Section 2.1 Intercepts; Symmetry; Graphing Key Equations
|
|
|
- Lorraine Harper
- 7 years 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 x-axis. 2. The x-coordinate of a point at which the graph crosses or touches the x-axis. 3. The x-intercepts of the graph of an equation are those x-values for which y = 0. y intercept is 1. The point where the line crosses (or intercepts) the y-axis. 2. The y-coordinate of a point at which the graph crosses or touches the y-axis. 3. The y-intercepts of the graph of an equation are those y-values for which x = 0. Notice that a point could be both the x-intercept and y-intercept. Finding x-intercepts Step 1: Substitute 0 for y or. Step 2: Solve for the x variable. Finding y-intercepts 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 x-axis 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 y-axis 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 Cheon-Sig Lee Page 1
2 Exercises (Solution 2) The x-intercepts of the graph of an equation are those x-values for which y = 0. The y-intercepts of the graph of an equation are those y-values for which x = (Solution 3) For every point, ; Symmetric with respect to the y-axis is, Symmetric with respect to the x-axis is, Symmetric with respect to the origin is, 4. (Solution 4) 4 is an x-intercept of this graph 4, 0 Symmetric with respect to the y-axis is 4, 0 Symmetric with respect to the x-axis is 4, 0 Symmetric with respect to the origin is 4, 0 Cheon-Sig Lee Page 2
3 5. (Solution 5) 3, 4 If the graph is symmetric with respect to the y-axis, -value changes, so 3, 4 If the graph is symmetric with respect to the x-axis, -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 x-intercepts of the graph of an equation, let y = 0 and solve for x. To find y-intercepts of the graph of an equation, let x = 0 and solve for y. 7. (Solution 7) The x-coordinate of a point at which the graph crosses or touches the x-axis is an x-intercept. The y-coordinate of a point at which the graph crosses or touches the y-axis is a y-intercept. 8. (Solution 8) The statement is false because a graph of a circle is symmetric with respect to the x-axis, y-axis, and origin. Cheon-Sig Lee Page 3
4 9. (Solution 9) x-intercept y-intercept 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, y-intercept is 0, So, x-ints are 2,0, 2,0 10. Plotting 3, 2 Cheon-Sig Lee Page 4
5 11. (Solution 11) (a) (b) x-intercepts are 3,0 and 3,0 y-intercept 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. y-axis symmetry: Origin symmetry: Cheon-Sig Lee Page 5
6 12. (Solution 12) Finding x-intercept(s) Substitute 0 for y, then solve for x Let m be, then 0 and we have By ac-method, we have or Since m = x 2, m is a positive for all x. Thus, discard the proposed solution Therefore, the x-intercepts are 4,0, 4,0 Finding y-intercept(s) Substitute 0 for x, then solve for y Therefore, the y-intercept is 0, 96 x-intercept y-intercept x-intercept x-axis 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 x-axis y-axis 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 y-axis symmetry is different from the original equation, so the graph is not symmetric with respect to the origin. Cheon-Sig Lee Page 6
7 13. (Solution 13) x-intercept(s) Substitute 0 for y, then solve for x or Therefore, the x-intercepts 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. y-intercept(s) Substitute 0 for x, then solve for y Thus, the y-intercept is 0,0 Therefore, the intercepts are 0,0, 7,0, and 7,0 x-axis 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 x-axis. y-axis 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 y-axis. Graph the function. Step 1: Plot all intercepts obtained. Step 2: Find behavior of the end points. Cheon-Sig 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 ac-method 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 Cheon-Sig Lee Page 8
9 Factoring by ac-method is used for trinomials and some binomials Factoring ac-method 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 Cheon-Sig 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 Cheon-Sig Lee Page 10
1.2 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 y-intercepts of graphs of equations. Use symmetry to sketch graphs
Lecture 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
Graphing 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
ALGEBRA 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
Vocabulary 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
Factoring 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
This 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
Section 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
FACTORING 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.
Algebra 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
expression 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
MSLC 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
1.3 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.
1.6 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
( ) 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
Name 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
Section 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
Algebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
Chapter 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
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students
Answer 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,
Definitions 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
2.5 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: [
Graphing 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 y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the
Factoring 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
NSM100 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
M 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
PARABOLAS 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
MATH 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.
Factoring 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,
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
Procedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + 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
Florida 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
Algebra 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
Factoring 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
Mathematics Placement
Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.
Algebra 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 real-life problem. algebraic expression - An expression with variables.
6.1 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
Math 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
A synonym is a word that has the same or almost the same definition of
Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given
Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
Factoring 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
3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
What 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
Algebra 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: A-APR.3: Identify zeros of polynomials
Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
BEST 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
MATH 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
Tool 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
Graphing Linear Equations in Two Variables
Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
Zeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
Factoring 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
Basic Graphing Functions for the TI-83 and TI-84
Basic Graphing Functions for the TI-83 and TI-84 The primary benefits of the TI-83 and TI-84 are the abilities to graph functions and to identify properties those functions possess. The purpose of this
A Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
AIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
TI-83/84 Plus Graphing Calculator Worksheet #2
TI-83/8 Plus Graphing Calculator Worksheet #2 The graphing calculator is set in the following, MODE, and Y, settings. Resetting your calculator brings it back to these original settings. MODE Y Note that
Factoring 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
In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
MA107 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
A. 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!
6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
Elements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on
Math 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
Algebra 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
Algebra 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
Graphing - Slope-Intercept Form
2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,
Higher 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
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
is the degree of the polynomial and is the leading coefficient.
Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...
Unit 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
HIBBING 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,
In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
MATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 )
Quiz: Factoring by Graphing Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 ) (x-3)(x-6), (x-6)(x-3), (1x-3)(1x-6), (1x-6)(1x-3), (x-3)*(x-6), (x-6)*(x-3), (1x- 3)*(1x-6),
In the Herb Business, Part III Factoring and Quadratic Equations
74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original
Manhattan 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
Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.
Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line
1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
Factoring Algebra- Chapter 8B Assignment Sheet
Name: Factoring Algebra- Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
CRLS 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
x 4-1 = (x²)² - (1)² = (x² + 1) (x² - 1) = (x² + 1) (x - 1) (x + 1)
Factoring Polynomials EXAMPLES STEP 1 : Greatest Common Factor GCF Factor out the greatest common factor. 6x³ + 12x²y = 6x² (x + 2y) 5x - 5 = 5 (x - 1) 7x² + 2y² = 1 (7x² + 2y²) 2x (x - 3) - (x - 3) =
QUADRATIC 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
Lesson 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 y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
Factoring a Difference of Two Squares. Factoring a Difference of Two Squares
284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this
EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
Copy 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,...}
CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
South Carolina College- and Career-Ready (SCCCR) Algebra 1
South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process
Introduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
Algebra 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
FACTORING POLYNOMIALS
296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
ALGEBRA 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
Warm-Up 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
Week 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: 2.1-2.2, and Chapter