# 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.

Save this PDF as:

Size: px
Start display at page:

Download "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."

## Transcription

1 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 2 + bx + c, a 0 This form is the standard form of a quadratic function. When a quadratic function is in the standard form y = ax 2 + bx + c, a 0, the value of the leading coefficient, a, determines the direction that the parabola opens. When a > 0 the parabola opens upward. When a < 0 the parabola opens downward. Positive Quadratic y = x 2 Negative Quadratic y = -x 2 The point at which the parabola changes directions is called the vertex of the parabola. If the parabola opens upward the vertex is the lowest point or minimum value of the function. If the parabola opens downward the vertex is the highest point or maximum value of the function. The domain of a quadratic function f(x) = ax 2 + bx + c, a 0, is the set of all real numbers. If the parabola opens upward the range of the function will be all values greater than or equal to the minimum value. If the parabola opens downward the range of the function will be all values less than or equal to the maximum value. You can fold the graph of any quadratic function y = ax 2 + bx + c, a 0 at the vertical line through its vertex, and the two halves match exactly. This fold line is called the axis of symmetry. The vertex of a parabola is the only point on the parabola that is on the axis of symmetry. 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.

2 Chapter 10-2 Graph Quadratic Functions: Parabola You can graph a quadratic function by making a function table. It is helpful to always include the vertex as one of the points in the table when graphing a quadratic function. For the graph of a quadratic function f(x) = ax 2 + bx + c, or y = ax 2 + bx + c, where a, b and c are real numbers and a 0. x = -(b/2a) is the equation of its axis of symmetry The x-coordinate of its vertex is -(b/2a), to find the y-value of the vertex substitute the x-value into the original equation. To graph a quadratic function: Find the equation of the axis of symmetry Find the coordinates of the vertex Make a function table. Select three x values greater than the x-coordinate of the vertex and three x values less than the x-coordinate of the vertex. The more points in your table the more accurate your graph. Example: Graph y = x 2 8x Find the axis of symmetry x = -(-8/2*1)) = 8/2 = 4 2. Find the coordinates of the vertex. The x coordinate = 4, the y-coordinate is: y = (4) 2 8(4) +12 = = -4 Vertex is (4,-4) 3. Make a function table 4. Graph the ordered pairs in the table on the coordinate plane. Draw a smooth curve through them. x y = x 2 8x + 12 (x,y) 1 (1) 2 8(1) +12 = 5 (1,5) 2 (2) 2 8(2) +12 = 0 (2,0) 3 (3) 2 8(3) + 12 = -3 (3,-3) 4 (4) 2 8(4) +12 = -4 (4,-4) 5 (5) 2 8(5) + 12 = -3 (5,-3) 6 (6) 2 8(6) + 12 = 0 (6,0) 7 (7) 2 8(7) + 12 = 5 (7,5)

4 Chapter 10.4 Solve Verbal Problems Involving the Quadratic Equations Mrs. Baca's art class is painting a mural on the front of the school. The mural is 3 meters wider than it is high and has an area of 10m 2. What are the dimensions of the mural. To solve this problem you may want to begin by drawing a sketch of the mural. We can let x = the height of the of the mural, so the width will be x+3. The area is 10m 2. So we can set up the equation 10m 2 = x(x+3) If we multiply this we get 10 = x 2 + 3x. We then want to write the equation in standard form by subtracting 10 from each side of the equation. x 2 + 3x 10 = 0 Now we want to factor the quadratic (x+5)(x-2) = 0 Now we want to apply the Zero Product Property x+5 = 0 or x-2 = 0 x = -5 or x = 2 You know that since x represents the height of the mural so x = -5 can be eliminated as a solution for x since you cannot have a negative height. The height of the mural will be 2 meters and the width will be 5 meters. You can write a quadratic equation, given its roots, by working backward. From the roots write the binomial factors Multiply the factors to write the equation in standard form. Example: Write an equation with given roots {-3,5} If the roots are -3 and 5 then the factors are (x+3) and (x-5) (x+3)(x-5) = 0 x 2 +3x-5x-15 = 0 x 2 2x 15 = 0 is an equation with roots {-3,5}

5 Chapter 10.5 Solve Quadratic Equations by Completing the Square To solve a quadratic equation by completing the square you must make the quadratic expression on one side of the equation into a perfect square. After completing the square you can solve the equation by taking the square root of each side. Completing the square is often a good method to use for solving a quadratic equation when the equation is not factorable using intergers. Completing the Square to solve ax 2 + bx + c = 0 where a, b and c are real numbers and a 0: Write the equation so that the constant term, c, is isolated on the right side. Divide each side of the equation by a; x 2 +(b/a)x = c/a Find the square of one half the coefficient of x. Add that number to each side of the equation. Factor the left side of the equation. The results should have the form (x + r) 2 where r is a constant. Take the square root of each side. Then solve for x, and simplify the solutions. Example: Solve 3x 2 7x +2 = 0 3x 2 7x = -2 x 2 7/3x = -2/3 ½ of 7/3 = 7/6 x 2 7/3x + 49/36 = -2/3 + 49/36 (x-7/6) 2 = -24/ /36 (x-7/6) 2 = 25/36 (x-7/6) 2 = 25/36 rewrite equation with constant isolated on right side of equation divide both sides of the equation by a Find ½ the value of the x term add the squared value of ½ x to both sides of the equation. make like denominators and add the values on the side of equation rewrite the trinomial as the square of a binomial take the square root of both sides of the equation x-7/6 = 5/6 or x 7/6 = 5/6 write and solve two equations. x-7/6 + 7/6 = 5/6 +7/6 x 7/6 + 7/6 = -5/6 + 7/6 x = 12/6 = 2 x = 2/6 = 1/3 So, x = 2 or x = 1/3 It is important to remember that every positive real number has two square roots one positive and one negative.

6 Chapter 10.6 The Quadratic Formula and the Discriminant The Quadratic Formula is a formula that can be used to solve any quadratic equation. It is derived by solving the standard form of a quadratic equation, ax 2 + bx + c = 0, for x by completing the square. The Quadratic Formula is as follows: If ax 2 + bx + c = 0, where a 0, then x = -b +/- b 2-4ac 2a The expression b 2-4ac is called the discriminant of the quadratic equation, and it provides important information about the roots of the equation. The symbol +/- indicates that the discriminant will be both added and subtracted, so the equation will have two roots. If b 2-4ac = 0 then the only root of the equation is -b/2a, which is a rational number. If b 2-4ac > 0 then b +/- positive number. If b 2-4ac is a perfect square, both roots will be 2a rational. If it is not, the two roots will be irrational. If b 2-4ac < 0 then x = -b +/- negative number. The equation will have no real roots because 2a the square root of a negative number is not a real number.

7 Chapter 10.7 Solve Quadratic Equations with the Quadratic Formula The equation 3x x 275 can be solved by factoring or completing the square, however a more practical method would be to use the Quadratic Formula. Remember that the Quadratic Formula is: If ax 2 + bx + c = 0, where a 0, then x = -b +/- b 2-4ac 2a In our example: a = 3, b = 40 and c = -275 x = -40 +/- (40) 2 4(3)(-275) 2(3) x = -40 +/ (-3300) 6 x = -40 +/ x = -40 +/ x = = 30 or x = = x = 5 or x = -55/3

8 Chapter 10.8 Solve Linear-Quadratic Systems One way to solve a system of linear-quadratic equations is by substitution. Solve the linear equation for y and substitute that value into the quadratic equation. Let s try it: y = x 2 + 5x + 18 y = -2x + 18 y = -2x + 18 is already solved for y so all we have to do is substitute the value for y in the linear equation into the quadratic equation. -2x + 18 = x 2 + 5x x +2x addition property of equality 18 = x 2 + 7x subtraction property of equality 0 = x 2 + 7x 0 = x(x+7) distributive property x = 0 or x+7 = 0 zero product property x = 0 or x = -7 Now to find y we have to substitute the values for x into either equation. If x = 0 then y = -2(0) + 18, y = 18 If x = -7, then y = -2(-7) + 18, y = , y = 32 (0,18) is a solution and (7,-32) is a solution Systems of linear-quadratic equations can have two solutions (two points of intersection), one solution (the parabola and line intersect at one point), or no solutions (no intersection). When you solve a linear-quadratic system you must check your solution in both equations in the system. What you are doing when you solve a system is finding the point or points where the two graphs intersect. That point or points when substituted into the original equations will make them both true. If the point does not make both equations true it is not a solution to the system. When solving a linear-quadratic system of equations: Solve one equation for one variable in terms of the other Substitute into the equation not used in step 1 Solve the equation Substitute to find the corresponding values of the other variable Check the solutions

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

### 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

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### 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

### 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

### 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

### 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

### 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

### 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

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Solving Quadratic Equations by Completing the Square

9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

### 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

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

### 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

### Quadratic Functions [Judy Ahrens, Pellissippi State Technical Community College]

Quadratic unctions [Judy Ahrens, Pellissippi State Technical Community College] A quadratic function may always e written in the form f(x)= ax + x + c, where a 0. The degree of the function is (the highest

### 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

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Warm-Up Oct. 22. Daily Agenda:

Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Algebra 1-2. A. Identify and translate variables and expressions.

St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

### The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}.

Use a table of values to graph each equation. State the domain and range. 1. y = x 2 + 3x + 1 x y 3 1 2 1 1 1 0 1 1 5 2 11 Graph the ordered pairs, and connect them to create a smooth curve. The parabola

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Park Forest Math Team. Meet #5. Algebra. Self-study Packet

Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number

### 3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models

3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1-1 Polynomial Function A polynomial function of degree n, where n

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

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

### 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

### Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

### BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best

### Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

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

### 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

Concept: Quadratic Functions Name: You should have completed Equations Section 5 Part A: Problem Solving before beginning this handout. PART B: COMPUTER COMPONENT Instructions : Login to UMath X Hover

74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### 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

### Unit 7: Radical Functions & Rational Exponents

Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### Introduction Assignment

PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Tool 1. Greatest Common Factor (GCF)

Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)

CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic

### 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.

### ALGEBRA I / ALGEBRA I SUPPORT

Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

### 1.3. Maximum or Minimum of a Quadratic Function. Investigate A

< P1-6 photo of a large arched bridge, similar to the one on page 292 or p 360-361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

What Does Your Quadratic Look Like? EXAMPLES 1. An equation such as y = x 2 4x + 1 descries a type of function known as a quadratic function. Review with students that a function is a relation in which

### ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### MATH 21. College Algebra 1 Lecture Notes

MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

### MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

### 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

### Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2

Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear

### Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

### ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Algebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations

Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,

### Rationale/Lesson Abstract: Students will be able to solve a Linear- Quadratic System algebraically and graphically.

Grade Level/Course: Algebra 1 Lesson/Unit Plan Name: Linear- Quadratic Systems Rationale/Lesson Abstract: Students will be able to solve a Linear- Quadratic System algebraically and graphically. Timeframe:

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### 1 Shapes of Cubic Functions

MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:

In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### The Product Property of Square Roots states: For any real numbers a and b, where a 0 and b 0, ab = a b.

Chapter 9. Simplify Radical Expressions Any term under a radical sign is called a radical or a square root expression. The number or expression under the the radical sign is called the radicand. The radicand

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are