Section 2.3. Learning Objectives. Graphing Quadratic Functions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Section 2.3. Learning Objectives. Graphing Quadratic Functions"

Transcription

1 Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic Functions The general technique for graphing quadratics is the same as for graphing linear equations. However, since quadratics graph as curvy lines (called parabolas ), rather than the straight lines generated by linear equations.

2 Quadratic Function A quadratic function, in mathematics, is a polynomial function of the form: The graph of a quadratic function is a parabola whose major axis is parallel to the y-axis. Definitions Every parabola has an axis of symmetry which is the line that runs down its center. This line divides the graph into two perfect halves. Axis of symmetry formula is x = - b/2a Definitions (Vertex) Parabolas always have a lowest point (or a highest point, if the parabola is upside-down). This point, where the parabola changes direction, is called the vertex. {- b/2a, f(-b/2a)} (Note: The a in the vertex form f(x) = a(x h) 2 + k of the quadratic is the same as the a in the common form of the quadratic equation, y = ax 2 + bx + c.)

3 Graphing Quadratic Functions positive quadratic y = x 2 negative quadratic y = x 2 Definitions (Vertex) Quadratic function in vertex form: f(x) = a(x h) 2 + k 1. Vertex is (h, k) 2. Axis of symmetry: x = h 3. y-intercept: Set x = 0 and solve for y. Or we can say, find f(0). 4. x-intercept: Set f(x) = 0 and solve for x. Solving a Quadratic Equation (Ex 1) Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Using the quadratic formula to solve this equation just substitute a, b, and c into the general formula:

4 Quadratic Equation (Ex 1) Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Solving a Quadratic Equation Example 1 Quadratic Equation: y = x² + 2x + 1 where a = 1, b = 2, and c = 1 Solving a Quadratic Equation (Ex 2) What is the range of y = x 2 4 x + 5 Step 1: Determine the min. values of the parabola Step 2: Use the formula: -b/2a => 4/2 = 2 Step 3: Substitute x into the equation y = 2 2 4(2) + 5 = 1 So the range is all points above the min. value or y 1

5 Solving a Quadratic Equation (Ex 2) Example 3 Find the x-intercepts and vertex of y = x 2 + 2x 4. To find the vertex, look at the coefficients: a = 1 and b = 2. Then: h = (2) / 2( 1) = 1 To find k, plug h in for x and simplify: k = (1) 2 + 2(1) 4 = = 2 5 = 3 (1, -3) Solution is the vertex Example 3 Now find some additional plot points using a T-chart:

6 Example 3 Pick x-values that were centered around the x-coordinate of the vertex and plot the parabola: Example 4 Solve the quadratic equation: x 2 = 32 4x 1. By factoring 2. With the quadratic equation 3. With your graphing calculator Example 4 Solve the quadratic equation: x 2 = 32 4x 1. By factoring (x 4)(x + 8) = 0 (4, - 8)

7 Example 4 Solve the quadratic equation: x 2 = 32 4x 1. With the quadratic equation Get a = 1; b = 4; and c =-32 Plug it in the equation and solve Example 4 Solve the quadratic equation: x 2 = 32 4x 1. With your graphing calculator Example 5 Given the function f(x) = x 2 4x + 2, complete the following parts. 1. State if the graph of f(x) opens upward or downward. 2. Find the vertex algebraically. Write answer in order pair. 3. State if the function has a maximum or min. Then give the max./min. value 4. Give the equation for the axis of symmetry. 5. Give the range of the function in interval notation.

8 Quadratic Function in Vertex Form (Ex 5) Example: Now sketch the graph of f(x) = x 2 4x + 2 Breakeven Analysis (Ex 6) The CFO of a company that makes Blackberry s has a revenue (in millions of dollars) and cost functions for x millions Blackberrys are given: R(x) = x(94.8 5x) C(x) = x Both have a domain 1 x 15 Use your graphing calculator to find the BE points to the nearest thousand Blackberrys. Breakeven Analysis (Ex 6) Graph the Profit function P(x) =

9 Quadratic Regression Example 7 A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression to fit a linear model to the data, we would use a quadratic regression on the graphing calculator to find the function of the form y = ax 2 + bx + c that best fits the data. Go to stat, choose 5. QuadReg Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). x Mileage Using the quadratic regression on the graphing calculator, find the quadratic function that best fits the date. Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). x Mileage Using the quadratic regression on the graphing calculator, find the quadratic function that best fits the date.

10 Example of Quadratic Regression (Ex 7) An tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). What is the slope? What is the vertex? y = x x Graphing Quadratic Functions (Ex 8) The most basic quadratic is y = x 2. Graphing Quadratic Functions (Ex 8) The most basic quadratic is y = x 2.

11 More General Functions In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x 2 4x + 7 is a polynomial, but x 2 4/x + 7x 3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number. Graphs of Even Degree Polynomials with a positive leading coefficient with a negative leading coefficient Graphs of Odd Degree Polynomials with a positive leading coefficient with a negative leading coefficient

12 Why, why, why? Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions.

Quadratic Functions and Models

Quadratic Functions and Models Quadratic Functions and Models MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: analyze the graphs of quadratic functions, write

More information

3.1 Quadratic Functions

3.1 Quadratic Functions 3.1 Quadratic Functions and Models Copyright Cengage Learning. All rights reserved. Objectives Graphing Quadratic Functions Using the Standard Form Maximum and Minimum Values of Quadratic Functions Modeling

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

Lesson 36 MA 152, Section 3.1

Lesson 36 MA 152, Section 3.1 Lesson 36 MA 5, Section 3. I Quadratic Functions A quadratic function of the form y = f ( x) = ax + bx + c, where a, b, and c are real numbers (general form) has the shape of a parabola when graphed. The

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

Functions in Standard Form

Functions in Standard Form 2-2 Properties of of Quadratic Functions in Functions in Warm Up Lesson Presentation Lesson Quiz Algebra 2 2-1 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point

More information

TEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x - h) 2 + k symbolic representations.

TEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x - h) 2 + k symbolic representations. Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum

More information

Precalculus Workshop - Functions

Precalculus Workshop - Functions Introduction to Functions A function f : D C is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set C. D is called the domain of f. C is called the codomain of f.

More information

Objective 1: Identify the characteristics of a quadratic function from its graph

Objective 1: Identify the characteristics of a quadratic function from its graph Section 8.2 Quadratic Functions and Their Graphs Definition Quadratic Function A quadratic function is a second-degree polynomial function of the form, where a, b, and c are real numbers and. Every quadratic

More information

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function)

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function) Outcome R3 Quadratic Functions McGraw-Hill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard

More information

CH 9. Quadratic Equations and Functions

CH 9. Quadratic Equations and Functions 9.1: Graph 9.2: Graph 9.3: Solve Quadratic Equations by Graphing 9.4: Use Square Roots to Solve Quadratic Equations 9.5: Solve Quadratic Equations by Completing the Square 9.6: Solve Quadratic Equations

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

More information

Week 1: Functions and Equations

Week 1: Functions and Equations Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

More information

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to:

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to: FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4.

More information

Graphing Quadratics using Transformations 5-1

Graphing Quadratics using Transformations 5-1 Graphing Quadratics using Transformations 5-1 5-1 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point ( 2, 5), give the coordinates of the translated point. 1.

More information

Quadratic Function Parabola Shape

Quadratic Function Parabola Shape Axis of Symmetry MA 158100 Lesson 8 Notes Summer 016 Definition: A quadratic function is of the form f(x) = y = ax + bx + c; where a, b, and c are real numbers and a 0. This form of the quadratic function

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Section 2.3 Quadratic Functions

Section 2.3 Quadratic Functions Section 2.3 Quadratic Functions DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the Standard

More information

7. The solutions of a quadratic equation are called roots. SOLUTION: The solutions of a quadratic equation are called roots. The statement is true.

7. The solutions of a quadratic equation are called roots. SOLUTION: The solutions of a quadratic equation are called roots. The statement is true. State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The axis of symmetry of a quadratic function can be found by using the equation x =. The

More information

The graphs of quadratic functions are so popular that they were given their own name. They are called parabolas.

The graphs of quadratic functions are so popular that they were given their own name. They are called parabolas. DETAILED SOLUTIONS AND CONCEPTS - QUADRATIC FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

PARABOLAS AND THEIR FEATURES

PARABOLAS AND THEIR FEATURES STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

More information

5-1. Lesson Objective. Lesson Presentation Lesson Review

5-1. Lesson Objective. Lesson Presentation Lesson Review 5-1 Using Transformations to Graph Quadratic Functions Lesson Objective Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x h) 2 + k. Lesson Presentation Lesson

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

Algebra. Indiana Standards 1 ST 6 WEEKS

Algebra. Indiana Standards 1 ST 6 WEEKS Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

More information

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s). 1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find

More information

QUADRATIC EQUATIONS AND FUNCTIONS

QUADRATIC EQUATIONS AND FUNCTIONS Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide

More information

f is a parabola whose vertex is the point (h,k). The parabola is symmetric with

f is a parabola whose vertex is the point (h,k). The parabola is symmetric with Math 1014: Precalculus with Transcendentals Ch. 3: Polynomials and Rational Functions Sec. 3.1 Quadratic Functions I. Quadratic Functions A. Definition 1. A quadratic function is a function of the form

More information

Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under

Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

Portable Assisted Study Sequence ALGEBRA IIA

Portable Assisted Study Sequence ALGEBRA IIA SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

More information

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets

More information

Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions

Pre-AP Algebra 2 Unit 3 Lesson 1 Quadratic Functions Unit 3 Lesson 1 Quadratic Functions Objectives: The students will be able to Identify and sketch the quadratic parent function Identify characteristics including vertex, axis of symmetry, x-intercept,

More information

Quadratic Functions. vertex. a > 0 opens up. a < 0 opens down. Definition: If a, b, c, h, and k are real numbers with a 0, then the functions

Quadratic Functions. vertex. a > 0 opens up. a < 0 opens down. Definition: If a, b, c, h, and k are real numbers with a 0, then the functions Math 141-copyright Joe Kahlig Page 1 Quadratic Functions Definition: If a, b, c, h, and k are real numbers with a 0, then the functions y = ax 2 +bx+c y = a(x h) 2 +k standard form vertex form both represent

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved. 1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

More information

Mathematics Chapter 8 and 10 Test Summary 10M2

Mathematics Chapter 8 and 10 Test Summary 10M2 Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus

More information

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2 Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

quadratic function, ( ) General and Standard forms of the equation of Quadratic Functions. Definition. 2 Graphs of Quadratic Functions.

quadratic function, ( ) General and Standard forms of the equation of Quadratic Functions. Definition. 2 Graphs of Quadratic Functions. CH 3.1(PART I). Quadratic Functions. Lectures #1 and #13 General and Standard forms of the equation of Quadratic Functions. Definition. f ( x) = ax + bx + c, where a, b, c are real numbers, a 0 is called

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

Student Resource Book Unit 2

Student Resource Book Unit 2 Student Resource Book Unit 2 1 2 3 4 5 6 7 8 9 10 ISBN 978-0-8251-7336-3 Copyright 2013 J. Weston Walch, Publisher Portland, ME 04103 www.walch.com Printed in the United States of America WALCH EDUCATION

More information

4.3-4 Quadratic Functions and Models

4.3-4 Quadratic Functions and Models 4.3-4 Quadratic Functions and Models I. Definition A quadratic function is a function of the form (standard form) where a, b, and c are real numbers and a 0. The domain of a quadratic function consists

More information

1.01 b) Operate with polynomials.

1.01 b) Operate with polynomials. 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of exponents. There are a few rules that simplify our dealings with exponents. Given the same base, there are ways

More information

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

N.CN.7, A.CED.1, 2, 3, N.Q.2, A.SSE.1,

N.CN.7, A.CED.1, 2, 3, N.Q.2, A.SSE.1, Learning Targets: I can solve interpret key features of quadratic functions from different form. I can choose a method to solve, and then, solve a quadratic equation and explain my reasoning. #1 4 For

More information

) represents the original function, will a, r 1

) represents the original function, will a, r 1 Transformations of Quadratic Functions Focus on Content: Download Problem Sets Problem Set #1 1. In the graph below, the quadratic function displayed on the right has been reflected over the y-axis producing

More information

Algebra II Semester Exam Review Sheet

Algebra II Semester Exam Review Sheet Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

More information

Pre-Calculus III Linear Functions and Quadratic Functions

Pre-Calculus III Linear Functions and Quadratic Functions Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3

More information

Chapter 1 Notes: Quadratic Functions

Chapter 1 Notes: Quadratic Functions 1 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form axis

More information

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology

More information

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots.

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots. 9.1 Solving Quadratic Equations by Finding Square Roots 1. Evaluate and approximate square roots. 2. Solve a quadratic equation by finding square roots. Key Terms Square Root Radicand Perfect Squares Irrational

More information

Section 2.1 Intercepts; Symmetry; Graphing Key Equations

Section 2.1 Intercepts; Symmetry; Graphing Key Equations Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

More information

AND RATIONAL FUNCTIONS. 2 Quadratic. Polynomials

AND RATIONAL FUNCTIONS. 2 Quadratic. Polynomials UNIT 5 POLYNOMIAL In prior units of Core-Plus Mathematics, you developed understanding and skill in the use of linear, quadratic, and inverse variation functions. Those functions are members of two larger

More information

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

More information

with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

with a, b and c representing real numbers, and a is not equal to zero. 3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

More information

Understanding Quadratic Functions Using Real World Problems and IT Abstract Introduction COL 111 (Mathematical Modeling with Functions)

Understanding Quadratic Functions Using Real World Problems and IT Abstract Introduction COL 111 (Mathematical Modeling with Functions) Understanding Quadratic Functions Using Real World Problems and IT Nakhshin A. Karim, Bsc, Msc Department of Mathematics and Statistics Zayed University Abu Dhabi, United Arab Emirates Nakhshin_karim@zu.ac.ae

More information

Pre-Calculus 20 Chapter 3 Notes

Pre-Calculus 20 Chapter 3 Notes Section 3.1 Quadratic Functions in Vertex Form Pre-Calculus 20 Chapter 3 Notes Using a table of values, graph y = x 2 y x y=x 2-2 4-2 4 x Using a table of values, graph y = -1x 2 (or y = -x 2 ) y x y=-x

More information

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved. 2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

BROCK UNIVERSITY MATHEMATICS MODULES

BROCK UNIVERSITY MATHEMATICS MODULES BROCK UNIVERSITY MATHEMATICS MODULES 11A.4: Maximum or Minimum Values for Quadratic Functions Author: Kristina Wamboldt WWW What it is: Maximum or minimum values for a quadratic function are the largest

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved. 1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

More information

Section 3.1 Quadratic Functions and Models

Section 3.1 Quadratic Functions and Models Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the

More information

Introduction to Quadratic Functions

Introduction to Quadratic Functions Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

More information

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

More information

This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0).

This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0). This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12)

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12) Course Name: College Algebra 001 Course Code: R3RK6-CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics

More information

Just Remember, The Vertex is the highest or lowest point of the graph.

Just Remember, The Vertex is the highest or lowest point of the graph. Quadratic Functions Quadratic functions are any functions that may be written in the form y = ax 2 + bx + c where a, b, and c are real coefficients and a 0. For example, y = 2x 2 is a quadratic function

More information

CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

More information

1.2. Mathematical Models: A Catalog of Essential Functions

1.2. Mathematical Models: A Catalog of Essential Functions 1.2. Mathematical Models: A Catalog of Essential Functions Mathematical model A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon

More information

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2 10-5 Parabolas Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Given, solve for p when c = Find each distance. 2. from (0, 2) to (12, 7) 13 3. from the line y = 6 to (12, 7) 13 Objectives Write the

More information

Math 150: Summer 2011 Test 3, Form: A

Math 150: Summer 2011 Test 3, Form: A Math 150: Summer 2011 Test 3, Form: A Name: Read all of the following information before starting the exam: It is to your advantage to answer ALL of the questions. There are 15 multiple choice and 5 short

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Algebra Nation MAFS Videos and Standards Alignment Algebra 2

Algebra Nation MAFS Videos and Standards Alignment Algebra 2 Section 1, Video 1: Linear Equations in One Variable - Part 1 Section 1, Video 2: Linear Equations in One Variable - Part 2 Section 1, Video 3: Linear Equations and Inequalities in Two Variables Section

More information

Math 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick Chapter 3 Test Review

Math 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick Chapter 3 Test Review Math 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/page/2434 Name: Date: Students Will Be Able To: Chapter 3 Test Review Use the quadratic formula to

More information

Algebra II A Final Exam

Algebra II A Final Exam Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

More information

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1 Test Bank Exercises in CHAPTER 4 Exercise Set 4.1 1. Graph the quadratic function f(x) = x 2 2x 3. Indicate the vertex, axis of symmetry, minimum 2. Graph the quadratic function f(x) = x 2 2x. Indicate

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT1X Intermediate Algebra Workshop I Quadratic Functions LEARNING CENTER Overview Workshop I Quadratic Functions General Form Domain and Range Some of the effects of the leading coefficient a The vertex

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

Equations of Lines Derivations

Equations of Lines Derivations Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

More information

CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions

CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions 4.5 Completing The Square 1) Solve quadratic equations using the square root property 2) Generate perfect square trinomials by completing the square 3) Solve quadratic equations by completing the square

More information

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 8}.

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 8}. Use a table of values to graph each equation. State the domain and range. 1. y = 2x 2 + 4x 6 x y = 2x 2 + 4x 6 (x, y) 3 y = 2( 3) 2 + 4( 3) 6 = 0 ( 3,0) 2 y = 2( 2) 2 + 4( 2) 6 = ( 2, 6) 6 1 y = 2( 1)

More information

This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0).

This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0). This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

Polynomial Expressions and Equations

Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

More information

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0 College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

More information

Algebra II Quadratic Functions and Equations- Transformations Unit 05a

Algebra II Quadratic Functions and Equations- Transformations Unit 05a Previous Knowledge: (What skills do the need to have to succeed?) Squares and Square Roots Simplif Radical Expressions Multipl Binomials Solve Multi-Step Equations Identif and graph linear functions Transform

More information

Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information