PRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0

Size: px
Start display at page:

Download "PRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0"

Transcription

1 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following form: ax bx c OR f ( x) ax bx c where a, b, and c are constants and a 0 When ou graph a quadratic function, ou get a curve, called a. B. PRACTICE Which of the following are quadratic functions? Hint: You ma need to rearrange the equation so it looks like a quadratic function.) 1. x x 4. g( x) x 5. h ( t) ( t 1) 5 3. f ( x) (x 5)( x 6) C. VOCABULARY Ever parabola can be divided into two smmetrical halves. This line is called the and is designated b a dotted line in the graph. The point where the line of smmetr intersects the parabola is called the. The - is where the parabola intersects the -axis and the - are the two points where the parabola intersects the x-axis. Chapter Quadratic Functions 1

2 Recall that the is the set of x-values represented b the graph or the equation of a function. The is the set of -values for which the graph or equation is true. Example 1: Graph the function defined b the equation x 7x 10. Determine the -intercept, the x- intercepts, the equation of the axis of smmetr, the coordinates of the vertex, and the domain and range. x intercept: x-intercepts: Chapter Quadratic Functions

3 axis of smmetr: vertex: domain: range: D. QUADRATIC EQUATIONS A quadratic equation is an equation that can be written in the following form: ax bx c 0 where a, b, and c are constants and a 0 The x-intercepts of a quadratic equation are called the of the equation (whereas the x- intercepts of a quadratic function are called the ). Example : Using algebra, determine the roots of the quadratic equation x 7x Example 3: Solve the quadratic equation 3x 11x 4 0. Chapter Quadratic Functions 3

4 1 Example 4: Write the equation of a quadratic function that has zeroes and. 3 Example 5: The zeroes of a quadratic function are 4 3 and 5. Write the equation for this function. Chapter Quadratic Functions 4

5 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson Modelling Real Situations Using Quadratic Functions (.) Date The use of quadratic functions in real life can be seen when studing projectile motion in phsics. We also see quadratic functions in situations where a certain quantit is the product of two other quantities. For example, the area of a rectangle is the product of its length and its width. When one quantit increases, the other decreases. Example 1: You have 40 m of fencing to make a rectangular pen for our dogs. a) Represent the area of the pen as a function of the length of one side of the pen. b) Graph the function x c) What dimensions provide an area greater than 90 m? Chapter Quadratic Functions 5

6 Example : A compan makes canoes, and then sells them for $500 each. At this price, it can sell 60 canoes in a season, generating revenue of $ To increase revenue, management is planning to increase the selling price. It estimates that for ever $50 increase in price, the number of canoes sold will drop b 4. a) Represent the number of canoes sold as a function of the selling price. b) Represent the revenue as a function of the selling price. c) Sketch the function without making a table of values. (Hint: Determine the roots and the coordinates of the vertex.) d) What selling price will provide the maximum revenue? What is the maximum revenue? e) What range of selling prices will provide revenue greater than $30 000? Chapter Quadratic Functions 6

7 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 3 Graphing a( x p) q (.3) Date The simplest quadratic function is Example 1: Graph x. x. x The graph parabola is: x can be expanded or compressed or moved horizontall or verticall. The general equation for a a( x p) q Chapter Quadratic Functions 7

8 Example : Graph the following functions and label them. x x x x 1 x 1 x Example 3: Graph the following functions and label them. x ( x 4) ( x 4) Example 4: Graph the following functions and label them. x x x Chapter Quadratic Functions 8

9 Example 5: Graph the function ( x 3). Does the parabola open up or down? x Is the parabola expanded or compressed? The coordinates of the vertex is The equation of the axis of smmetr is Example 6: Graph the function f ( x) ( x 4) 3. Does the parabola open up or down? x Is the parabola expanded or compressed? The coordinates of the vertex is The equation of the axis of smmetr is Example 7: Determine the equation of the following graph of a quadratic function. Chapter Quadratic Functions 9

10 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 4 Graphing ax bx c (.4) Date A. COMPLETING THE SQUARE Recall: The general equation for a parabola is: a( x p) q Also, the equation for a parabola can be written in the form: ax bx c It is easier to graph the first equation rather than the second one. However, sometimes it is necessar to graph functions that are written in the second form. To do this, we need to complete the square. Example 1: Write the following functions in the form a) x 8x 9 a( x p) q. Step 1: If necessar, remove the coefficient of x as a common factor from the first two terms. Step : Take the coefficient of x (including the negative sign, if present), divide b and then square it. Add and subtract this number inside the brackets. Step 3: Remove the last term from the brackets and combine with the constant term. Step 4: Factor the expression in the brackets as a complete square. b) ( x) 3x 1x 8 f c) f ( x) 5x 30x 7 Chapter Quadratic Functions 10

11 Example : Write x 1x 11 in the form a x p) q (, then sketch the graph. Complete the square: x B. MAXIMUM AND MINIMUM VALUES The vertex represents the maximum or minimum value of a function. You can determine this value without drawing a graph. Simpl look at the equation in the form a( x p) q. Example 3: Compare the following equations and determine their maximum or minimum values. a) 3( x 4) 7 What are the coordinates of the vertex? Is a positive or negative? Does the graph open upward or downward? Is the vertex the maximum or minimum value of the graph? b) f ( x) ( x 3) 8 What are the coordinates of the vertex? Is a positive or negative? Does the graph open upward or downward? Is the vertex the maximum or minimum value of the graph? Chapter Quadratic Functions 11

12 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 5 Maximum and Minimum Problems (.5) Date In man problems involving the maximum or minimum value of a function, the function is not given. It must be determined using the information that is given in the problem. Example 1: Two numbers have a difference of 10. Their product is a minimum. What are the numbers? Step 1: Write our let statements. Step : Write an algebraic expression. Step 3: The algebraic expression must contain onl one variable. If it contains more that one variable, substitute equivalent expressions into the equation. Step 4: Determine whether the quadratic function has a maximum or minimum value. Then complete the square to determine this value and where it occurs. Step 5: Answer the question in the problem. Chapter Quadratic Functions 1

13 Example : A rectangular piece of land is bounded on one side b a river and on the other three sides b a total of 80 m of fencing. Determine the dimensions of the largest possible piece of land. Chapter Quadratic Functions 13

14 Example 3: Computer software programs are sold to students for $0 each. Three hundred students are willing to bu them at that price. For ever $5 increase in price, there are 30 fewer students willing to bu the software. What is the maximum revenue? Chapter Quadratic Functions 14

15 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 6 The Inverse of a Linear Function (.6) Date The inverse of a function is a relation whose rule is obtained from that of a function b interchanging x and. To determine the inverse of a function: Interchange x and in the equation of the function. Solve the resulting equation for. Example 1: Determine the equation of the inverse of the linear function 3x. When x and are interchanged in the equation of a function: The x and coordinates of the points that satisf the equation of the function are interchanged. The graph of the function is reflected in the line x. Example : Determine the equation of the inverse of the linear function x 4. Then make a table of values for both linear functions and graph then on the same graph. x 4 : Inverse of x 4 : x x Chapter Quadratic Functions 15

16 To express the inverse of a linear function f (x) in function notation, we use the smbol 1 ( x ) f (x) f of x f 1 ( x) f inverse of x f. Example 3: Determine the inverse of the function f ( x) x 5. 1 Example 4: Given the graph of f (x), graph f 1 ( x ) on the same grid. Chapter Quadratic Functions 16

17 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 7 The Inverse of a Quadratic Function (.7) Date Last lesson, we looked at the inverse of a linear function. To determine the inverse of a quadratic function, we use the same steps. Note: We do not use the notation f 1 ( x ) necessaril be a function! to represent the inverse of a quadratic function because it ma not Example 1: Consider the function f ( x) x 4. a) Determine the inverse of f (x). Graph f (x) and its inverse on the same grid. f ( x) x 4 : x b) Is the inverse of f (x) a function? Explain. Chapter Quadratic Functions 17

18 Sometimes it is convenient to restrict the domain of a quadratic function so that its inverse is a function. Example : Show two was to restrict the domain of f ( x) x 4 so that its inverse is a function. Illustrate each wa with a graph. Chapter Quadratic Functions 18

19 PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 8 Solving Linear-Quadratic Sstems Algebraicall Date To solve a linear-quadratic sstem algebraicall, we need to follow these steps: Step 1: Solve the linear equation for one variable. (Usuall, we solve for.) Step : Substitute into the quadratic equation and solve for the other variable. Step 3: Substitute the results from Step into the linear equation and solve for the first variable. Example 1: Find the coordinates of the points of intersection of the circle 10 Check the solution. x and the line 3 6 x. Chapter Quadratic Functions 19

20 Example : Solve the following linear-quadratic sstem: x x 6 Example 3: Solve the following linear-quadratic sstem: x 1 x 4 Chapter Quadratic Functions 0

21 Example 4: Solve the following linear-quadratic sstem: x x 1 Example 5: Solve the following linear-quadratic sstem: x x 3 Chapter Quadratic Functions 1

1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. (c) ( )

1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. (c) ( ) Name: Date: QUADRATIC FUNCTION REVIEW FLUENCY Algebra II 1. Without the use of our calculator, evaluate each of the following quadratic functions for the specified input values. (a) g( x) g g ( 5) ( 3)

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

Lesson 9 Exploring Graphs of Quadratic Functions

Lesson 9 Exploring Graphs of Quadratic Functions Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point

More information

TEST REVIEW QUADRATICS EQUATIONS Name: 2. Which of the following statements is true about the graph of the function?

TEST REVIEW QUADRATICS EQUATIONS Name: 2. Which of the following statements is true about the graph of the function? Chapter MATHEMATICS 00 TEST REVIEW QUADRATICS EQUATIONS Name:. Which equation does not represent a quadratic function?. Which of the following statements is true about the graph of the function? it has

More information

RF2 Unit Test # 2 Review Quadratics (Chapter 6) 1. What is the degree of a quadratic function?

RF2 Unit Test # 2 Review Quadratics (Chapter 6) 1. What is the degree of a quadratic function? RF Unit Test # Review Quadratics (Chapter 6) 1. What is the degree of a quadratic function? Name: a. 1 b. c. 3 d. 0. What is the -intercept for = 3x + x 5? a. 5 b. 5 c. d. 3 3. Which set of data is correct

More information

Chapter 1 Graph of Functions

Chapter 1 Graph of Functions Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane

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

Prerequisite Skills Pg. 2 # 1 7. Properties of Graphs of Functions Pg. 23 # 1 3, 5, Sketching Graphs of Functions Pg.

Prerequisite Skills Pg. 2 # 1 7. Properties of Graphs of Functions Pg. 23 # 1 3, 5, Sketching Graphs of Functions Pg. UNIT FUNCTIONS I Date Lesson Text TOPIC Homework & Video Lesson.0 ().0 Prerequisite Skills Pg. #. (). Functions Pg. # abce,, ace, ace, abc,, 8, 8. (). Absolute Value Pg. # & WS. acegikn 9. (). Properties

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

The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. Mathematics 10 Page 1 of 8 Quadratic Relations in Vertex Form The expression y ax p q defines a quadratic relation in form. The coordinates of the of the corresponding parabola are p, q. If a > 0, 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

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint. Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

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

QUADRATIC FUNCTIONS AND MODELS

QUADRATIC FUNCTIONS AND MODELS QUADRATIC FUNCTIONS AND MODELS What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum and

More information

Quadratics in Vertex Form Unit 1

Quadratics in Vertex Form Unit 1 1 U n i t 1 11C Date: Name: Tentative TEST date Quadratics in Verte Form Unit 1 Reflect previous TEST mark, Overall mark now. Looking back, what can ou improve upon? Learning Goals/Success Criteria Use

More information

MATH College Algebra Review for Test 2

MATH College Algebra Review for Test 2 MATH 34 - College Algebra Review for Test 2 Sections 3. and 3.2. For f (x) = x 2 + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the

More information

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

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

Graph the linear system and estimate the solution. Then check the solution algebraically.

Graph the linear system and estimate the solution. Then check the solution algebraically. (Chapters and ) A. Linear Sstems (pp. 6 0). Solve a Sstem b Graphing Vocabular Solution For a sstem of linear equations in two variables, an ordered pair (x, ) that satisfies each equation. Consistent

More information

Algebra 2 Unit 2 Practice

Algebra 2 Unit 2 Practice Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of

More information

Quadratic Functions. and Equations

Quadratic Functions. and Equations Name: Quadratic Functions and Equations 1. + x 2 is a parabola 2. - x 2 is a parabola 3. A quadratic function is in the form ax 2 + bx + c, where a and is the y-intercept 4. Equation of the Axis of Symmetry

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

10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.

10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities. Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =

More information

Solving Linear-Quadratic Systems

Solving Linear-Quadratic Systems 36 LESSON Solving Linear-Quadratic Sstems UNDERSTAND A sstem of two or more equations can include linear and nonlinear equations. In a linear-quadratic sstem, there is one linear equation and one quadratic

More information

Math 1050 REVIEW for Exam 1. Use synthetic division to find the quotient and the remainder. 1) x3 - x2 + 6 is divided by x + 2

Math 1050 REVIEW for Exam 1. Use synthetic division to find the quotient and the remainder. 1) x3 - x2 + 6 is divided by x + 2 Math 0 REVIEW for Eam 1 Use snthetic division to find the quotient and the remainder. 1) 3-2 + 6 is divided b + 2 Use snthetic division to determine whether - c is a factor of the given polnomial. 2) 3-32

More information

Simultaneous equations

Simultaneous equations Get started Simultaneous equations This unit will help ou to solve two equations simultaneousl, where one equation is linear and the other non-linear. AO1 Fluenc check 1 Make the subject of each equation.

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question Midterm Review 0 Precalculu Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question ) A graph of a function g is shown below. Find g(0). (-, ) (-, 0) - -

More information

MATH College Algebra Review for Test 2

MATH College Algebra Review for Test 2 MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of

More information

3.1 Graph Quadratic Functions

3.1 Graph Quadratic Functions 3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

More information

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0) Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,

More information

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1. 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

More information

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations MA 800, Summer 06 Lesson Notes Solving a Sstem of Equations b substitution (or elimination) Matrices Consider the graphs of the two equations below. A Sstem of Equations From our mathematics eperience,

More information

Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.

Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2. - Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative)

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. 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

LHS Algebra Pre-Test

LHS Algebra Pre-Test Your Name Teacher Block Grade (please circle): 9 10 11 12 Course level (please circle): Honors Level 1 Instructions LHS Algebra Pre-Test The purpose of this test is to see whether you know Algebra 1 well

More information

What Did You Learn? Key Terms. Key Concepts. 158 Chapter 1 Functions and Their Graphs

What Did You Learn? Key Terms. Key Concepts. 158 Chapter 1 Functions and Their Graphs 333371_010R.qxp 12/27/0 10:37 AM Page 158 158 Chapter 1 Functions and Their Graphs Ke Terms What Did You Learn? equation, p. 77 solution point, p. 77 intercepts, p. 78 slope, p. 88 point-slope form, p.

More information

Fixed Perimeter Rectangles

Fixed Perimeter Rectangles Rectangles You have a flexible fence of length L = 13 meters. You want to use all of this fence to enclose a rectangular plot of land of at least 8 square meters in area. 1. Determine a function for the

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a - Attributes and Transformations of Quadratic Functions TEKS FCUS VCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction

More information

Mathematics 20-1 Final Exam Review. Directions: Identify the choice that best completes the statement or answers the question.

Mathematics 20-1 Final Exam Review. Directions: Identify the choice that best completes the statement or answers the question. Mathematics 0- Final Exam Review Name: ate: irections: Identif the choice that best completes the statement or answers the question.. Which of the following numbers occurs in the sequence,,, 0,,...?. The

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

Lesson 13: More Factoring Strategies for Quadratic Equations & Expressions

Lesson 13: More Factoring Strategies for Quadratic Equations & Expressions : More Factoring Strategies for Quadratic Equations & Expressions Opening Exploration Looking for Signs In the last lesson, we focused on quadratic equations where all the terms were positive. Juan s examples

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

Mathematics 2201 Common Mathematics Assessment June 12, 2013

Mathematics 2201 Common Mathematics Assessment June 12, 2013 Common Mathematics Assessment June 1, 013 Name: Mathematics Teacher: 8 Selected Response 8 marks 13 Constructed Response marks FINAL 70 Marks TIME: HOURS NOTE Diagrams are not necessaril drawn to scale.

More information

MATHEMATICAL METHODS UNIT 1 CHAPTER 3 ALGEBRAIC FOUNDATIONS

MATHEMATICAL METHODS UNIT 1 CHAPTER 3 ALGEBRAIC FOUNDATIONS E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 dt dt MATHEMATICAL METHODS UNIT 1 CHAPTER 3 ALGEBRAIC FOUNDATIONS Key knowledge Factorization patterns, the quadratic formula and discriminant,

More information

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections 3.1, 3.3, and 3.5 Department of Mathematics, University of Wisconsin-Madison Math 11 Worksheet Sections 3.1, 3.3, and 3.5 1. For f(x) = 5x + (a) Determine the slope and the y-intercept. f(x) = 5x + is of the form y = mx

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

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

A) (-1, -1, -2) B) No solution C) Infinite solutions D) (1, 1, 2) A) (6, 5, -3) B) No solution C) Infinite solutions D) (1, -3, -7)

A) (-1, -1, -2) B) No solution C) Infinite solutions D) (1, 1, 2) A) (6, 5, -3) B) No solution C) Infinite solutions D) (1, -3, -7) Algebra st Semester Final Exam Review Multiple Choice. Write an equation that models the data displayed in the Interest-Free Loan graph that is provided. y = x + 80 y = -0x + 800 C) y = 0x 00 y = 0x +

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

MATH 115: Review for Chapter 6

MATH 115: Review for Chapter 6 MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING

More information

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear

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

Skills Practice Skills Practice for Lesson 3.1

Skills Practice Skills Practice for Lesson 3.1 Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Define each term in our own words.. quadratic function. vertical motion Problem

More information

Quadratic Functions Lesson #5

Quadratic Functions Lesson #5 Quadratic Functions Lesson #5 Axes Of Symmetry And Vertex Axis Of Symmetry As we have seen from our previous exercises: o the equation of the axis of symmetry of y = ax + bx+ c is x =. a The problem with

More information

32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary.

32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary. Pre-Calculus A Final Review Part 2 Calculator Name 31. The price p and the quantity x sold of a certain product obey the demand equation: p = x + 80 where r = xp. What is the revenue to the nearest dollar

More information

SECTION 5.1: Polynomials

SECTION 5.1: Polynomials 1 SECTION 5.1: Polynomials Functions Definitions: Function, Independent Variable, Dependent Variable, Domain, and Range A function is a rule that assigns to each input value x exactly output value y =

More information

3.1. QUADRATIC FUNCTIONS AND MODELS

3.1. QUADRATIC FUNCTIONS AND MODELS 3.1. QUADRATIC FUNCTIONS AND MODELS 1 What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Find minimum

More information

A2.MidtermRev2015. Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts)

A2.MidtermRev2015. Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts) Name: UNIT 1 Algebra 2 Midterm Exam Review Part 1: Multiple Choice (75pts) Patterns & Expressions 1. Which of the following is the seventh term in the pattern below? 2. Which of the following is the eighth

More information

10.3 Solving Nonlinear Systems of Equations

10.3 Solving Nonlinear Systems of Equations 60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6

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

Section A Plotting Straight Line Graphs Grade D / C

Section A Plotting Straight Line Graphs Grade D / C Name: Teacher Assessment Section A Plotting Straight Line Graphs Grade D / C 1. (a) Complete the table of values for = 3x + x 0 1 3 5 10 16 19 (b) On the grid draw the graph of = 3x + for values of x from

More information

Name K ^2u0g1U7d pkhutt[ad nsqojfctrwlatraep klylect.r _ ma]lulm yruibg^hxttst mrne\sue]rvveerdj. 2) h(n) = -n 2-3n; Find h(n - 2) -n 2 + n + 2

Name K ^2u0g1U7d pkhutt[ad nsqojfctrwlatraep klylect.r _ ma]lulm yruibg^hxttst mrne\sue]rvveerdj. 2) h(n) = -n 2-3n; Find h(n - 2) -n 2 + n + 2 Advanced Algebra II Name K ^u0gu7d pkhutt[ad nsqojfctrwlatraep klylect.r _ ma]lulm ruibg^httst mrne\sue]rvveerdj. Semester Review Evaluate each function. ) w() = - + ; Find w(-7) ) h(n) = -n - n; Find

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

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator,

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator, GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The 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

Ch. 7 Absolute Value and Reciprocal Functions Notes

Ch. 7 Absolute Value and Reciprocal Functions Notes First Name: Last Name: Block: Ch. 7 Absolute Value and Reciprocal Functions Notes 7. ABSOLUTE VALUE Ch. 7. HW: p. 364 # 7 odd letters, 9, 3 7. PRE-REQUISITES - GRAPH OF LINEAR FUNCTIONS 4 7. PRE-REQUISITES

More information

UNIT 1 UNIT 1: QUADRATIC FUNCTIONS. By the end of this unit, I can. Name:

UNIT 1 UNIT 1: QUADRATIC FUNCTIONS. By the end of this unit, I can. Name: UNIT 1: QUADRATIC FUNCTIONS UNIT 1 By the end of this unit, I can Draw the graph of a function using different methods Explain the meaning of the term function and distinguish between a function and a

More information

PART A CALCULATOR ACTIVE: Maximum Time: 35 Minutes

PART A CALCULATOR ACTIVE: Maximum Time: 35 Minutes Algebra II: Chapter 5 Unit Test 2 Name: PART A CALCULATOR ACTIVE: Maximum Time: 35 Minutes Fill in the blanks: Put answers in the space provided. 1. The value of k that makes x 2 + kx + 25 4 a perfect

More information

H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.

H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function. H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify

More information

8 Systems of Linear Equations

8 Systems of Linear Equations 8 Systems of Linear Equations 8.1 Systems of linear equations in two variables To solve a system of linear equations of the form { a1 x + b 1 y = c 1 x + y = c 2 means to find all its solutions (all pairs

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

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks) 1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of

More information

MATH 115: Review for Chapter 3

MATH 115: Review for Chapter 3 MATH : Review for Chapter Can ou use the Zero-Product Propert to solve quadratic equations b factoring? () Solve each equation b factoring. 6 7 8 + = + ( ) = 8 7p ( p ) p ( p) = = c = c = + Can ou solve

More information

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x 5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able

More information

Name Class Date. Solving by Graphing and Algebraically

Name Class Date. Solving by Graphing and Algebraically Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem

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

1.5. 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

1.5. 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 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

3.6 Curbside Rivalry. A Solidify Understanding Task

3.6 Curbside Rivalry. A Solidify Understanding Task Carlos and Clarita have a brilliant idea for how they will earn money this summer. Since the community in which they live includes many high schools, a couple of universities, and even some professional

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

More information

d. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36

d. 2x 3 7x 2 5x 2 2x 2 3x 1 x 2x 3 3x 2 1x 2 4x 2 6x 2 3. a. x 5 x x 2 5x 5 5x 25 b. x 4 2x 2x 2 8x 3 3x 12 c. x 6 x x 2 6x 6 6x 36 Vertices: (.8, 5.), (.37, 3.563), (.6, 0.980), (5.373, 6.66), (.8, 7.88), (.95,.) Graph the equation for an value of P (the second graph shows the circle with P 5) and imagine increasing the value of P,

More information

4Cubic. polynomials UNCORRECTED PAGE PROOFS

4Cubic. polynomials UNCORRECTED PAGE PROOFS 4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review

More information

NAME DATE PERIOD. Study Guide and Intervention

NAME DATE PERIOD. Study Guide and Intervention NAME DATE PERID Stud Guide and Intervention Graph To graph a quadratic inequalit in two variables, use the following steps: 1. Graph the related quadratic equation, = a 2 + b + c. Use a dashed line for

More information

Lesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions

Lesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions (A) Lesson Context BIG PICTURE of this UNIT: CONTEXT of this LESSON: How do we analyze and then work with a data set that shows both increase and decrease What is a parabola and what key features do they

More information

MATH HIGH SCHOOL QUADRATIC FUNCTIONS EXERCISES

MATH HIGH SCHOOL QUADRATIC FUNCTIONS EXERCISES MATH HIGH SCHOOL QUADRATIC FUNCTIONS CONTENTS LESSON 1: ZOOMING IN ON PARABOLAS... 5 LESSON : QUADRATIC FUNCTIONS... 7 LESSON 3: REAL-WORLD PROBLEMS... 13 LESSON 4: GRAPHING QUADRATICS... 15 LESSON 5:

More information

Vertex. March 23, Ch 9 Guided Notes.notebook

Vertex. March 23, Ch 9 Guided Notes.notebook 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

More information

Given the table of values, determine the equation

Given the table of values, determine the equation 3.1 Properties of Quadratic Functions Recall: Standard Form f(x) = ax 2 + bx + c Factored Form f(x) = a(x r)(x s) Vertex Form f(x) = a(x h) 2 + k Given the table of values, determine the equation x y 1

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

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

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function

The Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function 8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line

More information

Chapter 9 Quadratic Graphs

Chapter 9 Quadratic Graphs Chapter 9 Quadratic Graphs Lesson 1: Graphing Quadratic Functions Lesson 2: Vertex Form & Shifts Lesson 3: Quadratic Modeling Lesson 4: Focus and Directrix Lesson 5: Equations of Circles and Systems Lesson

More information

Math 20-1 Year End Review

Math 20-1 Year End Review M0- Year End Review.docx Name: Math 0- Year End Review hapter Sequences & Series... Pages hapter Trigonometr... Pages hapters & Quadratic Functions & Equations... Pages hapter Radicals... Pages hapter

More information

FLC Ch 1-3 (except 1.4, 3.1, 3.2) Sec 1.2: Graphs of Equations in Two Variables; Intercepts, Symmetry

FLC Ch 1-3 (except 1.4, 3.1, 3.2) Sec 1.2: Graphs of Equations in Two Variables; Intercepts, Symmetry Math 370 Precalculus [Note to Student: Read/Review Sec 1.1: The Distance and Midpoint Formulas] Sec 1.2: Graphs of Equations in Two Variables; Intercepts, Symmetry Defns A graph is said to be symmetric

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

1.2. Characteristics of Polynomial Functions. What are the key features of the graphs of polynomial functions?

1.2. Characteristics of Polynomial Functions. What are the key features of the graphs of polynomial functions? 1.2 Characteristics of Polnomial Functions In Section 1.1, ou explored the features of power functions, which are single-term polnomial functions. Man polnomial functions that arise from real-world applications

More information

A2 HW Imaginary Numbers

A2 HW Imaginary Numbers Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest

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