add, subtract and multiply polynomials. To divide polynomials. To use the remainder theorem, factor theorem and rational-root theorem to identify
More informationPure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions
Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant
More information3 UNIT 4: QUADRATIC FUNCTIONS -- NO CALCULATOR
Name: Algebra Final Exam Review, Part 3 UNIT 4: QUADRATIC FUNCTIONS -- NO CALCULATOR. Solve each of the following equations. Show your steps and find all solutions. a. 3x + 5x = 0 b. x + 5x - 9 = x + c.
More informationUnit 1: Polynomial Functions SuggestedTime:14 hours
Unit 1: Polynomial Functions SuggestedTime:14 hours (Chapter 3 of the text) Prerequisite Skills Do the following: #1,3,4,5, 6a)c)d)f), 7a)b)c),8a)b), 9 Polynomial Functions A polynomial function is an
More informationSect Polynomial and Rational Inequalities
158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationChapter 2 Notes: Polynomials and Polynomial Functions
39 Algebra 2 Honors Chapter 2 Notes: Polynomials and Polynomial Functions Section 2.1: Use Properties of Exponents Evaluate each expression (3 4 ) 2 ( 5 8 ) 3 ( 2) 3 ( 2) 9 ( a2 3 ( y 2 ) 5 y 2 y 12 rs
More informationObjective Mathematics
Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given
More informatione. some other answer 6. The graph of the parabola given below has an axis of symmetry of: a. y = 5 b. x = 3 c. y = 3 d. x = 5 e. Some other answer.
Intermediate Algebra Solutions Review Problems Final Exam MTH 099 December, 006 1. True or False: (a + b) = a + b. True or False: x + y = x + y. True or False: The parabola given by the equation y = x
More information