The Rational Zero Theorem
|
|
- Chad Greer
- 6 years ago
- Views:
Transcription
1 The Rational Zero Theorem
2 Our goal in this section is to learn how we can ind the rational zeros o the polynomials. For example: x = x 4 + x x x + ( ) We could randomly try some actors and use synthetic division and know by the actor theorem that i the remainder is 0 then we have a actor. We might be trying things all day and not hit a actor so in this section we ll learn some techniques to help us narrow down the things to try.
3 Let ( x) = be a a n ( x) = x 4 + x x x + x polynomial unction o n + a n 1 x 1 We d need to try a lot o positive or negative numbers until we ound one that had 0 remainder. To help we have: The Rational Zeros Theorem n a x + a where each coeicient is an integer. 1 degree1or higher o 0, p, in lowest terms, is a q rational zero o, then p must be a actor o a0, and q must be a actor o an. What this tells us is that we can get a list o the POSSIBLE rational zeros that might work by taking actors o the constant divided by actors o the leading coeicient. Factors o the constant Factors o the leading coeicient I a n 0, a 0 ± 0 1, 1 the orm Both positives and negatives would work or actors
4 REMEMBER! Descartes Rule o Signs Let denote a polynomial unction written in standard orm. The number o positive real zeros o either equals the number o sign changes o (x) or else equals that number less an even integer. The number o negative real zeros o either equals the number o sign changes o (-x) or else equals that number less an even integer. 1 starts Pos. changes Neg. changes Pos. ( x) = x 4 + x x x + There are sign changes so this means there could be or 0 positive real zeros to the polynomial.
5 Descartes Rule o Signs Let denote a polynomial unction written in standard orm. The number o positive real zeros o either equals the number o sign changes o (x) or else equals that number less an even integer. The number o negative real zeros o either equals the number o sign changes o (-x) or else equals that number less an even integer. 1 ( x) = x 4 + x x x + starts Pos. changes Neg. changes Pos. ( ) ( ) 4 ( ) ( ) ( x = ) x + x x ( x) + x = x 4 x x + x + There simpliy are (-x) sign changes so this means there could be or 0 negative real zeros to the polynomial.
6 ± 1, So a list o possible things to try would be 1 any number rom the top divided by any rom the bottom with a + or - on it. In this case that just leaves us with ± 1 or ± ( x) = x 4 + x x x + Let s try ( )( ) x 1 x + x x Since 1 is a zero, we can write the actor x - 1, and use the quotient to write the polynomial actored. YES! It is a zero since the remainder is 0 We ound a positive real zero so Descartes Rule tells us there is another one
7 ± 1, 1 We could try, the other positive possible. IMPORTANT: Just because 1 worked doesn t mean it won t work again since it could have a multiplicity. ( ) ( )( ) x = x 1 x + x x x Let s try 1 again, but we try it on the actored version or the remaining actor (once you have it partly actored use that to keep going--- don't start over with the original). YES! the remainder is 0 ( + )( + 1) + x + = x x Once you can get it down to numbers here, you can put the variables back in and actor or use the quadratic ormula, we are done with trial and error.
8 Let s take our polynomial then and write all o the actors we ound: = x 4 + x x x + ( x) = ( x 1) ( x + )( x + 1) There ended up being two positive real zeros, 1 and 1 and two negative real zeros, -, and -1. In this actored orm we can ind intercepts and let and right hand behavior and graph the polynomial Let & right hand behavior Rough graph Plot intercepts Touches at 1 crosses at -1 and -.
9 Let s try another one rom start to inish using the theorems and rules to help us. ( x) = x 4 + 1x + 9x + 7x + 9 Using the rational zeros theorem let's ind actors o the constant over actors o the leading coeicient to know what numbers to try. 1,, 9 ± actors o constant 1, So possible rational zeros are all possible combinations o numbers on top with numbers on bottom: 1 ± 1, ±, ±, ±, ± 9, ± actors o leading coeicient 9
10 starts Pos. Stays positive ( x) = x 4 + 1x + 9x + 7x + 9 Let s see i Descartes Rule helps us narrow down the choices. 1 ± 1, ±, ±, ±, ± 9, ± 9 No sign changes in (x) so no positive real zeros---we just ruled out hal the choices to try so that helps! 1 4 starts Pos. changes Neg. changes Pos. Changes Neg. Changes Pos. ( x) = x 4 1x + 9x 7x sign changes so 4 or or 0 negative real zeros.
11 ( x) = x 4 + 1x + 9x + 7x + 9 Let s try Let s try -1 again x + 9x + 9 = x + x + ( )( ) 1 1,,,, 9, 9 Yes! We ound a zero. Let s work with reduced polynomial then. Yes! We ound another one. We are done with trial and error since we can put variables back in and solve the remaining quadratic equation. So remaining zeros ound by setting these actors = 0 are -/ and -. Notice these were in our list o choices.
12 ( x) = x 4 + 1x + 9x + 7x + 9 So our polynomial actored is: ( x) = ( x + 1) ( x + )( x + ) Its zeros are: -1,-1,-/,-.
Rational Functions. Definition A rational function can be written in the form. where N(x) and D(x) are
Rational Functions Deinition A rational unction can be written in the orm () N() where N() and D() are D() polynomials and D() is not the zero polynomial. *To ind the domain o a rational unction we must
More information9.8 Graphing Rational Functions
9. Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm P where P and Q are polynomials. Q An eample o a simple rational unction
More information5.2 Properties of Rational functions
5. Properties o Rational unctions A rational unction is a unction o the orm n n1 polynomial p an an 1 a1 a0 k k1 polynomial q bk bk 1 b1 b0 Eample 3 5 1 The domain o a rational unction is the set o all
More informationUNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS
UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS KEY FEATURES OF POLYNOMIALS Intercepts of a function o x-intercepts - a point on the graph where y is zero {Also called the zeros of the function.} o y-intercepts
More information2.6: Rational Functions and Their Graphs
2.6: Rational Functions and Their Graphs Rational Functions are quotients of polynomial functions. The of a rational expression is all real numbers except those that cause the to equal. Example 1 (like
More informationUNIT #2 TRANSFORMATIONS OF FUNCTIONS
Name: Date: UNIT # TRANSFORMATIONS OF FUNCTIONS Part I Questions. The quadratic unction ollowing does,, () has a turning point at have a turning point? 7, 3, 5 5, 8. I g 7 3, then at which o the The structure
More information1. How many white tiles will be in Design 5 of the pattern? Explain your reasoning.
Algebra 2 Semester 1 Review Answer the question for each pattern. 1. How many white tiles will be in Design 5 of the pattern Explain your reasoning. 2. What is another way to represent the expression 3.
More informationGraphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2
Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching
More information3.5D Graphing Rational Functions
3.5D Graphing Rational Functions A. Strategy 1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function. 2. Find the and intercepts. 3. Plot the and intercepts, draw
More informationToday is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class Back board says your name if you are on my roster. I need parent financial
More informationFinal Exam MAT 100 JS 2018
Final Exam MAT 100 JS 2018 Miles College T Dabit MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Tell which set or sets the number belongs to: natural
More informationDomain: The domain of f is all real numbers except those values for which Q(x) =0.
Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such
More informationRational Functions Video Lecture. Sections 4.4 and 4.5
Rational Functions Video Lecture Sections 4.4 and 4.5 Course Learning Objectives: 1)Demonstrate an understanding of functional attributes such as domain and range. Determine these attributes for a function
More informationMath 083 Final Exam Practice
Math 083 Final Exam Practice Name: 1. Simplify the expression. Remember, negative exponents give reciprocals.. Combine the expressions. 3. Write the expression in simplified form. (Assume the variables
More informationThe Graph of an Equation Graph the following by using a table of values and plotting points.
Calculus Preparation - Section 1 Graphs and Models Success in math as well as Calculus is to use a multiple perspective -- graphical, analytical, and numerical. Thanks to Rene Descartes we can represent
More informationPolynomial and Rational Functions
Chapter 3 Polynomial and Rational Functions Review sections as needed from Chapter 0, Basic Techniques, page 8. Refer to page 187 for an example of the work required on paper for all graded homework unless
More informationChapter 9 Review. By Charlie and Amy
Chapter 9 Review By Charlie and Amy 9.1- Inverse and Joint Variation- Explanation There are 3 basic types of variation: direct, indirect, and joint. Direct: y = kx Inverse: y = (k/x) Joint: y=kxz k is
More informationMath 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where
2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Begin b graphing the standard quadratic function f() =. Then use transformations of this
More informationChapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions
Chapter 2 Polynomial and Rational Functions 2.2 Quadratic Functions 1 /27 Chapter 2 Homework 2.2 p298 1, 5, 17, 31, 37, 41, 43, 45, 47, 49, 53, 55 2 /27 Chapter 2 Objectives Recognize characteristics of
More informationMath 121. Graphing Rational Functions Fall 2016
Math 121. Graphing Rational Functions Fall 2016 1. Let x2 85 x 2 70. (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More informationPolynomial Graph Features: 1. Must be a smooth continuous curve (no holes, or sharp corners) Graphing Polynomial Functions
3.4 - Graphing Polynomial Functions 1. Notice that the graph is a smooth continuous curve. 2. The graph also has several "turning points", which are local maximums and minimums. P(x)=(1/30)(x+3)(x-2) 2
More informationALGEBRA 2 W/ TRIGONOMETRY MIDTERM REVIEW
Name: Block: ALGEBRA W/ TRIGONOMETRY MIDTERM REVIEW Algebra 1 Review Find Slope and Rate of Change Graph Equations of Lines Write Equations of Lines Absolute Value Functions Transformations Piecewise Functions
More informationES 240: Scientific and Engineering Computation. a function f(x) that can be written as a finite series of power functions like
Polynomial Deinition a unction () that can be written as a inite series o power unctions like n is a polynomial o order n n ( ) = A polynomial is represented by coeicient vector rom highest power. p=[3-5
More informationSection 3.7 Notes. Rational Functions. is a rational function. The graph of every rational function is smooth (no sharp corners)
Section.7 Notes Rational Functions Introduction Definition A rational function is fraction of two polynomials. For example, f(x) = x x + x 5 Properties of Rational Graphs is a rational function. The graph
More informationUNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation:
UNIT 8: SOLVING AND GRAPHING QUADRATICS 8-1 Factoring to Solve Quadratic Equations Zero Product Property For all numbers a & b Solve each equation: If: ab 0, 1. (x + 3)(x 5) = 0 Then one of these is true:
More informationLarger K-maps. So far we have only discussed 2 and 3-variable K-maps. We can now create a 4-variable map in the
EET 3 Chapter 3 7/3/2 PAGE - 23 Larger K-maps The -variable K-map So ar we have only discussed 2 and 3-variable K-maps. We can now create a -variable map in the same way that we created the 3-variable
More informationNO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED
Algebra II (Wilsen) Midterm Review NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED Remember: Though the problems in this packet are a good representation of many of the topics that will be on the exam, this
More information16 Rational Functions Worksheet
16 Rational Functions Worksheet Concepts: The Definition of a Rational Function Identifying Rational Functions Finding the Domain of a Rational Function The Big-Little Principle The Graphs of Rational
More informationLimits at Infinity. as x, f (x)?
Limits at Infinity as x, f (x)? as x, f (x)? Let s look at... Let s look at... Let s look at... Definition of a Horizontal Asymptote: If Then the line y = L is called a horizontal asymptote of the graph
More informationGRAPHING RATIONAL FUNCTIONS DAY 2 & 3. Unit 12
1 GRAPHING RATIONAL FUNCTIONS DAY 2 & 3 Unit 12 2 Warm up! Analyze the graph Domain: Range: Even/Odd Symmetry: End behavior: Increasing: Decreasing: Intercepts: Vertical Asymptotes: Horizontal Asymptotes:
More information6.1 Evaluate Roots and Rational Exponents
VOCABULARY:. Evaluate Roots and Rational Exponents Radical: We know radicals as square roots. But really, radicals can be used to express any root: 0 8, 8, Index: The index tells us exactly what type of
More informationMore Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a
More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing
More information3.1 Investigating Quadratic Functions in Vertex Form
Math 2200 Date: 3.1 Investigating Quadratic Functions in Vertex Form Degree of a Function - refers to the highest exponent on the variable in an expression or equation. In Math 1201, you learned about
More informationSection Rational Functions and Inequalities. A rational function is a quotient of two polynomials. That is, is a rational function if
Section 6.1 --- Rational Functions and Inequalities A rational function is a quotient of two polynomials. That is, is a rational function if =, where and are polynomials and is not the zero polynomial.
More information2-4 Graphing Rational Functions
2-4 Graphing Rational Functions Factor What are the zeros? What are the end behaviors? How to identify the intercepts, asymptotes, and end behavior of a rational function. How to sketch the graph of a
More information1 Review of Functions Symmetry of Functions; Even and Odd Combinations of Functions... 42
Contents 0.1 Basic Facts...................................... 8 0.2 Factoring Formulas.................................. 9 1 Review of Functions 15 1.1 Functions.......................................
More information2-3 Graphing Rational Functions
2-3 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph
More informationFactor the following completely:
Factor the following completely: 1. 3x 2-8x+4 (3x-2)(x-2) 2. 11x 2-99 11(x+3)(x-3) 3. 16x 3 +128 16(x+2)(x 2-2x+4) 4. x 3 +2x 2-4x-8 (x-2)(x+2) 2 5. 2x 2 -x-15 (2x+5)(x-3) 6. 10x 3-80 10(x-2)(x 2 +2x+4)
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More information4.3 Quadratic functions and their properties
4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept:
More informationWeek 10. Topic 1 Polynomial Functions
Week 10 Topic 1 Polnomial Functions 1 Week 10 Topic 1 Polnomial Functions Reading Polnomial functions result from adding power functions 1 together. Their graphs can be ver complicated, so the come up
More informationFinding Asymptotes KEY
Unit: 0 Lesson: 0 Discontinuities Rational functions of the form f ( are undefined at values of that make 0. Wherever a rational function is undefined, a break occurs in its graph. Each such break is called
More informationChapter 2: Polynomial and Rational Functions Power Standard #7
Chapter 2: Polynomial and Rational s Power Standard #7 2.1 Quadratic s Lets glance at the finals. Learning Objective: In this lesson you learned how to sketch and analyze graphs of quadratic functions.
More information3.6-Rational Functions & Their Graphs
.6-Rational Functions & Their Graphs What is a Rational Function? A rational function is a function that is the ratio of two polynomial functions. This definition is similar to a rational number which
More information3x - 5 = 22 4x - 12 = 2x - 9
3. Algebra Solving Equations ax + b = cx + d Algebra is like one big number guessing game. I m thinking of a number. If you multiply it by 2 and add 5, you get 21. 2x + 5 = 21 For a long time in Algebra
More information1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation
1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation functions vertical line test function notation evaluate
More information4.3 Rational Thinking
RATIONAL EXPRESSIONS & FUNCTIONS -4.3 4.3 Rational Thinking A Solidify Understanding Task The broad category of functions that contains the function!(#) = & ' is called rational functions. A rational number
More informationMath 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
More informationPreCalculus 300. Algebra 2 Review
PreCalculus 00 Algebra Review Algebra Review The following topics are a review of some of what you learned last year in Algebra. I will spend some time reviewing them in class. You are responsible for
More information1. Divide by using long division. (8x 3 + 6x 2 + 7) (x + 2)
Bellwork 0-7-4. Divide by using long division. (8x + 6x 2 + 7) (x + 2) Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic
More informationMath 1: Solutions to Written Homework 1 Due Friday, October 3, 2008
Instructions: You are encouraged to work out solutions to these problems in groups! Discuss the problems with your classmates, the tutors and/or the instructors. After working doing so, please write up
More informationLaboratory One Distance and Time
Laboratory One Distance and Time Student Laboratory Description Distance and Time I. Background When an object is propelled upwards, its distance above the ground as a function of time is described by
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationf( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving
The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already
More informationUnit 1 and Unit 2 Concept Overview
Unit 1 and Unit 2 Concept Overview Unit 1 Do you recognize your basic parent functions? Transformations a. Inside Parameters i. Horizontal ii. Shift (do the opposite of what feels right) 1. f(x+h)=left
More informationIron County Schools. Yes! Less than 90 No! 90 No! More than 90. angle: an angle is made where two straight lines cross or meet each other at a point.
Iron County Schools 1 acute angle: any angle that is less than 90. Yes! Less than 90 No! 90 No! More than 90 acute triangle: a triangle where all the angles are less than 90 angle: an angle is made where
More informationStudent Exploration: Quadratics in Polynomial Form
Name: Date: Student Exploration: Quadratics in Polynomial Form Vocabulary: axis of symmetry, parabola, quadratic function, vertex of a parabola Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
More informationCollege Prep Algebra II Summer Packet
Name: College Prep Algebra II Summer Packet This packet is an optional review which is highly recommended before entering CP Algebra II. It provides practice for necessary Algebra I topics. Remember: When
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More information(-,+) (+,+) Plotting Points
Algebra Basics +y (-,+) (+,+) -x +x (-,-) (+,-) Plotting Points -y Commutative Property of Addition/Multiplication * You can commute or move the terms * This only applies to addition and multiplication
More informationMath-3 Lesson 1-7 Analyzing the Graphs of Functions
Math- Lesson -7 Analyzing the Graphs o Functions Which unctions are symmetric about the y-axis? cosx sin x x We call unctions that are symmetric about the y -axis, even unctions. Which transormation is
More informationMath-2. Lesson 3-1. Equations of Lines
Math-2 Lesson 3-1 Equations of Lines How can an equation make a line? y = x + 1 x -4-3 -2-1 0 1 2 3 Fill in the rest of the table rule x + 1 f(x) -4 + 1-3 -3 + 1-2 -2 + 1-1 -1 + 1 0 0 + 1 1 1 + 1 2 2 +
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationMath Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:
1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain
More informationMaximum and Minimum Slopes Wilfrid Laurier University
Maximum and Minimum Slopes Wilfrid Laurier University Wilfrid Laurier University December 12, 2014 In this document, you ll learn: In this document, you ll learn: how to determine the uncertainties in
More informationLesson 19: The Graph of a Linear Equation in Two Variables Is a Line
The Graph of a Linear Equation in Two Variables Is a Line Classwork Exercises THEOREM: The graph of a linear equation yy = mmmm + bb is a non-vertical line with slope mm and passing through (0, bb), where
More informationChapter 1 An Introduction to Computer Science. INVITATION TO Computer Science 1
Chapter 1 An Introduction to Computer Science INVITATION TO Computer Science 1 Q8. Under what conditions would the well-known quadratic formula not be effectively computable? (Assume that you are working
More information= ( )= To find the domain, we look at the vertical asymptote(s) (where denominator equals zero) , =0
Precalculus College Algebra Review for Final Name It is also a good idea to go back through your old tests and quizzes to review. 1. Find (+1) given ()=3 +1 2. Determine () given ()=+2 and ()= (+1)=3(+1)
More information. As x gets really large, the last terms drops off and f(x) ½x
Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be
More informationLesson 19: The Graph of a Linear Equation in Two Variables is a Line
Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Classwork Exercises Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b),
More informationMath 370 Exam 1 Review Name. Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
Math 370 Exam 1 Review Name Determine whether the relation is a function. 1) {(-6, 6), (-6, -6), (1, 3), (3, -8), (8, -6)} Not a function The x-value -6 corresponds to two different y-values, so this relation
More informationIntegers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not.
What is an INTEGER/NONINTEGER? Integers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not. What is a REAL/IMAGINARY number? A real number is
More informationQuadratics Functions: Review
Quadratics Functions: Review Name Per Review outline Quadratic function general form: Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum
More informationSolving Quadratics Algebraically Investigation
Unit NOTES Honors Common Core Math 1 Day 1: Factoring Review and Solving For Zeroes Algebraically Warm-Up: 1. Write an equivalent epression for each of the problems below: a. ( + )( + 4) b. ( 5)( + 8)
More informationMath-3. Lesson 6-8. Graphs of the sine and cosine functions; and Periodic Behavior
Math-3 Lesson 6-8 Graphs o the sine and cosine unctions; and Periodic Behavior What is a unction? () Function: a rule that matches each input to eactly one output. What is the domain o a unction? Domain:
More informationMATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation
MATHS METHODS QUADRATICS REVIEW LAWS OF EXPANSION A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation a) b) c) d) e) FACTORISING Exercise 4A Q6ace,7acegi
More informationRational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ
Rational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ Definition of Rational Functions Rational Functions are defined as the quotient of two polynomial functions. This means any rational function can
More informationIntegers and Rational Numbers
A A Family Letter: Integers Dear Family, The student will be learning about integers and how these numbers relate to the coordinate plane. The set of integers includes the set of whole numbers (0, 1,,,...)
More informationSection 2.3 Rational Numbers. A rational number is a number that may be written in the form a b. for any integer a and any nonzero integer b.
Section 2.3 Rational Numbers A rational number is a number that may be written in the form a b for any integer a and any nonzero integer b. Why is division by zero undefined? For example, we know that
More informationAlgebra II Chapter 4: Quadratic Functions and Factoring Part 1
Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Chapter 4 Lesson 1 Graph Quadratic Functions in Standard Form Vocabulary 1 Example 1: Graph a Function of the Form y = ax 2 Steps: 1. Make
More informationName: Chapter 7 Review: Graphing Quadratic Functions
Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers
More information3x 2 + 7x + 2. A 8-6 Factor. Step 1. Step 3 Step 4. Step 2. Step 1 Step 2 Step 3 Step 4
A 8-6 Factor. Step 1 3x 2 + 7x + 2 Step 2 Step 3 Step 4 3x 2 + 7x + 2 3x 2 + 7x + 2 Step 1 Step 2 Step 3 Step 4 Factor. 1. 3x 2 + 4x +1 = 2. 3x 2 +10x + 3 = 3. 3x 2 +13x + 4 = A 8-6 Name BDFM? Why? Factor.
More informationObjectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function
SECTIONS 3.5: Rational Functions Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function I. Rational Functions A rational function is a
More informationAlgebra II Radical Equations
1 Algebra II Radical Equations 2016-04-21 www.njctl.org 2 Table of Contents: Graphing Square Root Functions Working with Square Roots Irrational Roots Adding and Subtracting Radicals Multiplying Radicals
More informationExample 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs
Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have
More informationMultiplying and Dividing Rational Expressions
Multiplying and Dividing Rational Expressions Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 Factor each expression. 2. y 3 y 3 y 6 x 4 4. y 2 1 y 5 y 3 5.
More informationSchool Year:
School Year: 2010 2011 1 McDougal Littell CA Math Algebra 1 Pacing Guide Begin First Semester During the first two weeks of school, teachers will work with students on study skills and diagnostic assessments
More informationSection 2-7. Graphs of Rational Functions
Section 2-7 Graphs of Rational Functions Section 2-7 rational functions and domain transforming the reciprocal function finding horizontal and vertical asymptotes graphing a rational function analyzing
More informationRational number operations can often be simplified by converting mixed numbers to improper fractions Add EXAMPLE:
Rational number operations can often be simplified by converting mixed numbers to improper fractions Add ( 2) EXAMPLE: 2 Multiply 1 Negative fractions can be written with the negative number in the numerator
More informationSec. 3.7 Rational Functions and their Graphs. A rational function is of the form: where P(x) and Q(x) are Polynomials
Sec. 3.7 Rational Functions and their Graphs A rational function is of the form: where P(x) and Q(x) are Polynomials The Domain of r(x) is all values of x where Q (x) is not equal to zero. The simplest
More informationCSE 215: Foundations of Computer Science Recitation Exercises Set #4 Stony Brook University. Name: ID#: Section #: Score: / 4
CSE 215: Foundations of Computer Science Recitation Exercises Set #4 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 7: Direct Proof Introduction 1. The statement below is true. Rewrite the
More informationThe Rational Number System: Investigate Rational Numbers: Play Answer Sheet
Name _ Date _ The Rational Number System: Investigate Rational Numbers: Play Answer Sheet Selected Response Items Indicate letter or letters only.. 2.... 6. 7. Fill-in-the-Blank Items 8. 9. 0.. 2.. Discovery
More informationScientific Method and Graphing
Scientific Method and Graphing Objectives - Students will be able to: 1.Explain what an independent and a dependent variable are. 2.Properly label a data table and graph 3.Create a graph from a data table
More informationThe domain of any rational function is all real numbers except the numbers that make the denominator zero or where q ( x)
We will look at the graphs of these functions, eploring their domain and end behavior. College algebra Class notes Rational Functions with Vertical, Horizontal, and Oblique Asymptotes (section 4.) Definition:
More informationPRECALCULUS MR. MILLER
PRECALCULUS MR. MILLER I. COURSE DESCRIPTION This course requires students to use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study
More informationAmphitheater School District End Of Year Algebra II Performance Assessment Review
Amphitheater School District End Of Year Algebra II Performance Assessment Review This packet is intended to support student preparation and review for the Algebra II course concepts for the district common
More informationQuadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
More information2.3 Graph Sketching: Asymptotes and Rational Functions Math 125
.3 Graph Sketching: Asymptotes and Rational Functions Math 15.3 GRAPH SKETCHING: ASYMPTOTES AND RATIONAL FUNCTIONS All the functions from the previous section were continuous. In this section we will concern
More information