Vertex. March 23, Ch 9 Guided Notes.notebook

Size: px
Start display at page:

Download "Vertex. March 23, Ch 9 Guided Notes.notebook"

Transcription

1 March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function is =. Since all graphs are a variation of this one we called it the "parent" function for quadratics. Dec :7 PM Dec :00 PM The graph is smmetric. (It is the same on both sides.) It will have a minimum or a maimum. The line where ou could fold the graph onto itself is called the ais of smmetr. The ais of smmetr alwas passes through the verte. Apr 9 :07 AM Dec :00 PM What makes the graph open up? What makes the graph skinn? What makes the graph open down? What makes the graph wide? Dec : PM Apr 9 :0 AM

2 March, 07 What moves the graph up? Give three properties of the graphs of the equations below. What moves the graph down? Apr 9 :0 AM Apr 9 :0 AM Domain and Range of a Quadratic Function Domain values All quadratic function have a domain of all real numbers. ou can plug an number in for. Range values Look at the graph. See the ma/min and use that to create our range. The range is never all real numbers for quadratics Domain: Range: Find the domain and range of the following quadratic Domain: Range: Mar 0 0: AM Mar 0 0: AM 9 Graphing Quadratic Functions One important feature of a quadratic function is the location of the ais of smmetr. Find the equation of the ais of smmetr of the following quadratic function. The equation for the ais of smmetr is: Feb 7 :7 AM Feb 7 : AM

3 March, 07 Find the equation of the ais of smmetr of the following quadratic function. Knowing the location of the ais of smmetr helps ou find the verte of the parabola. ) Find the ais of smmetr. ) Plug our answer from # into the equation for to find the value of. Be sure to use parentheses around negative numbers. ) (,) from and is the location of the verte. Find the verte of the following quadratic function. Feb 7 : AM Feb 7 :9 AM Find the verte of the following quadratic function. The intercept of the quadratic is just the value of. Find the intercept of the following parabola. Feb 7 9:7 AM Feb 7 9:7 AM Find the intercept of the following parabola. Put everthing together. Here are the steps to graphing a quadratic function. a) Find ais of smmetr using = b/a, and then draw it on the graph. b) Plug our answer from part a into the given equation to find the value of the verte. Put the verte in the middle of the table and on our graph. c) The intercept is the value of c. Put it on our graph and table. Mark the matching point on the other side of the graph and table. d) Find one more point b choosing an value not et used on our table. Plug it into the given equation. Put the point on our graph and table. Mark the matching point on the other side. e) Make sure our table has points and our graph has points. Feb 7 9: AM Feb 7 9: AM

4 March, 07 Graph the quadratic equation be filling in points on the table AND points on the graph. Graph the quadratic equation be filling in points on the table AND points on the graph. 0 0 Feb 7 9: AM Feb 7 9: AM 9 Solving Quadratics Equations when b=0 When can we use this method? When there is no b value. How to use: ) Get the b itself. (add/sub then mult/div) ) Take the square root of both sides ) Answer is the + and of that number The square root of a negative number is not real... thus the answer would be: No solution How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb :9 PM Feb : PM How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb : PM Feb : PM

5 March, 07 Not ever equation will have a solution: Eample: How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Ma 7 : PM Ma 7 :9 PM How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb : PM Feb : PM 9 Solving Quadratics Using the Factoring Method Zero Product Propert When two numbers multipl to be 0, then one of the numbers must be 0. We should write all quadratic problems with a value of b in their factored form so that we can use this propert. This onl works when the equation is =0. Ma : PM Ma :0 PM

6 March, 07 Ma :0 PM Ma :0 PM How are these different than the ones we did esterda? esterda there was no b value. Toda there is a b value. Ma : PM Ma : PM How do the answers relate to the graph? The answers we get are where the graph, when equal to 0, will cross the -ais. Solve the equation Check the answers Sketch the graph Ma :7 PM Apr 7 0: AM

7 March, 07 Summar of how to solve when there is a b-value: ) Make sure one side is =0 (add/sub to make it happen) ) Factor using the ac method ) Set each factor (thing in parentheses) equal to 0 9 Completing the Square and Verte Form Section we used square roots to solve. That section was simple in comparison to section. Eample: = *Don't forget plus and minus Apr 7 0: AM Feb 7 9:0 AM So we can appl that same knowledge to the following equation. (+) = *Don't forget plus and minus AFTER ou square root both sides. If possible write the following as a quantit squared. What if the equation is NOT written as a quantit squared? Then we need to force it to be that we so that we can appl our simple process from section. Let's take a look at writing epressions as "something" squared. Feb 7 0: AM Apr :9 PM So what do we do with the things that were not squared values. MAKE them squared while still following our rules of algebra. Completing the Square Steps: ) Make sure a = (if not div b a to all terms) ) Get c b itself on the other side ) Add to both sides. ) Factor must be the same factors (will alwas be b/) ) Square Root ( + or ) ) Solve resulting two equations. Feb 7 0: AM Feb 7 0: AM 7

8 March, 07 Solve b completing the square. Solve b completing the square. Feb 7 : PM Feb 7 : PM Solve b completing the square. Solve b completing the square. Ma : PM Feb 7 :0 AM Verte Form: Find the verte: (h,k) is the verte of the quadratic Ma :7 PM Ma : PM

9 March, 07 Find the verte: Find the verte: Ma : PM Ma : PM If not in the right form, then ou must complete the square to find the verte. Write in verte form and state the verte: Write in verte form and state the verte: Ma : PM Ma : PM 9 The Quadratic Formula Solve using the quadratic formula. The equation must be =0 in order to use. a is b is c is There are two seperate answers here. The first time ou do + the square root and get an answer. The second time ou do the square root and get an answer. Feb 7 : AM Apr :00 PM 9

10 March, 07 Solve using the quadratic formula. (Must be set = 0 to use the quadratic formula.) Solve using the quadratic formula. Feb 9 : PM Feb 9 :0 PM 9 7 Linear, Quadratic, and Eponential Models Linear Quadratic Eponential Are the following graphs linear, quadratic or eponential? Linear the same amount each time Quadratic The differences are the same. Eponential the same amount each time Feb 7 : AM Feb 7 :9 PM Are the following graphs linear, quadratic or eponential? Are the following graphs linear, quadratic or eponential? Feb 7 :9 PM Feb 7 :9 PM 0

11 March, 07 Which tpe of function best models the data in each table? Which tpe of function best models the data in each table? Tpe of function: Equation from calc: Tpe of function: Equation from calc: Feb 7 :0 PM Feb 7 :0 PM Which tpe of function best models the data in each table? Tpe of function: Equation from calc: 9 Solving Sstems Using a Graphing Device ) Plug both equations into calculator. (Be sure each equation is solved for or ou can't plug it in.) ) Adjust window so that ou can see the intersections of the graphs. ) Find the intersections of the graphs. Remember the value of where the cross is our answer. 0 Feb 7 :0 PM Mar 7: AM Use a graphing device to solve for the value(s) of. Use a graphing device to solve for the value(s) of. Mar 7:7 AM Mar 7: AM

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots. Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to

More information

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve

More information

Vertex form of a quadratic equation

Vertex form of a quadratic equation Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #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 information

RELATIONS AND FUNCTIONS through

RELATIONS AND FUNCTIONS through RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or

More information

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.

Ch 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 information

Modeling Revision Questions Set 1

Modeling Revision Questions Set 1 Modeling Revision Questions Set. In an eperiment researchers found that a specific culture of bacteria increases in number according to the formula N = 5 2 t, where N is the number of bacteria present

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

QUADRATIC FUNCTION REVIEW

QUADRATIC FUNCTION REVIEW Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

More information

Unit 2 Notes Packet on Quadratic Functions and Factoring

Unit 2 Notes Packet on Quadratic Functions and Factoring Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

More information

Unit 10 - Graphing Quadratic Functions

Unit 10 - Graphing Quadratic Functions Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

More information

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape

More information

3.7 Linear and Quadratic Models

3.7 Linear and Quadratic Models 3.7. Linear and Quadratic Models www.ck12.org 3.7 Linear and Quadratic Models Learning Objectives Identif functions using differences and ratios. Write equations for functions. Perform eponential and quadratic

More information

Mt. Douglas Secondary

Mt. Douglas Secondary Foundations of Math 11 Section 7.1 Quadratic Functions 31 7.1 Quadratic Functions Mt. Douglas Secondar Quadratic functions are found in everda situations, not just in our math classroom. Tossing a ball

More information

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a. Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of

More information

Maintaining Mathematical Proficiency

Maintaining 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 information

Rev Name Date. Solve each of the following equations for y by isolating the square and using the square root property.

Rev Name Date. Solve each of the following equations for y by isolating the square and using the square root property. Rev 8-8-3 Name Date TI-8 GC 3 Using GC to Graph Parabolae that are Not Functions of Objectives: Recall the square root propert Practice solving a quadratic equation f Graph the two parts of a hizontal

More information

Secondary Math 2 Honors Unit 4 Graphing Quadratic Functions

Secondary Math 2 Honors Unit 4 Graphing Quadratic Functions SMH Secondary Math Honors Unit 4 Graphing Quadratic Functions 4.0 Forms of Quadratic Functions Form: ( ) f = a + b + c, where a 0. There are no parentheses. f = 3 + 7 Eample: ( ) Form: f ( ) = a( p)( q),

More information

Writing Quadratic Functions in Standard Form

Writing Quadratic Functions in Standard Form Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval

More information

Systems of Linear Equations: Solving by Graphing

Systems 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 information

EOC Review. Algebra I

EOC Review. Algebra I EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.

More information

17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96

17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96 Section.3 Zeros of the Quadratic 473.3 Eercises In Eercises 1-8, factor the given quadratic polnomial. 1. 2 + 9 + 14 2. 2 + 6 + 3. 2 + + 9 4. 2 + 4 21. 2 4 6. 2 + 7 8 7. 2 7 + 12 8. 2 + 24 In Eercises

More information

Quadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.

Quadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3. Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

First Semester Final Review NON-Graphing Calculator

First Semester Final Review NON-Graphing Calculator Algebra First Semester Final Review NON-Graphing Calculator Name:. 1. Find the slope of the line passing through the points ( 5, ) and ( 3, 7).. Find the slope-intercept equation of the line passing through

More information

Algebra Final Exam Review Packet

Algebra Final Exam Review Packet Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

More information

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.

More information

5.2 Solving Quadratic Equations by Factoring

5.2 Solving Quadratic Equations by Factoring Name. Solving Quadratic Equations b Factoring MATHPOWER TM, Ontario Edition, pp. 78 8 To solve a quadratic equation b factoring, a) write the equation in the form a + b + c = b) factor a + b + c c) use

More information

Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra 2/Honors Algebra 2 Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

Graphs and Solutions for Quadratic Equations

Graphs and Solutions for Quadratic Equations Format y = a + b + c where a 0 Graphs and Solutions for Quadratic Equations Graphing a quadratic equation creates a parabola. If a is positive, the parabola opens up or is called a smiley face. If a is

More information

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept. Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of

More information

Maintaining Mathematical Proficiency

Maintaining 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 information

National 5 Mathematics

National 5 Mathematics St Andrew s Academ Mathematics Department National 5 Mathematics UNIT 4 ASSESSMENT PREPARATION St Andrew's Academ Maths Dept 016-17 1 Practice Unit Assessment 4A for National 5 1. Simplif, giving our answer

More information

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of

More information

Unit 12 Study Notes 1 Systems of Equations

Unit 12 Study Notes 1 Systems of Equations You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve

More information

Section 2.5: Graphs of Functions

Section 2.5: Graphs of Functions Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree

More information

Reteaching (continued)

Reteaching (continued) Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression

More information

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula

Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and

More information

Algebra 2 Semester Exam Review

Algebra 2 Semester Exam Review Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation

More information

Worksheet #1. A little review.

Worksheet #1. A little review. Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves

More information

Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions

Math 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 information

Keira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:

Keira 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 information

Algebra 1 Skills Needed for Success in Math

Algebra 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 information

6.4 graphs OF logarithmic FUnCTIOnS

6.4 graphs OF logarithmic FUnCTIOnS SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS

More information

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric

More information

A. Simplifying Polynomial Expressions

A. Simplifying Polynomial Expressions A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

Solve Quadratics Using the Formula

Solve Quadratics Using the Formula Clip 6 Solve Quadratics Using the Formula a + b + c = 0, = b± b 4 ac a ) Solve the equation + 4 + = 0 Give our answers correct to decimal places. ) Solve the equation + 8 + 6 = 0 ) Solve the equation =

More information

CC Algebra 2H Transforming the parent function

CC Algebra 2H Transforming the parent function CC Algebra H Transforming the Name: March. Open up the geometer s sketchpad document on Mr. March s website (It s under CC Algebra Unit Algebra Review). Make sure ou maimize both windows once ou open the

More information

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10 CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.

More information

Math Review Packet #5 Algebra II (Part 2) Notes

Math Review Packet #5 Algebra II (Part 2) Notes SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of

More information

a > 0 parabola opens a < 0 parabola opens

a > 0 parabola opens a < 0 parabola opens Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(

More information

12x y (4) 2x y (4) 5x y is the same as

12x y (4) 2x y (4) 5x y is the same as Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents

More information

H.Algebra 2 Summer Review Packet

H.Algebra 2 Summer Review Packet H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,

More information

TRANSFORMATIONS OF f(x) = x Example 1

TRANSFORMATIONS OF f(x) = x Example 1 TRANSFORMATIONS OF f() = 2 2.1.1 2.1.2 Students investigate the general equation for a famil of quadratic functions, discovering was to shift and change the graphs. Additionall, the learn how to graph

More information

AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs

AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs AQA Level Further mathematics Number & algebra Section : Functions and their graphs Notes and Eamples These notes contain subsections on: The language of functions Gradients The equation of a straight

More information

Answers Investigation 2

Answers Investigation 2 Applications 1. a. Square Side Area 4 16 2 6 36 7 49 64 9 1 10 100 11 121 Rectangle Length Width Area 0 0 9 1 9 10 2 20 11 3 33 12 4 4 13 6 14 6 4 1 7 10 b. (See Figure 1.) c. The graph and the table both

More information

Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square

Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square Chapter Notes Alg. H -A (Lesson -) Solving Quadratic Equations b Finding the Square Root and Completing the Square p. *Calculator Find the Square Root: take the square root of. E: Solve b finding square

More information

Using Intercept Form

Using 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 information

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand

More information

College Algebra Final, 7/2/10

College Algebra Final, 7/2/10 NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Algebra II Midterm Exam Review Packet

Algebra II Midterm Exam Review Packet Algebra II Midterm Eam Review Packet Name: Hour: CHAPTER 1 Midterm Review Evaluate the power. 1.. 5 5. 6. 7 Find the value of each epression given the value of each variable. 5. 10 when 5 10 6. when 6

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

More information

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology. Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

MATH 152 COLLEGE ALGEBRA AND TRIGONOMETRY UNIT 1 HOMEWORK ASSIGNMENTS

MATH 152 COLLEGE ALGEBRA AND TRIGONOMETRY UNIT 1 HOMEWORK ASSIGNMENTS 0//0 MATH COLLEGE ALGEBRA AND TRIGONOMETRY UNIT HOMEWORK ASSIGNMENTS General Instructions Be sure to write out all our work, because method is as important as getting the correct answer. The answers to

More information

Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry

Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function = - -1 0 1 1 Eample 1: Make a table,

More information

Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 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 information

Section 4.1 Increasing and Decreasing Functions

Section 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

2 nd Semester Final Exam Review Block Date

2 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 information

Fair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal

Fair 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 information

CHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis

CHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis ADDITIONAL MATHEMATICS MODULE 5 QUADRATIC FUNCTIONS CHAPTER 3 : QUADRARIC FUNCTIONS MODULE 5 3.1 CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions 3 3.3 Graphs of quadratic functions 4 Eercise

More information

The American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet

The American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness

More information

Pure Core 1. Revision Notes

Pure Core 1. Revision Notes Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....

More information

Algebra 2 Honors Summer Packet 2018

Algebra 2 Honors Summer Packet 2018 Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

More information

3. TRANSLATED PARABOLAS

3. TRANSLATED PARABOLAS 3. TRANSLATED PARABOLAS The Parabola with Verte V(h, k) and Aes Parallel to the ais Consider the concave up parabola with verte V(h, k) shown below. This parabola is obtained b translating the parabola

More information

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember

More information

Quadratic equationsgraphs

Quadratic equationsgraphs All reasonable efforts have been made to make sure the notes are accurate. The author cannot be held responsible for an damages arising from the use of these notes in an fashion. Quadratic equationsgraphs

More information

One of the most common applications of Calculus involves determining maximum or minimum values.

One of the most common applications of Calculus involves determining maximum or minimum values. 8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..

More information

ALGEBRA 1 CP FINAL EXAM REVIEW

ALGEBRA 1 CP FINAL EXAM REVIEW ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8

More information

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3 1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which

More information

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

More information

y = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is

y = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,

More information

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100

Mathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100 Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation

More information

3.1 Graphing Quadratic Functions. Quadratic functions are of the form.

3.1 Graphing Quadratic Functions. Quadratic functions are of the form. 3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.

More information

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. 1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function

More information

x Radical Sign: Radicand: the number beneath the radical sign

x Radical Sign: Radicand: the number beneath the radical sign Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.

More information

Algebra 2 CPA Summer Assignment 2018

Algebra 2 CPA Summer Assignment 2018 Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

More information

One Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x

One Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have

More information

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

(b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens (2) If a is negative, it opens

(b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens (2) If a is negative, it opens Section.1 Graphing Quadratics Objectives: 1. Graph Quadratic Functions. Find the ais of symmetry and coordinates of the verte of a parabola.. Model data using a quadratic function. y = 5 I. Think and Discuss

More information

Equations for Some Hyperbolas

Equations for Some Hyperbolas Lesson 1-6 Lesson 1-6 BIG IDEA From the geometric defi nition of a hperbola, an equation for an hperbola smmetric to the - and -aes can be found. The edges of the silhouettes of each of the towers pictured

More information

9.12 Quadratics Review

9.12 Quadratics Review Algebra Name _ B2g0gD6L jkwudtaaa msvopfwtowiarneq CLOLXCa.I K `Awljla `rtiugohhtfs_ QrIefsfeYrZvtetdf. 9.2 Quadratics Review ) What is the difference between the two mathematical statements below? Then

More information

decreases as x increases.

decreases as x increases. Chapter Review FREQUENTLY ASKED Questions Q: How can ou identif an eponential function from its equation? its graph? a table of values? A: The eponential function has the form f () 5 b, where the variable

More information