ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form


 Collin Tate
 4 years ago
 Views:
Transcription
1 ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola The Ushaped graph of a quadratic function Vertex The lowest or highest point on a parabola Axis of symmetry The vertical line that divides the parabola into mirror images and passes through the vertex Minimum and maximum value For y= ax 2 + bx+ c, the vertex's ycoordinate is the minimum value of the function if a 0 and its maximum value if a 0. PARENT FUNCTION FOR QUADRATIC FUNCTIONS The parent function for the family of all quadratic functions is f(x) = x 2. The graph is shown below The lowest or highest point on a parabola is the vertex for f(x) = x2 is (0,0) The axis of symmetry divides the parabola into mirror images and passes through the vertex For f(x) = ax 2, and for any quadratic function g(x)= ax 2 + bx + c where b = 0, the vertex lies on the yaxis and the axis of symmetry is x = _0_.
2 Example 1 Graph a function of the form y = ax 2 + c Graph y = 2x Compare the graph with the graph of y = x 2. PROPERTIES OF THE GRAPH OF y = ax 2 + bx + c Characteristics of the graph of y = ax 2 + bx + c: The graph opens up if a _ _ 0 and opens down if a _ _ 0. The graph is narrower than the graph of y = x 2 if a _ _ 1 and wider if a _ _ 1. b b The axis of symmetry is x = and the vertex has xcoordinate. 2 a 2 a The yintercept is _c_. So, the point (0, _c_) is on the parabola. Example 2 Graph a function of the form y = ax 2 + bx + c Graph y = x 2 + 4x 3.
3 MINIMUM AND MAXIMUM VALUES Words For y = ax 2 + bx + c, the vertex's ycoordinate is the minimum value of the function if a _ _ 0 and the maximum value if a _ _ 0. Example 3 Find the minimum or maximum value Tell whether the function y = 3x x 6 has a minimum value or a maximum value. Then find the minimum or maximum value. Checkpoint Complete the following exercises. Graph the function. Label the vertex and axis of symmetry 1. y = x 2 2. y = x 2 4x Find the minimum value of y = 2x 2 6x + 6
4 4.2 Graph Quadratic Functions in Vertex or Intercept Form Goal Graph quadratic functions in vertex form or intercept form. VOCABULARY Vertex form A quadratic function written in the form y = a(x h) 2 + k Intercept form A quadratic function written in the form y = a(x p)(x q) GRAPH OF VERTEX FORM y = a(x h) 2 + k The graph of y = a(x h) 2 + k is the parabola y = ax 2 translated _horizontally_ h units and _vertically_ k units. The vertex is (_h_, _k_ ). The axis of symmetry is x = _h_. The graph opens up if a _ _ 0 and down if a _ _ 0. Example 1 Graph a quadratic function in vertex form and find the maximum or minimum value. 1 Graph y = (x + l) GRAPH OF INTERCEPT FORM y = a(x p)(x q): Characteristics of the graph y = a(x p)(x q): The xintercepts are _p_ and _q_. The axis of symmetry is halfway between ( _p, 0) and ( _q_, 0). It has equation x = p q 2 The graph opens up if a _>_ 0 and opens down if a _<_ 0.
5 Example 2 Graph a quadratic function in intercept form and find the maximum or minimum value. Graph y = 2(x 1)(x 5). Checkpoint Complete the following exercises. 1. Graph the function. Label the vertex and the axis of symmetry and find the maximum or minimum value. y = (x 3) Graph the function. Label the vertex, axis of symmetry, and the xintercepts and find the maximum or minimum value. y = (x 4)(x + 2)
6 FOIL METHOD Words To multiply two expressions that each contain two terms, add the products of the _First_ terms, the _Outer_ terms, the _Inner_ terms, and the _Last_ terms. Example F O I L (x + 4)(x + 7) = x 2 + 7x + 4x + 28 = x x + 28 Example 3 Change from intercept form to standard form Write y = 3(x + 2)(x 5) in standard form. Example 4 Change from vertex form to standard form Write f(x) = 5(x + 2) in standard form. Checkpoint Write the quadratic function in standard form. 3. y = 4(x 3) y = 3(x 7)(x + 6)
7 4.3 Solve x 2 bx c 0 by Factoring Goal Solve quadratic equations. VOCABULARY Monomial An expression that is either a number, a variable, or the product of a number and one or more variables Binomial The sum of two monomials Trinomial The sum of three monomials Quadratic equation An equation in one variable that can be written in the form ax 2 + bx + c = 0 where a 0 Root of an equation A solution of a quadratic function Zero of a function The numbers p and q of a function in intercept form are also called the zeros of the function. FACTOR 2 x bx c STEP 1: STEP 2: c Example 1 b Find two numbers whose product is c r r c and whose sum is b. r r b r c r 1 2 b Write in factored form. ( x r )( x r ) 1 2
8 Factor trinomials of the form x 2 bx c Factor the expression x 2 7x 8. SPECIAL FACTORING PATTERNS Pattern Name Difference of Two Squares Perfect Square Trinomial Perfect Square Trinomial a 2 b 2 = ( a + b )( a b ) x 2 4 = (x + 2)(x 2) a 2 + 2ab + b 2 = ( a + b ) 2 x 2 + 6x + 9 = (x + 3) 2 a 2 2ab + b 2 = ( a b ) 2 x 2 4x + 4 = (x 2) 2 Example 2 Factor with special patterns Checkpoint Factor the expression. If it cannot be factored, say so. 1. x 2 + 7x x 2 81
9 ZERO PRODUCT PROPERTY Words If the _product_ of two expressions is zero, then _one_ or _both_ of the expressions equals zero. Algebra If A and B are expressions and AB = _0_, then A = _0_ or B = _0_. Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = _ 5_ or x = _ 2_. Example 3 Find the roots of an equation Find the roots of the equation x 2 2x 15 = 0. Example 4 Find the zeros of a quadratic function Find the zeros of the function y = x 2 5x 6 by rewriting the function in intercept form. Checkpoint Complete the following exercises. 3. Find the roots of the equation x 2 3x + 2 = Find the zeros of the function y = x 2 + 3x 40 by rewriting the function in intercept form.
10 4.4 Solve ax 2 + bx + c = 0 by Factoring Goal Use factoring to solve equations of the form ax 2 + bx + c = 0. Step1: Step 2: ac Find two numbers whose product is c r r c 1 2 and whose sum is b. r r b 1 2 b r ac r 1 2 a a b Write in factored form. ( ax r )( ax r ) 1 2 Example 1 Factor ax 2 + bx + c where c > 0 Factor 2x 2 + 9x + 7. Example 2 Factor ax 2 + bx + c where c < 0 Factor 3x 2 x 2.
11 Checkpoint Factor the expression. If it cannot be factored, say so. 1. 3x 2 + 7x x 2 13x + 6 Example 3 Factor with special patterns Factor the expression. a. 16x 2 36 b. 9y y + 49 c. 25t 2 110t Checkpoint Factor the expression y 2 40y x 2 81
12 Example 4 Factor out monomials first Factor the expression. a. 4x 2 4 b. 3y 2 18y c. 4m 2 10m + 24 d. 5z 2 25z + 40 Checkpoint Factor the expression x p p 63 Example 5 Solve quadratic equations a. 2x 2 x 21 = 0
13 b. 4r 2 18r + 24 = 6r 12 Checkpoint Solve the equation. 7. 2x 2 + 4x 30 = 0 8. z z + 12 = 5z 4
14 4.5 Solve Quadratic Equations by Finding Square Roots Goal Solve quadratic equations by finding square roots. VOCABULARY Square root A number r is a square root of a number s if r 2 = s. Radical An expression of the form s where s is a number or expression Radicand The number s beneath the radical sign Rationalizing the denominator The process of eliminating a radical from the denominator of a fraction Conjugates The expressions a + b and a b, used to rationalize the denominator, whose product is always a rational number PROPERTIES OF SQUARE ROOTS (a > 0, b> 0) Product Property Example ab = a b Quotient Property a a 2 2 = 2 b b Example 1 Use properties of square roots a b. 9 64
15 Your Notes Example 2 Rationalize denominators of fractions Simplify (a) 7 3 and (b) Checkpoint Simplify the expression Example 3 Solve a quadratic equation 1 Solve (y 6) 2 = 8. 4
16 Example 4 Model a dropped object with a quadratic function Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk? h = 16t 2 + h 0 Checkpoint Complete the following exercises. 3. Solve the equation 2x 2 16 = In Example 4, suppose that the water balloon is dropped from a height of 27 feet. How long does it take for the balloon to hit the sidewalk?
17 4.6 Perform Operations with Complex Numbers Goal Perform operations with complex numbers. VOCABULARY Imaginary unit i The imaginary unit i is defined as i 1. Complex number A number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part. Imaginary number A complex number a + bi where b 0 Complex conjugates Two complex numbers of the form a + bi and a bi Complex plane A coordinate plane where each point (a, b) represents a complex number a + bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Absolute value of a complex number The absolute value of a complex number z = a + bi, denoted z, is a nonnegative real number defined as z 2 b 2 a. THE SQUARE ROOT OF A NEGATIVE NUMBER Property 1. If r is a positive real r number, i r. then Example 3 i 3 2. By Property (1), it follows that r 2 2 i r. i 3 2 i 3
18 Example 1 Solve a quadratic equation 2x = 35 Checkpoint Solve the equation. 1. 3x = 23 SUMS AND DIFFERENCES OF COMPLEX NUMBERS To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Difference of complex numbers: (a + bi) (c + di) = (a c) + (b d)i
19 Example 2 Add and subtract complex numbers Write as a complex number in standard form. a. (6 + 3i) (4 /) b. (2 + 5i) + (7 2i) Example 3 Multiply complex numbers Write the expression (2 + i)( 5 + 2i) as a complex number in standard form.. Example 4 Divide complex numbers Write the quotient 6 4 i in standard form. 2 i Checkpoint Write the expression as a complex number in standard form. 2. (12 2i) (16 + 3i) 3. 4i(9 + 5i)
20 i 3 i 5. (4+4i) + ( 6+3i) Example 5 Plot complex numbers Plot the complex numbers in the same complex plane. a i b. 5 4i ABSOLUTE VALUE OF A COMPLEX NUMBER The absolute value of a complex number z = a + bi, denoted z, is a _nonnegative_ real number defined as z = 2 2 a b. This is the distance of z from the _origin in the complex plane. Example 6 Find absolute values of complex numbers Find the absolute value of (a) 6 8i and (b) 6i.
21 Checkpoint Plot the complex numbers in the same complex plane. Then find the absolute value. 6. 4i i 8. 3+i
22 4.7 Complete the Square Goal Solve quadratic equations by completing the square. VOCABULARY Completing the square The process that allows you to write an expression of the form x 2 + bx as the square of a binomial COMPLETING THE SQUARE Words To complete the square for the expression x 2 + bx, add b. 2 Algebra x 2 b + bx + = 2 2 b b x x 2 2 = 2 x b 2 2 Example 1 Solving equations by square rooting. 2 x 6x 9 1 Checkpoint Solve the equation by finding square roots x 10x 25 4
23 Example 2 Make a perfect square trinomial Find the value of c that makes x 2 12x c a perfect square trinomial. Then write the expression as the square of a binomial. Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 2. x 2 24x + c 3. x x +c
24 Example 3 Solve ax 2 bx c = 0 when a = 1 Solve x 2 10x 13 0 by completing the square. Checkpoint Solve the equations by completing the square. 4. x 2 8x + 7 = 0 Example 4 Write a quadratic function in vertex form Write y = x 2 14x 44 in vertex form. Then identify the vertex.
25 Checkpoint Write the quadratic function in vertex form. Then identify the vertex. 5. y = x 2 12x y = x x +53 Example 5 (Day 2) Solve ax 2 bx c 0 when a 1 Solve 3x 2 12x 27 = 0 by completing the square. Checkpoint Solve the equations by completing the square. 5. 2x 2 20x + 24 = 0
26 4.8 Use the Quadratic Formula and the Discriminant Goal Solve quadratic equations using the quadratic formula. VOCABULARY Quadratic formula The formula that gives the solutions to any quadratic equation Discriminant The expression b 2 4ac under the radical sign of the quadratic formula THE QUADRATIC FORMULA Let a, b, and c be real numbers such that a 0. The solutions of the quadratic equation ax 2 + bx + c are: x = b 2 b 4 2 a ac Example 1 Solve an equation with two real solutions Solve x 2 7x 6. Example 2 Solve an equation with one real solution Solve 2x 2 8x 8 = 0.
27 Checkpoint Use the quadratic formula to solve the equation. 1. 2x x = x 2 13x = 7x x 2 6x + 6 = 0 4. x 2 3x + 3 = 0 USING THE DISCRIMINANT OF ax 2 + bx + c = 0 When b 2 4ac > 0, the equation has _two real solutions_. The graph has _two_xintercepts. When b 2 4ac = 0, the equation has _one real solution_. The graph has _one_xintercept. When b 2 4ac < 0, the equation has _two imaginary solutions_. The graph has _no_xintercepts.
28 Example 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x 2 + 6x + 5 = 0 b. x 2 + 6x + 9 = 0 c. x 2 + 6x + 13 = 0 Checkpoint Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 5. x 2 8x + 17 = 0 6. x 2 + 4x + 3 = 0 7. x 2 + 2x 1 = 0 8. x 2 + 6x + 4 = 0
7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationWarmUp Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x and yintercepts of graphs of equations. Use symmetry to sketch graphs
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationCONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)
CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationPARABOLAS AND THEIR FEATURES
STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: AAPR.3: Identify zeros of polynomials
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationAlgebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations
Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More information53 Polynomial Functions. not in one variable because there are two variables, x. and y
y. 53 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More informationGraphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x value and L be the yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationPolynomials and Quadratics
Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying
More informationManhattan Center for Science and Math High School Mathematics Department Curriculum
Content/Discipline Algebra 1 Semester 2: Marking Period 1  Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationAlgebra 2 Chapter 5 Practice Test (Review)
Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationALGEBRA I (Created 2014) Amherst County Public Schools
ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies
More informationDRAFT. Algebra 1 EOC Item Specifications
DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationSection 3.1 Quadratic Functions and Models
Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationMultiplying and Dividing Radicals
9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationQUADRATIC EQUATIONS AND FUNCTIONS
Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide
More informationMath 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction
Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationBEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.
BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. HrubikVulanovic email: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 HigherDegree Polynomial Functions... 1 Section 6.1 HigherDegree Polynomial Functions...
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationUnit 7: Radical Functions & Rational Exponents
Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationMath 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationMathematics Curriculum
Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 249x + 6 x  6 A) 1, x 6 B) 8x  1, x 6 x 
More information1.3. Maximum or Minimum of a Quadratic Function. Investigate A
< P16 photo of a large arched bridge, similar to the one on page 292 or p 360361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.12.2, and Chapter
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More information