Section 2.1 Intercepts; Symmetry; Graphing Key Equations


 Lorraine Harper
 10 months ago
 Views:
Transcription
1 Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the xaxis. 2. The xcoordinate of a point at which the graph crosses or touches the xaxis. 3. The xintercepts of the graph of an equation are those xvalues for which y = 0. y intercept is 1. The point where the line crosses (or intercepts) the yaxis. 2. The ycoordinate of a point at which the graph crosses or touches the yaxis. 3. The yintercepts of the graph of an equation are those yvalues for which x = 0. Notice that a point could be both the xintercept and yintercept. Finding xintercepts Step 1: Substitute 0 for y or. Step 2: Solve for the x variable. Finding yintercepts Step 1: Substitute 0 for x. Step 2: Solve for the y variable. Axis of symmetry is a line of symmetry for a graph If a graph has the xaxis symmetry, then. Step 1: Replace y by y in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the x axis if the equivalent equation results. If a graph has the yaxis symmetry, then. Step 1: Replace x by x in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the y axis if the equivalent equation results If a graph has the origin symmetry, then Step 1: Replace x by x and y by y in the equation. Step 2: Solve for y. Step 3: The graph of the equation in the step 2 is symmetric with respect to the origin if the equivalent equation results CheonSig Lee Page 1
2 Exercises (Solution 2) The xintercepts of the graph of an equation are those xvalues for which y = 0. The yintercepts of the graph of an equation are those yvalues for which x = (Solution 3) For every point, ; Symmetric with respect to the yaxis is, Symmetric with respect to the xaxis is, Symmetric with respect to the origin is, 4. (Solution 4) 4 is an xintercept of this graph 4, 0 Symmetric with respect to the yaxis is 4, 0 Symmetric with respect to the xaxis is 4, 0 Symmetric with respect to the origin is 4, 0 CheonSig Lee Page 2
3 5. (Solution 5) 3, 4 If the graph is symmetric with respect to the yaxis, value changes, so 3, 4 If the graph is symmetric with respect to the xaxis, value changes, so 3, 4 If the graph is symmetric with respect to the origin, both value and value change, so 3, 4 6. (Solution 6) To find xintercepts of the graph of an equation, let y = 0 and solve for x. To find yintercepts of the graph of an equation, let x = 0 and solve for y. 7. (Solution 7) The xcoordinate of a point at which the graph crosses or touches the xaxis is an xintercept. The ycoordinate of a point at which the graph crosses or touches the yaxis is a yintercept. 8. (Solution 8) The statement is false because a graph of a circle is symmetric with respect to the xaxis, yaxis, and origin. CheonSig Lee Page 3
4 9. (Solution 9) xintercept yintercept Therefore, the intercepts are 2,0, 2,0, 0, 8 Step 1: Substitute 0 for y. Step 1: Substitute 0 for x. and the graph is shown as below when you plot the obtained all three points Step 2: Solve for x. Step 2: Solve for y So, yintercept is 0, So, xints are 2,0, 2,0 10. Plotting 3, 2 CheonSig Lee Page 4
5 11. (Solution 11) (a) (b) xintercepts are 3,0 and 3,0 yintercept is 0,3 Thus, intercepts are 3,0, 3,0, 0,3 Actually, we do not have enough information to decide the symmetry. We must assume that the graph has the symmetry. yaxis symmetry: Origin symmetry: CheonSig Lee Page 5
6 12. (Solution 12) Finding xintercept(s) Substitute 0 for y, then solve for x Let m be, then 0 and we have By acmethod, we have or Since m = x 2, m is a positive for all x. Thus, discard the proposed solution Therefore, the xintercepts are 4,0, 4,0 Finding yintercept(s) Substitute 0 for x, then solve for y Therefore, the yintercept is 0, 96 xintercept yintercept xintercept xaxis symmetry Replace y by y, then solve for y is different from the original equation, so the graph is not symmetric with respect to the xaxis yaxis symmetry Origin symmetry Replace x by x, then solve for y. Replace x by x and y by y, then solve for y The equivalent equation results, so the graph has the yaxis symmetry is different from the original equation, so the graph is not symmetric with respect to the origin. CheonSig Lee Page 6
7 13. (Solution 13) xintercept(s) Substitute 0 for y, then solve for x or Therefore, the xintercepts are 0,0, 7,0 and 7,0 Origin symmetry Replace x by x and y by y, then solve for y The equivalent equation results, so the graph has the origin symmetry. yintercept(s) Substitute 0 for x, then solve for y Thus, the yintercept is 0,0 Therefore, the intercepts are 0,0, 7,0, and 7,0 xaxis symmetry Replace y by y, then solve for y is different from the original equation, so the graph is NOT symmetric with respect to the xaxis. yaxis symmetry Replace x by x, then solve for y is different from the original equation, so the graph is NOT symmetric with respect to the yaxis. Graph the function. Step 1: Plot all intercepts obtained. Step 2: Find behavior of the end points. CheonSig Lee Page 7
8 Solving Quadratic Equations By Factoring By Square Root Property and Completing the square By Quadratic Formula Solving Quadratic Equations by Factoring Step 1: Factor out GCF, if possible Step 2: Determine the number of terms in the polynomial Step 3: Determine a factoring technique as follows: If the given polynomial is a Binomial, factoring by one of the following If the given polynomial is a Trinomial, factoring by acmethod If the given polynomial has 4 or more terms, factoring by Grouping Greatest Common Factor (GCF): The GCF is an expression of the highest degree that divides each term of the polynomial. The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial. Finding GCF Step 1: Find the prime factorization of all integers and integer coefficients Step 2: List all the factors that are common to all terms, including variables Step 3: Choose the smallest power for each factor that is common to all terms Step 4: Multiply these powers to find the GCF NOTE: If there is no common prime factor or variable, then the GCF is 1 (Example) Find the GCF for the following set of algebraic terms: {30x 4 y, 45x 3 y, 75x 5 y 2 } Therefore, GCF is CheonSig Lee Page 8
9 Factoring by acmethod is used for trinomials and some binomials Factoring acmethod will be discussed with examples Examples for positive ac 1. Factor the given polynomial Factor the given polynomial 5 4 Step 1: Factor out GCF. GCF = 1, so DONE! Step 1: Factor out GCF. GCF = 1, so DONE! 1, 7, 10 1, 5, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 4: 1 4, 2 2 Since ac is positive, there are two case; Since ac is positive, there are two case; Case 1: Both factors are positive Case 1: Both factors are positive Case 2: Both factors are negative Case 2: Both factors are negative Since b is positive, both factors are positive Since b is negative, both factors are negative 1 10,, 2 2 Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. Then numbers found in step 4 as coefficients. Then factoring by grouping. factoring by grouping Factor the given polynomial Factor the given polynomial Step 1: Factor out GCF. GCF = 2 Step 1: Factor out GCF. GCF = , 11, 5 2, 9, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 14: 1 14, 2 7 Since ac is positive, there are two case; Since ac is positive, there are two case; Case 1: Both factors are positive Case 1: Both factors are positive Case 2: Both factors are negative Case 2: Both factors are negative Since b is positive, both factors are positive Since b is negative, both factors are negative, , Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. numbers found in step 4 as coefficients. Then, factoring by grouping. Then, factoring by grouping CheonSig Lee Page 9
10 Examples for negative ac 1. Factor the given polynomial Factor the given polynomial 3 10 Step 1: Factor out GCF. GCF = 1, so DONE! Step 1: Factor out GCF. GCF = 1, so DONE! 1, 3, 10 1, 3, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 10: 1 10, 2 5 Since ac is negative, one factor is positive Since ac is negative, one factor is positive and the other is negative and the other is negative Since b is positive, bigger factor is positive Since b is negative, bigger factors is negative 1 10, 1 10, Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. Then numbers found in step 4 as coefficients. Then factoring by grouping. factoring by grouping Factor the given polynomial Factor the given polynomial Step 1: Factor out GCF. GCF = 2 Step 1: Factor out GCF. GCF = , 11, 5 2, 5, Step 4: Find positive factors of ac whose products Step 4: Find positive factors of ac whose products Positive factors of 10: 1 10, 2 5 Positive factors of 14: 1 14, 2 7 Since ac is negative, one factor is positive Since ac is negative, one factor is positive and the other is negative and the other is negative Since b is positive, bigger factor is positive Since b is negative, bigger factors is negative, , Step 5: Rewrite the middle term, bx, using the two Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. numbers found in step 4 as coefficients. Then, factoring by grouping. Then, factoring by grouping Square Root Property If, then where a is a nonzero real number. Note: indicates that or CheonSig Lee Page 10
CH 9. Quadratic Equations and Functions
9.1: Graph 9.2: Graph 9.3: Solve Quadratic Equations by Graphing 9.4: Use Square Roots to Solve Quadratic Equations 9.5: Solve Quadratic Equations by Completing the Square 9.6: Solve Quadratic Equations
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More information7. The solutions of a quadratic equation are called roots. SOLUTION: The solutions of a quadratic equation are called roots. The statement is true.
State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The axis of symmetry of a quadratic function can be found by using the equation x =. The
More informationGraphing Quadratics using Transformations 51
Graphing Quadratics using Transformations 51 51 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point ( 2, 5), give the coordinates of the translated point. 1.
More information10.1 NotesGraphing Quadratics
Name: Period: 10.1 NotesGraphing Quadratics Section 1: Identifying the vertex (minimum/maximum), the axis of symmetry, and the roots (zeros): State the maximum or minimum point (vertex), the axis of symmetry,
More informationQuadratic Function Parabola Shape
Axis of Symmetry MA 158100 Lesson 8 Notes Summer 016 Definition: A quadratic function is of the form f(x) = y = ax + bx + c; where a, b, and c are real numbers and a 0. This form of the quadratic function
More informationTEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x  h) 2 + k symbolic representations.
Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum
More informationThe xintercepts of the graph are the xvalues for the points where the graph intersects the xaxis. A parabola may have one, two, or no xintercepts.
Chapter 101 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
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 informationKey Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function)
Outcome R3 Quadratic Functions McGrawHill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard
More informationMath 2: Algebra 2, Geometry and Statistics Ms. SheppardBrick Chapter 3 Test Review
Math 2: Algebra 2, Geometry and Statistics Ms. SheppardBrick 617.596.4133 http://lps.lexingtonma.org/page/2434 Name: Date: Students Will Be Able To: Chapter 3 Test Review Use the quadratic formula to
More informationGraphing Quadratic Functions
Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x
More information) represents the original function, will a, r 1
Transformations of Quadratic Functions Focus on Content: Download Problem Sets Problem Set #1 1. In the graph below, the quadratic function displayed on the right has been reflected over the yaxis producing
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
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
More informationMathematics Chapter 8 and 10 Test Summary 10M2
Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus
More informationSection summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2
Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2
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 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 informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
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 informationLesson 36 MA 152, Section 3.1
Lesson 36 MA 5, Section 3. I Quadratic Functions A quadratic function of the form y = f ( x) = ax + bx + c, where a, b, and c are real numbers (general form) has the shape of a parabola when graphed. The
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationThe graphs of quadratic functions are so popular that they were given their own name. They are called parabolas.
DETAILED SOLUTIONS AND CONCEPTS  QUADRATIC FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationPark Forest Math Team. Meet #5. Algebra. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
More informationCCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions
4.5 Completing The Square 1) Solve quadratic equations using the square root property 2) Generate perfect square trinomials by completing the square 3) Solve quadratic equations by completing the square
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationChapter 1 Notes: Quadratic Functions
1 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a Ushape called a. Standard Form Vertex Form axis
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationPreCalculus III Linear Functions and Quadratic Functions
Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3
More informationWhat's Going On? Quiz Tomorrow. What's My Vertex? Find my Vertex! What's my y Intercept?
What's Going On? Quiz Tomorrow What's My Vertex? Find my Vertex! What's my y Intercept? Learning Goal I will understand how to find the zeros or x intercepts of a quadratic equation. 1 What's my Vertex?
More informationFunctions in Standard Form
22 Properties of of Quadratic Functions in Functions in Warm Up Lesson Presentation Lesson Quiz Algebra 2 21 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point
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 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 informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
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 informationThis is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0).
This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationGRAPHING LINEAR EQUATIONS IN TWO VARIABLES
GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: SlopeIntercept Form: y = mx+ b In an equation
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 informationThe equation of the axis of symmetry is. Therefore, the xcoordinate of the vertex is 2.
1. Find the yintercept, the equation of the axis of symmetry, and the xcoordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. Here, a = 2, b = 8, and c
More informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
More informationJust Remember, The Vertex is the highest or lowest point of the graph.
Quadratic Functions Quadratic functions are any functions that may be written in the form y = ax 2 + bx + c where a, b, and c are real coefficients and a 0. For example, y = 2x 2 is a quadratic function
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationQuadratic Functions [Judy Ahrens, Pellissippi State Technical Community College]
Quadratic unctions [Judy Ahrens, Pellissippi State Technical Community College] A quadratic function may always e written in the form f(x)= ax + x + c, where a 0. The degree of the function is (the highest
More informationChapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
More information31 Exponential Functions
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 1. f (x) = 2 x 4 16 2 4 1 2 0 1 1 0.5
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 informationf is a parabola whose vertex is the point (h,k). The parabola is symmetric with
Math 1014: Precalculus with Transcendentals Ch. 3: Polynomials and Rational Functions Sec. 3.1 Quadratic Functions I. Quadratic Functions A. Definition 1. A quadratic function is a function of the form
More informationSection 8.1 System of Linear Equations: Substitution and Elimination
Identifying Linear Systems Linear Systems in two variables Solution, 0 1 1 0 Consistent/ Independent Consistent/ Dependent Inconsistent Exactly one solution Infinitely many solutions No solution Two lines
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationCHAPTER 3: GRAPHS OF QUADRATIC RELATIONS
CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology
More informationObjective 1: Identify the characteristics of a quadratic function from its graph
Section 8.2 Quadratic Functions and Their Graphs Definition Quadratic Function A quadratic function is a seconddegree polynomial function of the form, where a, b, and c are real numbers and. Every quadratic
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 informationPreAP Algebra 2 Unit 3 Lesson 1 Quadratic Functions
Unit 3 Lesson 1 Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, xintercept,
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationTitle: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under
Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly
More informationSection 2.3. Learning Objectives. Graphing Quadratic Functions
Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic
More information3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models
3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.11 Polynomial Function A polynomial function of degree n, where n
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 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 informationQuadratic Functions and Models
Quadratic Functions and Models MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: analyze the graphs of quadratic functions, write
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
More informationAnswer Key Building Polynomial Functions
Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,
More informationThis is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0).
This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationQuadratic. Functions
Quadratic Functions 5A Quadratic Functions and Complex Numbers Lab Explore Parameter Changes 51 Using Transformations to Graph Quadratic Functions 5 Properties of Quadratic Functions in Standard Form
More informationBROCK UNIVERSITY MATHEMATICS MODULES
BROCK UNIVERSITY MATHEMATICS MODULES 11A.4: Maximum or Minimum Values for Quadratic Functions Author: Kristina Wamboldt WWW What it is: Maximum or minimum values for a quadratic function are the largest
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 informationPreCalculus 20 Chapter 3 Notes
Section 3.1 Quadratic Functions in Vertex Form PreCalculus 20 Chapter 3 Notes Using a table of values, graph y = x 2 y x y=x 22 42 4 x Using a table of values, graph y = 1x 2 (or y = x 2 ) y x y=x
More informationSECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.
(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much
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 information2.1 Algebraic Expressions and Combining like Terms
2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More information2x  y 4 y 3x  6 y < 2x 5x  3y > 7
DETAILED SOLUTIONS AND CONCEPTS GRAPHICAL REPRESENTATION OF LINEAR INEQUALITIES IN TWO VARIABLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu.
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 informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a n x n + a n1 x n1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationMINI LESSON. Lesson 5b Solving Quadratic Equations
MINI LESSON Lesson 5b Solving Quadratic Equations Lesson Objectives By the end of this lesson, you should be able to: 1. Determine the number and type of solutions to a QUADRATIC EQUATION by graphing 2.
More informationThis is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0).
This is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/ 3.0/)
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationM 1310 4.1 Polynomial Functions 1
M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a
More informationRockhurst High School Algebra 1 Topics
Rockhurst High School Algebra 1 Topics Chapter 1 PreAlgebra Skills Simplify a numerical expression using PEMDAS. Substitute whole numbers into an algebraic expression and evaluate that expression. Substitute
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 information7.1 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 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 I Pacing Guide Days Units Notes 9 Chapter 1 ( , )
Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.11.4, 1.61.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
More informationA coordinate system is formed by the intersection of two number lines, the horizontal axis and the vertical axis. Consider the number line.
State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. An exponent indicates the number the base is to be multiplied by. The exponent indicates the
More informationwith "a", "b" and "c" representing real numbers, and "a" is not equal to zero.
3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,
More informationGraphing the Complex Roots of Quadratic Functions on a Three Dimensional Coordinate Space
IOSR Journal of Mathematics (IOSRJM) eissn: 22785728. Volume 5, Issue 5 (Mar.  Apr. 2013), PP 2736 Graphing the Complex Roots of Quadratic Functions on a Three Dimensional Coordinate Space Aravind
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More information2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.
2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results
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 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 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 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 informationUnit 10: Quadratic Equations Chapter Test
Unit 10: Quadratic Equations Chapter Test Part 1: Multiple Choice. Circle the correct answer. 1. The area of a square is 169 cm 2. What is the length of one side of the square? A. 84.5 cm C. 42.25 cm B.
More information