ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form"

Transcription

1 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 The U-shaped graph of a quadratic function Vertex The lowest or highest point on a parabola Axis of symmetry The vertical line that divides the parabola into mirror images and passes through the vertex Minimum and maximum value For y= ax 2 + bx+ c, the vertex's y-coordinate is the minimum value of the function if a 0 and its maximum value if a 0. PARENT FUNCTION FOR QUADRATIC FUNCTIONS The parent function for the family of all quadratic functions is f(x) = x 2. The graph is shown below The lowest or highest point on a parabola is the vertex for f(x) = x2 is (0,0) The axis of symmetry divides the parabola into mirror images and passes through the vertex For f(x) = ax 2, and for any quadratic function g(x)= ax 2 + bx + c where b = 0, the vertex lies on the y-axis and the axis of symmetry is x = _0_.

2 Example 1 Graph a function of the form y = ax 2 + c Graph y = 2x Compare the graph with the graph of y = x 2. PROPERTIES OF THE GRAPH OF y = ax 2 + bx + c Characteristics of the graph of y = ax 2 + bx + c: The graph opens up if a _ _ 0 and opens down if a _ _ 0. The graph is narrower than the graph of y = x 2 if a _ _ 1 and wider if a _ _ 1. b b The axis of symmetry is x = and the vertex has x-coordinate. 2 a 2 a The y-intercept is _c_. So, the point (0, _c_) is on the parabola. Example 2 Graph a function of the form y = ax 2 + bx + c Graph y = x 2 + 4x 3.

3 MINIMUM AND MAXIMUM VALUES Words For y = ax 2 + bx + c, the vertex's y-coordinate is the minimum value of the function if a _ _ 0 and the maximum value if a _ _ 0. Example 3 Find the minimum or maximum value Tell whether the function y = 3x x 6 has a minimum value or a maximum value. Then find the minimum or maximum value. Checkpoint Complete the following exercises. Graph the function. Label the vertex and axis of symmetry 1. y = x 2 2. y = x 2 4x Find the minimum value of y = 2x 2 6x + 6

4 4.2 Graph Quadratic Functions in Vertex or Intercept Form Goal Graph quadratic functions in vertex form or intercept form. VOCABULARY Vertex form A quadratic function written in the form y = a(x h) 2 + k Intercept form A quadratic function written in the form y = a(x p)(x q) GRAPH OF VERTEX FORM y = a(x h) 2 + k The graph of y = a(x h) 2 + k is the parabola y = ax 2 translated _horizontally_ h units and _vertically_ k units. The vertex is (_h_, _k_ ). The axis of symmetry is x = _h_. The graph opens up if a _ _ 0 and down if a _ _ 0. Example 1 Graph a quadratic function in vertex form and find the maximum or minimum value. 1 Graph y = (x + l) GRAPH OF INTERCEPT FORM y = a(x p)(x q): Characteristics of the graph y = a(x p)(x q): The x-intercepts are _p_ and _q_. The axis of symmetry is halfway between ( _p, 0) and ( _q_, 0). It has equation x = p q 2 The graph opens up if a _>_ 0 and opens down if a _<_ 0.

5 Example 2 Graph a quadratic function in intercept form and find the maximum or minimum value. Graph y = 2(x 1)(x 5). Checkpoint Complete the following exercises. 1. Graph the function. Label the vertex and the axis of symmetry and find the maximum or minimum value. y = (x 3) Graph the function. Label the vertex, axis of symmetry, and the x-intercepts and find the maximum or minimum value. y = (x 4)(x + 2)

6 FOIL METHOD Words To multiply two expressions that each contain two terms, add the products of the _First_ terms, the _Outer_ terms, the _Inner_ terms, and the _Last_ terms. Example F O I L (x + 4)(x + 7) = x 2 + 7x + 4x + 28 = x x + 28 Example 3 Change from intercept form to standard form Write y = 3(x + 2)(x 5) in standard form. Example 4 Change from vertex form to standard form Write f(x) = 5(x + 2) in standard form. Checkpoint Write the quadratic function in standard form. 3. y = 4(x 3) y = 3(x 7)(x + 6)

7 4.3 Solve x 2 bx c 0 by Factoring Goal Solve quadratic equations. VOCABULARY Monomial An expression that is either a number, a variable, or the product of a number and one or more variables Binomial The sum of two monomials Trinomial The sum of three monomials Quadratic equation An equation in one variable that can be written in the form ax 2 + bx + c = 0 where a 0 Root of an equation A solution of a quadratic function Zero of a function The numbers p and q of a function in intercept form are also called the zeros of the function. FACTOR 2 x bx c STEP 1: STEP 2: c Example 1 b Find two numbers whose product is c r r c and whose sum is b. r r b r c r 1 2 b Write in factored form. ( x r )( x r ) 1 2

8 Factor trinomials of the form x 2 bx c Factor the expression x 2 7x 8. SPECIAL FACTORING PATTERNS Pattern Name Difference of Two Squares Perfect Square Trinomial Perfect Square Trinomial a 2 b 2 = ( a + b )( a b ) x 2 4 = (x + 2)(x 2) a 2 + 2ab + b 2 = ( a + b ) 2 x 2 + 6x + 9 = (x + 3) 2 a 2 2ab + b 2 = ( a b ) 2 x 2 4x + 4 = (x 2) 2 Example 2 Factor with special patterns Checkpoint Factor the expression. If it cannot be factored, say so. 1. x 2 + 7x x 2 81

9 ZERO PRODUCT PROPERTY Words If the _product_ of two expressions is zero, then _one_ or _both_ of the expressions equals zero. Algebra If A and B are expressions and AB = _0_, then A = _0_ or B = _0_. Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = _ 5_ or x = _ 2_. Example 3 Find the roots of an equation Find the roots of the equation x 2 2x 15 = 0. Example 4 Find the zeros of a quadratic function Find the zeros of the function y = x 2 5x 6 by rewriting the function in intercept form. Checkpoint Complete the following exercises. 3. Find the roots of the equation x 2 3x + 2 = Find the zeros of the function y = x 2 + 3x 40 by rewriting the function in intercept form.

10 4.4 Solve ax 2 + bx + c = 0 by Factoring Goal Use factoring to solve equations of the form ax 2 + bx + c = 0. Step1: Step 2: ac Find two numbers whose product is c r r c 1 2 and whose sum is b. r r b 1 2 b r ac r 1 2 a a b Write in factored form. ( ax r )( ax r ) 1 2 Example 1 Factor ax 2 + bx + c where c > 0 Factor 2x 2 + 9x + 7. Example 2 Factor ax 2 + bx + c where c < 0 Factor 3x 2 x 2.

11 Checkpoint Factor the expression. If it cannot be factored, say so. 1. 3x 2 + 7x x 2 13x + 6 Example 3 Factor with special patterns Factor the expression. a. 16x 2 36 b. 9y y + 49 c. 25t 2 110t Checkpoint Factor the expression y 2 40y x 2 81

12 Example 4 Factor out monomials first Factor the expression. a. 4x 2 4 b. 3y 2 18y c. 4m 2 10m + 24 d. 5z 2 25z + 40 Checkpoint Factor the expression x p p 63 Example 5 Solve quadratic equations a. 2x 2 x 21 = 0

13 b. 4r 2 18r + 24 = 6r 12 Checkpoint Solve the equation. 7. 2x 2 + 4x 30 = 0 8. z z + 12 = 5z 4

14 4.5 Solve Quadratic Equations by Finding Square Roots Goal Solve quadratic equations by finding square roots. VOCABULARY Square root A number r is a square root of a number s if r 2 = s. Radical An expression of the form s where s is a number or expression Radicand The number s beneath the radical sign Rationalizing the denominator The process of eliminating a radical from the denominator of a fraction Conjugates The expressions a + b and a b, used to rationalize the denominator, whose product is always a rational number PROPERTIES OF SQUARE ROOTS (a > 0, b> 0) Product Property Example ab = a b Quotient Property a a 2 2 = 2 b b Example 1 Use properties of square roots a b. 9 64

15 Your Notes Example 2 Rationalize denominators of fractions Simplify (a) 7 3 and (b) Checkpoint Simplify the expression Example 3 Solve a quadratic equation 1 Solve (y 6) 2 = 8. 4

16 Example 4 Model a dropped object with a quadratic function Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk? h = 16t 2 + h 0 Checkpoint Complete the following exercises. 3. Solve the equation 2x 2 16 = In Example 4, suppose that the water balloon is dropped from a height of 27 feet. How long does it take for the balloon to hit the sidewalk?

17 4.6 Perform Operations with Complex Numbers Goal Perform operations with complex numbers. VOCABULARY Imaginary unit i The imaginary unit i is defined as i 1. Complex number A number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part. Imaginary number A complex number a + bi where b 0 Complex conjugates Two complex numbers of the form a + bi and a bi Complex plane A coordinate plane where each point (a, b) represents a complex number a + bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Absolute value of a complex number The absolute value of a complex number z = a + bi, denoted z, is a nonnegative real number defined as z 2 b 2 a. THE SQUARE ROOT OF A NEGATIVE NUMBER Property 1. If r is a positive real r number, i r. then Example 3 i 3 2. By Property (1), it follows that r 2 2 i r. i 3 2 i 3

18 Example 1 Solve a quadratic equation 2x = 35 Checkpoint Solve the equation. 1. 3x = 23 SUMS AND DIFFERENCES OF COMPLEX NUMBERS To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Difference of complex numbers: (a + bi) (c + di) = (a c) + (b d)i

19 Example 2 Add and subtract complex numbers Write as a complex number in standard form. a. (6 + 3i) (4 /) b. (2 + 5i) + (7 2i) Example 3 Multiply complex numbers Write the expression (2 + i)( 5 + 2i) as a complex number in standard form.. Example 4 Divide complex numbers Write the quotient 6 4 i in standard form. 2 i Checkpoint Write the expression as a complex number in standard form. 2. (12 2i) (16 + 3i) 3. 4i(9 + 5i)

20 i 3 i 5. (4+4i) + ( 6+3i) Example 5 Plot complex numbers Plot the complex numbers in the same complex plane. a i b. 5 4i ABSOLUTE VALUE OF A COMPLEX NUMBER The absolute value of a complex number z = a + bi, denoted z, is a _nonnegative_ real number defined as z = 2 2 a b. This is the distance of z from the _origin in the complex plane. Example 6 Find absolute values of complex numbers Find the absolute value of (a) 6 8i and (b) 6i.

21 Checkpoint Plot the complex numbers in the same complex plane. Then find the absolute value. 6. 4i i 8. 3+i

22 4.7 Complete the Square Goal Solve quadratic equations by completing the square. VOCABULARY Completing the square The process that allows you to write an expression of the form x 2 + bx as the square of a binomial COMPLETING THE SQUARE Words To complete the square for the expression x 2 + bx, add b. 2 Algebra x 2 b + bx + = 2 2 b b x x 2 2 = 2 x b 2 2 Example 1 Solving equations by square rooting. 2 x 6x 9 1 Checkpoint Solve the equation by finding square roots x 10x 25 4

23 Example 2 Make a perfect square trinomial Find the value of c that makes x 2 12x c a perfect square trinomial. Then write the expression as the square of a binomial. Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 2. x 2 24x + c 3. x x +c

24 Example 3 Solve ax 2 bx c = 0 when a = 1 Solve x 2 10x 13 0 by completing the square. Checkpoint Solve the equations by completing the square. 4. x 2 8x + 7 = 0 Example 4 Write a quadratic function in vertex form Write y = x 2 14x 44 in vertex form. Then identify the vertex.

25 Checkpoint Write the quadratic function in vertex form. Then identify the vertex. 5. y = x 2 12x y = x x +53 Example 5 (Day 2) Solve ax 2 bx c 0 when a 1 Solve 3x 2 12x 27 = 0 by completing the square. Checkpoint Solve the equations by completing the square. 5. 2x 2 20x + 24 = 0

26 4.8 Use the Quadratic Formula and the Discriminant Goal Solve quadratic equations using the quadratic formula. VOCABULARY Quadratic formula The formula that gives the solutions to any quadratic equation Discriminant The expression b 2 4ac under the radical sign of the quadratic formula THE QUADRATIC FORMULA Let a, b, and c be real numbers such that a 0. The solutions of the quadratic equation ax 2 + bx + c are: x = b 2 b 4 2 a ac Example 1 Solve an equation with two real solutions Solve x 2 7x 6. Example 2 Solve an equation with one real solution Solve 2x 2 8x 8 = 0.

27 Checkpoint Use the quadratic formula to solve the equation. 1. 2x x = x 2 13x = 7x x 2 6x + 6 = 0 4. x 2 3x + 3 = 0 USING THE DISCRIMINANT OF ax 2 + bx + c = 0 When b 2 4ac > 0, the equation has _two real solutions_. The graph has _two_x-intercepts. When b 2 4ac = 0, the equation has _one real solution_. The graph has _one_x-intercept. When b 2 4ac < 0, the equation has _two imaginary solutions_. The graph has _no_x-intercepts.

28 Example 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x 2 + 6x + 5 = 0 b. x 2 + 6x + 9 = 0 c. x 2 + 6x + 13 = 0 Checkpoint Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 5. x 2 8x + 17 = 0 6. x 2 + 4x + 3 = 0 7. x 2 + 2x 1 = 0 8. x 2 + 6x + 4 = 0

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

1.3 Algebraic Expressions

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,

More information

Warm-Up Oct. 22. Daily Agenda:

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

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

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

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

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

Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

More information

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

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

More information

Algebra 1 Course Title

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

More information

Review of Intermediate Algebra Content

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

More information

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

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.

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

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 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

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

Higher Education Math Placement

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

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS 4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

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

Lesson 9.1 Solving Quadratic Equations

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

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

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.

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

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

MATH 21. College Algebra 1 Lecture Notes

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

More information

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) 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

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

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

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:

More information

Answer Key for California State Standards: Algebra I

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.

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

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

Mathematics Placement

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.

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

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

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

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,

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

Zeros of Polynomial Functions

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

More information

5-3 Polynomial Functions. not in one variable because there are two variables, x. and y

5-3 Polynomial Functions. not in one variable because there are two variables, x. and y y. 5-3 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

Factoring Polynomials

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

More information

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

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

Graphing Quadratic Functions

Graphing Quadratic Functions Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

2.5 Transformations of Functions

2.5 Transformations of Functions 2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

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.

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Solving Quadratic Equations

Solving Quadratic Equations 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

More information

Polynomials and Quadratics

Polynomials and Quadratics Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

More information

Manhattan Center for Science and Math High School Mathematics Department Curriculum

Manhattan Center for Science and Math High School Mathematics Department Curriculum Content/Discipline Algebra 1 Semester 2: Marking Period 1 - Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Algebra 2 Chapter 5 Practice Test (Review)

Algebra 2 Chapter 5 Practice Test (Review) Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or

More information

FACTORING QUADRATICS 8.1.1 through 8.1.4

FACTORING QUADRATICS 8.1.1 through 8.1.4 Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

More information

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

More information

ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

More information

DRAFT. Algebra 1 EOC Item Specifications

DRAFT. Algebra 1 EOC Item Specifications DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

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

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

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Algebra 1. Curriculum Map

Algebra 1. Curriculum Map Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring

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

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

More information

Florida Math for College Readiness

Florida Math for College Readiness Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

More information

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Multiplying and Dividing Radicals

Multiplying and Dividing Radicals 9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated

More information

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

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

More information

POLYNOMIAL FUNCTIONS

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

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

Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

More information

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

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

More information

Algebra 2: Q1 & Q2 Review

Algebra 2: Q1 & Q2 Review Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

More information

Polynomial Operations and Factoring

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.

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

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.

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

More information

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

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

More information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

is the degree of the polynomial and is the leading coefficient.

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

More information

Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

More information

Factoring Polynomials and Solving Quadratic Equations

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

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

A. Factoring out the Greatest Common Factor.

A. Factoring out the Greatest Common Factor. DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

More information

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

More information

Tool 1. Greatest Common Factor (GCF)

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

More information

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

More information

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

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

More information

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

More information

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

More information

Unit 7: Radical Functions & Rational Exponents

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

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

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

More information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

Factoring Trinomials: The ac Method

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

More information

Mathematics Curriculum

Mathematics Curriculum Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,

More information

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

More information

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

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.

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

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary) Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

More information