Solving Polynomial Equations Exponential Growth in Factored Form
|
|
- Mitchell Marshall
- 6 years ago
- Views:
Transcription
1 7.5 Solving Polnomial Equations Eponential Growth in Factored Form is written in factored form? How can ou solve a polnomial equation that Two polnomial equations are equivalent when the have the same solutions. For instance, the following equations are equivalent because the onl solutions of each equation are = and =. Factored Form ( )( ) = 0 Standard Form + = 0 Nonstandard Form = Check this b substituting and for in each equation. ACTIVITY: Matching Equivalent Forms of an Equation Work with a partner. Match each factored form of the equation with two other forms of equivalent equations. Notice that an equation is considered to be in factored form onl when the product of the factors is equal to 0. Factored Form Standard Form a. ( )( ) = 0 A. = 0. b. ( )( ) = 0 B. + = 0. c. ( + )( ) = 0 C. 4 + = 0. d. ( )( + ) = 0 D = 0 4. e. ( + )( ) = 0 E. = 0 5. Nonstandard Form 5 = 6 ( ) = 4 = ( + ) = 4 = COMMON CORE Polnomial Equations In this lesson, ou will solve polnomial equations in factored form. Learning Standard A.REI.4b ACTIVITY: Writing a Conjecture Work with a partner. Substitute,,, 4, 5, and 6 for in each equation. Write a conjecture describing what ou discovered. a. ( )( ) = 0 b. ( )( ) = 0 c. ( )( 4) = 0 d. ( 4)( 5) = 0 e. ( 5)( 6) = 0 f. ( 6)( ) = 0 56 Chapter 7 Polnomial Equations and Factoring
2 ACTIVITY: Special Properties of 0 and Work with a partner. The numbers 0 and have special properties that are shared b no other numbers. For each of the following, decide whether the propert is true for 0,, both, or neither. Eplain our reasoning. a. If ou add to a number n, ou get n. b. If the product of two numbers is, then one or both numbers are 0. c. The square of is equal to itself. d. If ou multipl a number n b, ou get n. e. If ou multipl a number n b, ou get 0. f. The opposite of is equal to itself. 4 ACTIVITY: Writing About Solving Equations Math Practice Use Definitions What previous eamples, information, and definitions can ou use to repl to the student s comment? Work with a partner. Imagine that ou are part of a stud group in our algebra class. One of the students in the group makes the following comment. I don t see wh we spend so much time solving equations that are equal to zero. Wh don t we spend more time solving equations that are equal to other numbers? Write an answer for this student. 5. One of the properties in Activit is called the Zero-Product Propert. It is one of the most important properties in all of algebra. Which propert is it? Eplain how it is used in algebra and wh it is so important. 6. IN YOUR OWN WORDS How can ou solve a polnomial equation that is written in factored form? Use what ou learned about solving polnomial equations to complete Eercises 4 6 on page 60. Section 7.5 Solving Polnomial Equations in Factored Form 57
3 7.5 Lesson Lesson Tutorials Ke Vocabular factored form, p. 58 Zero-Product Propert, p. 58 root, p. 58 A polnomial is in factored form when it is written as a product of factors. Standard form Factored form + ( + ) ( )( + 8) When one side of an equation is a polnomial in factored form and the other side is 0, use the Zero-Product Propert to solve the polnomial equation. The solutions of a polnomial equation are also called roots. Zero-Product Propert Words If the product of two real numbers is 0, then at least one of the numbers is 0. Algebra If a and b are real numbers and ab = 0, then a = 0 or b = 0. EXAMPLE Solving Polnomial Equations Solve each equation. Check Substitute each solution in the original equation. 0(0 + 8) =? 0 0(8) =? 0 0 = 0 8( 8 + 8) =? 0 8(0) =? 0 0 = 0 a. ( + 8) = 0 ( + 8) = 0 Write equation. = 0 or + 8 = 0 Use Zero-Product Propert. = 8 Solve for. The roots are = 0 and = 8. b. ( + 6)( 5) = 0 ( + 6)( 5) = 0 Write equation. + 6 = 0 or 5 = 0 Use Zero-Product Propert. = 6 or = 5 Solve for. The roots are = 6 and = 5. Eercises 4 9. ( ) = 0. t(t + ) = 0. (z 4)(z 6) = 0 4. (b + 7) = 0 58 Chapter 7 Polnomial Equations and Factoring
4 EXAMPLE Solving a Polnomial Equation What are the solutions of (a + 7)(a 7) = 0? A 7 and 7 B 7 and 7 C and D 7 and 7 (a + 7)(a 7) = 0 Write equation. a + 7 = 0 or a 7 = 0 Use Zero-Product Propert. a = 7 or a = 7 Solve for a. The correct answer is B. EXAMPLE Real-Life Application The arch of a fireplace can be modeled b = ( + 8)( 8), 9 where and are measured in inches. The -ais represents the floor. Find the width of the arch at floor level. Use the -coordinates at floor level to find the width. At floor level, = 0. So, substitute 0 for and solve for. 5 = ( + 8)( 8) Write equation. 9 O 5 0 = ( + 8)( 8) Substitute 0 for. 9 0 = ( + 8)( 8) Multipl each side b = 0 or 8 = 0 Use Zero-Product Propert. = 8 or = 8 Solve for. The width is the distance between the -coordinates, 8 and 8. So, the width of the arch at floor level is 8 ( 8) = 6 inches. Eercises (p + 5)(p 5) = 0 6. ( 6) = 0 7. The entrance to a mine shaft can be modeled b = ( + 4)( 4), where and are measured in feet. The -ais represents the ground. Find the width of the entrance at ground level. Section 7.5 Solving Polnomial Equations in Factored Form 59
5 7.5 Eercises Help with Homework. REASONING Is = a solution of ( )( + 6) = 0? Eplain.. WRITING Describe how to solve ( )( + ) = 0 using the Zero-Product Propert.. WHICH ONE DOESN T BELONG? Which statement does not belong with the other three? Eplain our reasoning. (n 9)(n + ) (k + 5)(k ) (g + ) (-6)= +(-)= 4+(-9)= 9+(-)= 4. ( + 7) = 0 5. t(t 5) = 0 6. (s 9)(s ) = 0 7. (q + )(q ) = 0 8. (h 8) = 0 9. (m + 4) = 0 0. (5 k)(5 + k) = 0. ( g)(7 g) = 0. (p + 6) = 0. (4z ) = 0 4. ( + 4 ) ( 8) = 0 5. ( d ) ( d + ) = 0 6. ERROR ANALYSIS Describe and correct the error in solving the equation. 6( + 5) = = 0 = 5 The root is = 5. Find the -coordinates of the points where the graph crosses the -ais. 7. = ( 8)( + 8) 8. = ( 4)( 5) 9. = 0.( + )( 5) 5 9 O O O Chapter 7 Polnomial Equations and Factoring
6 0. CHOOSE TOOLS The entrance of a tunnel can be modeled b = ( 4)( 4), where 50 and are measured in feet. The -ais represents the ground. Find the width of the tunnel at ground level. 4 O 8. 5z(z + )(z ) = 0. w(w 6) = 0. (r 4)(r + 4)(r + 8) = 0 4. (p + )(p )(p + 7) = 0 00 O Tallest point GATEWAY ARCH The Gatewa Arch in St. Louis can be modeled b = ( + 5)( 5), 5 where and are measured in feet. The -ais represents the ground. a. Find the width of the arch at ground level. b. How tall is the arch? 6. Find the values of in terms of that are solutions of the equation. a. ( + )( ) = 0 b. ( )(4 + 6) = 0 Find the greatest common factor of the numbers. (Skills Review Handbook) 7. and 6 8. and , 75, and MULTIPLE CHOICE What is the slope of the line? (Section.) A B 4 4 C D 4 Section 7.5 Solving Polnomial Equations in Factored Form 6
Comparing Linear and Nonlinear Functions 5.5. ACTIVITY: Finding Patterns for Similar Figures. How can you recognize when a pattern
5.5 Comparing Linear and Nonlinear Functions in real life is linear or nonlinear? How can ou recognize when a pattern ACTIVITY: Finding Patterns for Similar Figures Work with a partner. Cop and complete
More informationHow can you determine the number of solutions of a quadratic equation of the form ax 2 + c = 0? ACTIVITY: The Number of Solutions of ax 2 + c = 0
9. Solving Quadratic Equations Using Square Roots How can ou determine the number of solutions of a quadratic equation of the form a + c = 0? ACTIVITY: The Number of Solutions of a + c = 0 Work with a
More informationHow can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines
.7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the
More informationSystems of Linear Inequalities
. Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit
More informationComparing Linear, Exponential, and Quadratic Functions
. Comparing Linear, Eponential, and Quadratic Functions How can ou compare the growth rates of linear, eponential, and quadratic functions? ACTIVITY: Comparing Speeds Work with a partner. Three cars start
More informationSolving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic
9. Solving Quadratic Equations b Graphing equation in one variable? How can ou use a graph to solve a quadratic Earlier in the book, ou learned that the -intercept of the graph of = a + b variables is
More informationSolving Systems of Linear Equations by Graphing
. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. The
More informationWriting Equations in Point-Slope Form
. Writing Equations in Point-Slope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch
More informationYou studied exponential growth and decay functions.
TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit
More informationName Date. Work with a partner. Each graph shown is a transformation of the parent function
3. Transformations of Eponential and Logarithmic Functions For use with Eploration 3. Essential Question How can ou transform the graphs of eponential and logarithmic functions? 1 EXPLORATION: Identifing
More informationFactoring Polynomials
5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with
More informationDiscrete and Continuous Domains
. Discrete and Continuous Domains How can ou decide whether the domain of a function is discrete or continuous? EXAMPLE: Discrete and Continuous Domains In Activities and in Section., ou studied two real-life
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationACTIVITY: Using a Table to Plot Points
.5 Graphing Linear Equations in Standard Form equation a + b = c? How can ou describe the graph of the ACTIVITY: Using a Table to Plot Points Work with a partner. You sold a total of $6 worth of tickets
More informationACTIVITY: Comparing Types of Decay
6.6 Eponential Deca eponential deca? What are the characteristics of 1 ACTIVITY: Comparing Tpes of Deca Work with a partner. Describe the pattern of deca for each sequence and graph. Which of the patterns
More informationSimplifying Rational Expressions
.3 Simplifying Rational Epressions What are the ecluded values of a rational epression? How can you simplify a rational epression? ACTIVITY: Simplifying a Rational Epression Work with a partner. Sample:
More informationACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.
7.9 Factoring Special Products special products? How can you recognize and factor 1 ACTIVITY: Factoring Special Products Work with a partner. Six different algebra tiles are shown below. 1 1 x x x 2 x
More informationCan a system of linear equations have no solution? Can a system of linear equations have many solutions?
5. Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with
More informationEssential Question How can you solve a nonlinear system of equations?
.5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.
More informationSEE the Big Idea. of a Falling Object (p. 400) Game Reserve (p. 394) Photo Cropping (p. 390) Gateway Arch (p. 382) Framing a Photo (p.
7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Solving Polynomial Equations in Factored Form 7.5 Factoring
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More information) approaches e
COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural
More information9.1.1 What else can I solve?
CCA Ch 9: Solving Quadratics and Inequalities Name Team # 9.1.1 What else can I solve? Solving Quadratic Equations 9-1. USE THE ZERO PRODUCT PROPERTY TO SOLVE FOR X. a. 9 3 2 4 6 b. 0 3 5 2 3 c. 2 6 0
More information9.1.1 What else can I solve?
CCA Ch 9: Solving Quadratics and Inequalities Name Team # 9.1.1 What else can I solve? Solving Quadratic Equations 9-1. USE THE ZERO PRODUCT PROPERTY TO SOLVE FOR X. a. 9 3 2 4 6 b. 0 3 5 2 3 c. 2 6 0
More informationFunctions. Essential Question What are some of the characteristics of the graph of a logarithmic function?
5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real
More informationSTAAR Category 2 Grade 7 Mathematics TEKS 7.7A. Student Activity 1. Complete the table below to show the slope and y-intercept of each equation.
STAAR Categor Grade Mathematics TEKS.A Student Activit Work with our partner to answer the following questions. Problem : Complete the table below to show the slope and -intercept of each equation. Equation
More information7.1 Practice A. w y represents the height of an object t seconds. Name Date
Name Date 7.1 Practice A In Eercises 1 3, find the degree of the monomial. 3 1. 7n. 1 w 5 3 3. 5 In Eercises 4 6, write the polnomial in standard form. Identif the degree and leading coefficient of the
More informationAlgebra 1 Skills Needed for Success in Math
Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationEssential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane?
10.7 Circles in the Coordinate Plane Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? The Equation of a Circle with Center at the Origin Work
More informationHow can you construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot
9. Scatter Plots How can ou construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot Work with a partner. The weights (in ounces) and circumferences C (in inches) of several sports
More informationEVALUATING POLYNOMIAL FUNCTIONS
Page 1 of 8 6.2 Evaluating and Graphing Polnomial Functions What ou should learn GOAL 1 Evaluate a polnomial function. GOAL 2 Graph a polnomial function, as applied in Eample 5. Wh ou should learn it To
More informationEssential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)
5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.
More information( 3x. Chapter Review. Review Key Vocabulary. Review Examples and Exercises 6.1 Properties of Square Roots (pp )
6 Chapter Review Review Ke Vocabular closed, p. 266 nth root, p. 278 eponential function, p. 286 eponential growth, p. 296 eponential growth function, p. 296 compound interest, p. 297 Vocabular Help eponential
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationModeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?
.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match
More information1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs
0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals
More informationFair Game Review. Chapter 7. Simplify the expression. Write an expression for the perimeter of the figure
Name Date Chapter 7 Simplify the expression. Fair Game Review 1. 5y + 6 9y. h + 11 + 3h 4 + + 4. 7 ( m + 8) 3. 8a 10 4a 6 a 5. 5 ( d + 3) + 4( d 6) 6. q ( q ) 16 + 9 + 7 Write an expression for the perimeter
More informationDomain, Range, and End Behavior
Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship
More informationLaurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
More information2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.
9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept
More informationMaintaining Mathematical Proficiency
Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +
More informationGraph Square Root and Cube Root Functions
TEKS 6.5 2A.4.B, 2A.9.A, 2A.9.B, 2A.9.F Graph Square Root and Cube Root Functions Before You graphed polnomial functions. Now You will graph square root and cube root functions. Wh? So ou can graph the
More informationAlgebra 1, Semester 2
Algebra, Semester for the WCSD Math Common Finals The are for students and teacher use and are aligned to the Math Common Final test blueprint for this course. When used as test practice, success on the
More informationFair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal
Name Date Chapter Graph the linear equation. Fair Game Review. =. = +. =. =. = +. = + Copright Big Ideas Learning, LLC Big Ideas Math Algebra Name Date Chapter Fair Game Review (continued) Evaluate the
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationGraphing Linear Functions The collection of all input values is called the of a function.
Math /7 NTES (9.3) Name Graphing Linear Functions The collection of all input values is called the of a function. The collection of all output values is called the of a function. Make a table for the function.
More informationMaintaining Mathematical Proficiency
Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on
More informationEssential Question How can you factor a polynomial completely?
REASONING ABSTRACTLY 7.8 To be proficient in math, ou need to know and flexibl use different properties of operations and objects. Factoring Polnomials Completel Essential Question How can ou factor a
More informationHow can you use inductive reasoning to observe patterns and write general rules involving properties of exponents?
0. Product of Powers Property How can you use inductive reasoning to observe patterns and write general rules involving properties of exponents? ACTIVITY: Finding Products of Powers Work with a partner.
More information4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
More information2.1 Evaluate and Graph Polynomial
2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of
More informationGraphing and Writing Linear Equations
Graphing and Writing Linear Equations. Graphing Linear Equations. Slope of a Line. Graphing Linear Equations in Slope-Intercept Form. Graphing Linear Equations in Standard Form. Writing Equations in Slope-Intercept
More information7.2 Connecting Intercepts and Linear Factors
Name Class Date 7.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationObjectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More informationMath 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions
1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains
More informationProperties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a
0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value
More information11.1 Inverses of Simple Quadratic and Cubic Functions
Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,
More informationSolving Quadratic Equations
9 Solving Quadratic Equations 9. Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square
More informationName Date Class Period. pencil straightedge graph paper How can you relate slope, y-intercept, and an equation?
Name Date Class Period Activit 8.5 Investigating Slope-Intercept Form MATERIALS QUESTION pencil straightedge graph paper How can ou relate slope, -intercept, and an equation? You can find the slope and
More informationAdding and Subtracting Polynomials
7.2 Adding and Subtracting Polynomials subtract polynomials? How can you add polynomials? How can you 1 EXAMPLE: Adding Polynomials Using Algebra Tiles Work with a partner. Six different algebra tiles
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
More informationSquare Root Functions as Inverses. Inverse of a Quadratic Function. y f 1 (x) x
6-1 Square Root Functions as Inverses TEKS FOCUS TEKS ()(C) Describe and analze the relationship between a function and its inverse (quadratic and square root, logarithmic and eponential), including the
More informationACTIVITY: Simplifying Algebraic Expressions
. Algebraic Expressions How can you simplify an algebraic expression? ACTIVITY: Simplifying Algebraic Expressions Work with a partner. a. Evaluate each algebraic expression when x = 0 and when x =. Use
More informationOriginal site. translation. transformation. Decide whether the red figure is a translation of the blue figure. Compare a Figure and Its Image
Page of 8 3.7 Translations Goal Identif and use translations. Ke Words translation image transformation In 996, New York Cit s Empire Theater was slid 70 feet up 2nd Street to a new location. Original
More informationSolving Linear Inequalities
Solving Linear Inequalities. Writing and Graphing Inequalities. Solving Inequalities Using Addition or Subtraction. Solving Inequalities Using Multiplication or Division. Solving Multi-Step Inequalities.5
More informationChapter Review. Review Key Vocabulary. Review Examples and Exercises. 4.1 Graphing Linear Equations (pp ) Graph y = 3x 1.
Chapter Review Review Ke Vocabular linear equation p. solution of a linear equation, p., p. 0 rise, p. 0 run, p. 0 Vocabular Help x-intercept, p. 8 -intercept, p. 8 -intercept form, p. 8 standard form,
More information15.2 Graphing Logarithmic
_ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b
More informationUNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:
UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay
More informationHow can you use multiplication or division to solve an inequality? ACTIVITY: Using a Table to Solve an Inequality
. Solving Inequalities Using Multiplication or Division How can you use multiplication or division to solve an inequality? 1 ACTIVITY: Using a Table to Solve an Inequality Work with a partner. Copy and
More informationR = { } Fill-in-the-Table with the missing vocabulary terms: 1) 2) Fill-in-the-blanks: Function
Name: Date: / / QUIZ DAY! Fill-in-the-Table with the missing vocabular terms: ) ) Input Fill-in-the-blanks: 3) Output Function A special tpe of where there is one and onl one range () value for ever domain
More informationThe Coordinate Plane and Linear Equations Algebra 1
Name: The Coordinate Plane and Linear Equations Algebra Date: We use the Cartesian Coordinate plane to locate points in two-dimensional space. We can do this b measuring the directed distances the point
More informationLEARN ABOUT the Math
1.5 Inverse Relations YOU WILL NEED graph paper graphing calculator GOAL Determine the equation of an inverse relation and the conditions for an inverse relation to be a function. LEARN ABOUT the Math
More informationFunctions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f.
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing
More informationCharacteristics of Quadratic Functions
. Characteristics of Quadratic Functions Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete
More informationMHF 4U Unit 1 Polynomial Functions Outline
MHF 4U Unit 1 Polnomial Functions Outline Da Lesson Title Specific Epectations 1 Average Rate of Change and Secants D1., 1.6, both D1.1A s - Instantaneous Rate of Change and Tangents D1.6, 1.4, 1.7, 1.5,
More informationPre-AP Algebra 2 Lesson 1-1 Basics of Functions
Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function
More informationSolve Quadratic Equations by Graphing
0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6.
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationFactoring Polynomials Using the GCF
7.6 Factoring Polynomials Using the GCF polynomial in factored form? How can you use common factors to write a 1 ACTIVITY: Finding Monomial Factors Work with a partner. Six different algebra tiles are
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationInverse of a Function
. Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to
More informationKeira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:
Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 Graphing
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationAlgebra 1 Skills Needed to be Successful in Algebra 2
Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed
More informationACTIVITY: Quarterback Passing Efficiency
3. Solving Inequalities Using Addition or Subtraction solve an inequality? How can you use addition or subtraction to 1 ACTIVITY: Quarterback Passing Efficiency Work with a partner. The National Collegiate
More informationCh 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations
Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4
More information7.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI.,
More informationName Date. Logarithms and Logarithmic Functions For use with Exploration 3.3
3.3 Logarithms and Logarithmic Functions For use with Eploration 3.3 Essential Question What are some of the characteristics of the graph of a logarithmic function? Every eponential function of the form
More information10.2 Graphing Exponential Functions
Name Class Date 10. Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Resource Locker Eplore Eploring Graphs of Eponential Functions Eponential
More informationFair Game Review. Chapter of a mile the next day. How. far will you jog over the next two days? How many servings does the
Name Date Chapter Evaluate the epression.. Fair Game Review 5 +. 3 3 7 3 8 4 3. 4 4. + 5 0 5 6 5. 3 6. 4 6 5 4 6 3 7. 5 8. 3 9 8 4 3 5 9. You plan to jog 3 4 of a mile tomorrow and 7 8 of a mile the net
More information2 nd Semester Final Exam Review Block Date
Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict
More informationFair Game Review. Chapter = How many calculators are sold when the profit is $425? Solve the equation. Check your solution.
Name Date Chapter 4 Fair Game Review Solve the equation. Check our solution.. 8 3 = 3 2. 4a + a = 2 3. 9 = 4( 3k 4) 7k 4. ( m) 2 5 6 2 = 8 5. 5 t + 8t = 3 6. 3 5h 2 h + 4 = 0 2 7. The profit P (in dollars)
More informationCh 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.
Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have
More informationIn everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
More informationFair Game Review. Chapter 5. Input, x Output, y. 1. Input, x Output, y. Describe the pattern of inputs x and outputs y.
Name Date Chapter Fair Game Review Describe the pattern of inputs and outputs.. Input, utput,. 8 Input, utput,. Input, 9. utput, 8 Input, utput, 9. The table shows the number of customers in hours. Describe
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > 0 and b 1 related to the graph of f () = log b? Resource Locker A.5.A Determine
More informationMath 141 Review for Midterm
Math 141 Review for Midterm There will be two parts to this test. Part 1 will have graph sketching, and no calculator is allowed. Part will have everthing else, and a calculator and/or graphing calculator
More information