UNIT 2: FACTOR QUADRATIC EXPRESSIONS. By the end of this unit, I will be able to:
|
|
- Beverley Payne
- 6 years ago
- Views:
Transcription
1 UNIT 2: FACTOR QUADRATIC EXPRESSIONS UNIT 2 By the end of this unit, I will be able to: o Represent situations using quadratic expressions in one variable o Expand and simplify quadratic expressions in one variable o Factor quadratic expressions in one variable using a variety of methods o Solve quadratic equations by selecting and applying an appropriate factoring method o Determine and describe the connection between the factors of a quadratic equation and the x-intercepts of the graph o Explain any restrictions on the domain and range of a quadratic function in contexts arising from real-world applications o Express the equation of a quadratic function in standard form Name: 1
2 2.1 EQUATION FORMS Success Criteria: I can - Analyze and evaluate different forms of the quadratic function Warm up: Find f(2) for the following functions: f(x) = 2x x + 10 f(x) = 2(x + 3) 2 8 f(x) = 2(x + 1)(x + 5) What do you notice? A quadratic function can be expressed in three different forms: Standard Form - f(x) = ax 2 + bx + c Vertex Form - f(x) = a(x h) 2 + k Factored Form - f(x) = a(x r)(x s) What information can we get from each form by inspection: Properties Direction of Opening Standard Form f(x) = ax 2 + bx + c Vertex Form f(x) = a(x h) 2 + k Factored Form f(x) = a(x r)(x s) Maximum or Minimum Vertex Axis of Symmetry Max/Min Value y-intercept x-intercept(s) 2
3 For the quadratic functions below, identify the following key features. EX 1 Standard Form: f(x) = 2x 2 + x 3 Direction of Opening: Maximum or Minimum: y-intercept: EX 2 Vertex Form: f x = 2 x Direction of Opening: Maximum or Minimum: Vertex: Axis of Symmetry: Max/Min Value: y-intercept: EX 3 If we are given a function in factored form, we are able to find more key features - Factored form: f x = (x 1)(x 5) Direction of Opening: Maximum or Minimum: x-intercepts: Axis of Symmetry: Vertex: Max/Min Value: y-intercept: 3
4 Practice Find the key features for the functions below: EX 1 Standard Form: f x = 5x 2 + 8x + 74 Direction of Opening: Maximum or Minimum: y-intercept: EX 2 Vertex Form: f x = 3 x Direction of Opening: Maximum or Minimum: Vertex: Axis of Symmetry: Max/Min Value: y-intercept: EX 3 Factored form: f x = (x + 6)(x 4) Direction of Opening: Maximum or Minimum: x-intercepts: Axis of Symmetry: Vertex: Max/Min Value: y-intercept: Homework: 4
5 2.2 GRAPHING FROM FACTORED FORM Success Criteria: I can Sketch the graph of a quadratic function in factored form Expand and simplify quadratic expressions in one variable Express the equation of a quadratic function in standard form Graphing from Factored Form A quadratic function can be graphed from factored form, y = a(x r)(x s) by: 1. Plotting the x-intercept(s) (x =, x = ) 2. Averaging the x-intercept(s) to find the axis of symmetry which occurs midway between the x-intercept(s) (x = h) 3. Substituting the x-value from (2) into the equation to find the max/min value (y = k) 4. Finding the y-intercept by letting x=0 and plotting it 5. Plotting the vertex (h, k) and connecting the points EX 1: Graph each quadratic function using the x-intercepts, vertex, and y-intercept. a) y = ½(x 3)(x + 5) b) f(x) = -x(x + 6) 5
6 Writing Equations in Standard Form A quadratic function in vertex or factored form can be converted into standard form by expanding and simplifying. This can be useful, as we know the c in standard form is the. f(x) = a(x h) 2 + k f(x) = a(x r)(x s) f(x) = ax 2 + bx + c EX 2: Write each quadratic function in standard form. Then, state the key features. a) f(x) = 4x(x 2) b) y = 2(x + 3) 2 9 c) f(x) = -(x + 2)(x 4) y-intercept: y-intercept: y-intercept: x-intercepts: x-intercepts axis of symmetry: axis of symmetry: axis of symmetry: vertex: vertex: vertex: Minimum or maximum? Minimum or maximum? Minimum or maximum? * Now we know how to convert from factored & vertex to standard, but how do we convert from standard to factored? 6
7 2.3 FACTORING POLYNOMIALS Success Criteria: I can Determine the greatest common factor for general expressions 7
8 2.4 FACTORING QUADRATIC FUNCTIONS Day 1 Success Criteria: I can Factor quadratic expressions in one variable using a variety of methods To factor a polynomial means to write it as a product. The opposite of factoring is expanding. factoring expanding 4(x + y) 4x + 4y There are several methods of factoring: A) COMMON FACTORING *Always try 1 st - Find the GCF for all terms (biggest # and variable that divides into all the terms) - Remove the GCF (by dividing each te X 1: Factor by common factoring: a) 4x+ 16x 2 b) 12m mn+ 3m c) 10x 3 25x 2 d) 6x 2 y 18xy + 9y 8
9 coefficient of 1 B) SIMPLE TRINOMIAL FACTORING, x 2 + bx + c - Find two numbers that add to b and multiply to c - Write as (x + ) (x + ) EX 2: Factor a) x 2 + 5x + 6 b) x 2 x 20 c) 2x 2 22x + 36 C) DIFFERENCE OF SQUARES, a 2 b 2 - Identify perfect squares o A perfect square is a number found by squaring another number. o E.g. 1, 4, 9, 25, 36, 49, 64, 81, 100, - Write as (a b)(a + b) EX 3: Factor a) x 2 4 b) 9m
10 c) y d) 3p 2 75 Practice: Factor the following: 1) x 2 + 8x + 7 2) 3n 2 30n ) g ) 9y 2-16 Homework: 10
11 2.5 FACTORING QUADRATIC FUNCTIONS Day 2 Success Criteria: I can Factor quadratic expressions in one variable using a variety of methods D) FACTORING TRICKY TRINOMIALS, ax 2 + bx + c (a 1) Replace middle term with two terms that add to b and multiply to ac; Group and common factor each pair of terms Remove common binomial factor EX 4: Factor a) 3x x + 8 b) 2m 2 + m 15 c) 4y 2 7y + 3 d) 6n 2 16y
12 Recall: A perfect square is a number found by squaring another number. E.g. 1, 4, 9, 25, 36, 49, 64, 81, 100, E) PERFECT SQUARE TRINOMIALS, a 2 ± 2ab + b 2 - Identify perfect squares - Write as (a ± b) 2 Square of a binomial EX 6: Factor Twice product of square roots of first and last terms a) y y + 25 b) x 2 8xy + 16y 2 c) 4k k + 9 d) y 2 8y - 16 e) 16t 2 40t + 25 f) 49x xy + 4y 2 Homework: 12
13 Factoring Practice When choosing what factoring strategy to use, always start with GCF and work from there. Remember some expressions will be non factorable, meaning you cannot factor them m 3 + 5m 2. n 3 10n a b 3 2b x m 3 + 2m b 3 40b x x y 3 15y x B + 6x 13
14 Factoring: More Practice a) 7x b) 5x x 40 c) 12x 2 20x d) 4x 2 9x - 9 e) 2x 2 18x + 36 f) -9x 2 15x + 6 g) 3x 2 6x + 12 h) 25x 2 9 i) 6x x 14
15 2.6 SOLVING QUADRATIC EQUATIONS BY FACTORING Success Criteria: I can Solve quadratic equations by selecting and applying an appropriate factoring method Determine and describe the connection between the factors of a quadratic equation and the x-intercepts of the graph Explain any restrictions on the domain and range of a quadratic function in contexts arising from real-world applications To solve a quadratic function means to find the,, or. To find these, we: For example, y = x 2 4 Find the x-intercepts: To prove that these are the zero(s) or root(s) of this equation, sub the x-intercepts into the original equation. If y=0, these are the zero(s) or root(s) of this equation. EX 1: Solve by factoring to find the x-intercept(s). a) y = 2x 2 14x b) f(x) =-3x 2 12x + 36 c) y = 2x 2 5x 12 d) f(x) = x 2 10x
16 Solving quadratic equations is useful in application questions. EX 2: A stone is tossed from a bridge. Its height above the water is modeled by the equation, h(t) = - 5t 2 + 5t + 60, where h(t) is the height of the stone above the water, in metres, and t is the time, in seconds. Find the zero(s) of the function and explain their significance. EX 3: A relief package is released from a helicopter at 1600 feet. The height of the package can be modeled by the equation h(t) = -16t , where h is the height of the package in feet and t is the time in seconds. The pilot wants to know how long it will take for the package to hit the ground. 16
17 17
18 Practice: Sama is kicking a soccer ball on the top Emily Carr to his cousin who is on the soccer field. The height of the ball as a function of time is given by h(t)= -5t 2 +5t+10, where t is the time, in seconds, and h(t) is the height of the stone above the ground, in metres, at time t. a) Find the zeros of the function and explain their significance. b) Use the x-intercepts to find the vertex. What does it represent in this scenario? c) What is the height of the school? How do you know? d) Sketch a graph of the parabola: Homework: 18
7.1 Solving Quadratic Equations by Graphing
Math 2201 Date: 7.1 Solving Quadratic Equations by Graphing In Mathematics 1201, students factored difference of squares, perfect square trinomials and polynomials of the form x 2 + bx + c and ax 2 + bx
More informationSM3 Lesson 2-3 (Intercept Form Quadratic Equation)
SM3 Lesson 2-3 (Intercept Form Quadratic Equation) Factor the following quadratic expressions: x 2 + 11x + 30 x 2 10x 24 x 2 8x + 15 Standard Form Quadratic Equation (x + 5)(x + 6) (x 12)(x + 2) (x 5)(x
More informationRoots of Quadratic Functions
LESSON 12 Roots of Quadratic Functions LEARNING OBJECTIVES Today I am: sketching parabolas with limited information. So that I can: identify the strengths of each form of a quadratic equation. I ll know
More informationSect 4.5 Inequalities Involving Quadratic Function
71 Sect 4. Inequalities Involving Quadratic Function Objective #0: Solving Inequalities using a graph Use the graph to the right to find the following: Ex. 1 a) Find the intervals where f(x) > 0. b) Find
More informationPART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to find key features. The table of values is shown below:
Math (L-3a) Learning Targets: I can find the vertex from intercept solutions calculated by quadratic formula. PART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to
More information2.3 BUILDING THE PERFECT SQUARE
16 2.3 BUILDING THE PERFECT SQUARE A Develop Understanding Task Quadratic)Quilts Optimahasaquiltshopwhereshesellsmanycolorfulquiltblocksforpeoplewhowant tomaketheirownquilts.shehasquiltdesignsthataremadesothattheycanbesized
More informationStudent Exploration: Quadratics in Factored Form
Name: Date: Student Exploration: Quadratics in Factored Form Vocabulary: factored form of a quadratic function, linear factor, parabola, polynomial, quadratic function, root of an equation, vertex of a
More informationDetermine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 A) Even B) Odd C) Neither
Assignment 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 1) A)
More informationVOCABULARY WORDS. quadratic equation root(s) of an equation zero(s) of a function extraneous root quadratic formula discriminant
VOCABULARY WORDS quadratic equation root(s) of an equation zero(s) of a function extraneous root quadratic formula discriminant 1. Each water fountain jet creates a parabolic stream of water. You can represent
More informationFactored Form When a = 1
Lesson 4 Hart Interactive Algebra Lesson 4: Factored Form When a = Opening Activity Graph Exchange Your group will need: one quadratic graph. A. For your given graph, circle the graph number on the table
More informationSection 6.3: Factored Form of a Quadratic Function
Section 6.3: Factored Form of a Quadratic Function make the connection between the factored form of a quadratic and the x-intercepts of the graph Forms of a Quadratic Function (i) Standard Form (ii) Factored
More information6.1.2: Graphing Quadratic Equations
6.1.: Graphing Quadratic Equations 1. Obtain a pair of equations from your teacher.. Press the Zoom button and press 6 (for ZStandard) to set the window to make the max and min on both axes go from 10
More informationSECONDARY 2H ~ UNIT 5 (Into to Quadratics)
SECONDARY 2H ~ UNIT 5 (Into to Quadratics) Assignments from your Student Workbook are labeled WB Those from your hardbound Student Resource Book are labeled RB. Do all work from the Student Resource Book
More informationFor Questions 1-15, NO CALCULATOR!
For Questions 1-15, NO CALCULATOR! 1. Identify the y-intercept: Identify the vertex: 2. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation Identify
More informationLesson 16. Opening Exploration A Special Case
Opening Exploration A Special Case 1. Consuela ran across the quadratic equation y = 4x 2 16 and wondered how it could be factored. She rewrote it as y = 4x 2 + 0x 16. A. Use one of the methods you ve
More informationChapter 8. Lesson a. (2x+3)(x+2) b. (2x+1)(3x+2) c. no solution d. (2x+y)(y+3) ; Conclusion. Not every expression can be factored.
Chapter 8 Lesson 8.1.1 8-1. a. (x+4)(y+x+) = xy+x +6x+4y+8 b. 18x +9x 8-. a. (x+3)(x+) b. (x+1)(3x+) c. no solution d. (x+y)(y+3) ; Conclusion. Not every expression can be factored. 8-3. a. (3x+1)(x+5)=6x
More informationSECONDARY 2H ~ UNIT 5 (Intro to Quadratics)
SECONDARY 2H ~ UNIT 5 (Intro to Quadratics) Assignments from your Student Workbook are labeled WB Those from your hardbound Student Resource Book are labeled RB. Do all work from the Student Resource Book
More informationMath 10C Chapter 3 Factors and Products Review Notes
Math 10C Chapter Factors and Products Review Notes Prime Factorization Prime Numbers: Numbers that can only be divided by themselves and 1. The first few prime numbers:,, 5,, 11, 1, 1, 19,, 9. Prime Factorization:
More informationLength of a Side (m)
Quadratics Day 1 The graph shows length and area data for rectangles with a fixed perimeter. Area (m ) 450 400 350 300 50 00 150 100 50 5 10 15 0 5 30 35 40 Length of a Side (m) 1. Describe the shape of
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Squares and More Using Patterns to Generate Algebraic Functions Vocabulary Match each word with its corresponding definition. 1. linear function a.
More informationLesson 3.4 Completing the Square
Lesson 3. Completing the Square Activity 1 Squares of Binomials 1. a. Write a formula for the square of a binomial: ÐB :Ñ œ Notice that the constant term of the trinomial is coefficient of the linear term
More information5.1N Key Features of Rational Functions
5.1N Key Features of Rational Functions A. Vocabulary Review Domain: Range: x-intercept: y-intercept: Increasing: Decreasing: Constant: Positive: Negative: Maximum: Minimum: Symmetry: End Behavior/Limits:
More information2. Questions on Classwork and Homework form yesterday. 4. Completing the square to solve quadratic
1. Warm -up word problem - 2. Questions on Classwork and Homework form yesterday 3. Number Sense. 4. Completing the square to solve quadratic equations 1 2 3 Apr 12 12:35 PM 4 Apr 13 2:12 PM 5 6 7 factors
More informationYear 11 Graphing Notes
Year 11 Graphing Notes Terminology It is very important that students understand, and always use, the correct terms. Indeed, not understanding or using the correct terms is one of the main reasons students
More informationA slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
More informationUp and Down or Down and Up
Lesson.1 Assignment Name Date Up and Down or Down and Up Exploring Quadratic Functions 1. The citizens of Herrington County are wild about their dogs. They have an existing dog park for dogs to play, but
More informationPre Calc. Conics.
1 Pre Calc Conics 2015 03 24 www.njctl.org 2 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections Parabolas Circles Ellipses Hyperbolas
More informationMATH 150 Pre-Calculus
MATH 150 Pre-Calculus Fall, 2014, WEEK 5 JoungDong Kim Week 5: 3B, 3C Chapter 3B. Graphs of Equations Draw the graph x+y = 6. Then every point on the graph satisfies the equation x+y = 6. Note. The graph
More informationIntermediate Mathematics League of Eastern Massachusetts
Meet #5 April 2003 Intermediate Mathematics League of Eastern Massachusetts www.imlem.org Meet #5 April 2003 Category 1 Mystery You may use a calculator 1. In his book In an Average Lifetime, author Tom
More informationPREREQUISITE/PRE-CALCULUS REVIEW
PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which
More informationOutcome 9 Review Foundations and Pre-Calculus 10
Outcome 9 Review Foundations and Pre-Calculus 10 Level 2 Example: Writing an equation in slope intercept form Slope-Intercept Form: y = mx + b m = slope b = y-intercept Ex : Write the equation of a line
More informationChapter 9 Linear equations/graphing. 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane
Chapter 9 Linear equations/graphing 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane Rectangular Coordinate System Quadrant II (-,+) y-axis Quadrant
More informationSect Linear Equations in Two Variables
99 Concept # Sect. - Linear Equations in Two Variables Solutions to Linear Equations in Two Variables In this chapter, we will examine linear equations involving two variables. Such equations have an infinite
More informationore C ommon Core Edition APlgebra Algebra 1 ESTS RACTICE PRACTICE TESTS Topical Review Book Company Topical Review Book Company
C ommon Core ommon Edition C ore Edition Algebra 1 APlgebra 1 T RACTICE ESTS Answer Keys PRACTICE TESTS Topical Review Book Company Topical Review Book Company TEST 1 Part I 1. 3 5. 2 9. 4 13. 1 17. 4
More informationPre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas
Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 160 Review of Midpoint and Distance Formulas Intro to Conic Sections
More informationLesson Objectives. Simplifying Algebraic Expressions with Polynomials Multiplying Monomials and Binomials
UDM11L04BLM/AK_61519 8/11/03 5:15 PM Page 29 Lesson Objectives Find the product of two monomials. Find the product of a monomial and a binomial. Find the product of two binomials using the Distributive
More informationRadical Expressions and Graph (7.1) EXAMPLE #1: EXAMPLE #2: EXAMPLE #3: Find roots of numbers (Objective #1) Figure #1:
Radical Expressions and Graph (7.1) Find roots of numbers EXAMPLE #1: Figure #1: Find principal (positive) roots EXAMPLE #2: Find n th roots of n th powers (Objective #3) EXAMPLE #3: Figure #2: 7.1 Radical
More informationAlgebra 1 B Semester Exam Review
Algebra 1 B 014 MCPS 013 014 Residual: Difference between the observed (actual) value and the predicted (regression) value Slope-Intercept Form of a linear function: f m b Forms of quadratic functions:
More informationAN5_Grade 10 AN5 Factoring concretely when a is not equal to 1.notebook
April 7, 2015 Can we use algebra tiles to show the factors of trinomials when a >1? ax 2 + bx + c Let's begin exploring trinomials with a>1 and b and c both positive integers. TAKE NOTICE: ALWAYS look
More information5.1, 5.2, 5.3 Properites of Exponents last revised 12/28/2010
48 5.1, 5.2, 5.3 Properites of Exponents last revised 12/28/2010 Properites of Exponents 1. *Simplify each of the following: a. b. 2. c. d. 3. e. 4. f. g. 5. h. i. j. Negative exponents are NOT considered
More informationMaxima and Minima. Terminology note: Do not confuse the maximum f(a, b) (a number) with the point (a, b) where the maximum occurs.
10-11-2010 HW: 14.7: 1,5,7,13,29,33,39,51,55 Maxima and Minima In this very important chapter, we describe how to use the tools of calculus to locate the maxima and minima of a function of two variables.
More informationSelected Answers for Core Connections Algebra
Selected Answers for Core Connections Algebra Lesson 8.1.1 8-6. (2x 3)(x + 2y 4) = 2x 2 + 4xy 11x 6y +12 8-7. a: 12x 2 +17x 5 b: 4x 2 28x + 49 8-8. a: t(n) = 500 +1500(n 1) b: t(n) = 30!5 n 1 8-9. a: b:
More informationYou identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas.
You identified, analyzed, and graphed quadratic functions. (Lesson 1 5) Analyze and graph equations of parabolas. Write equations of parabolas. conic section degenerate conic locus parabola focus directrix
More informationDiscussion 8 Solution Thursday, February 10th. Consider the function f(x, y) := y 2 x 2.
Discussion 8 Solution Thursday, February 10th. 1. Consider the function f(x, y) := y 2 x 2. (a) This function is a mapping from R n to R m. Determine the values of n and m. The value of n is 2 corresponding
More informationAnthony Chan. September, Georgia Adult Education Conference
Anthony Chan September, 2018 1 2018 Georgia Adult Education Conference Attendees will be able to: Make difficult math concepts simple and help their students discover math principles on their own. This
More informationE. Slope-Intercept Form and Direct Variation (pp )
and Direct Variation (pp. 32 35) For any two points, there is one and only one line that contains both points. This fact can help you graph a linear equation. Many times, it will be convenient to use the
More information3.1 Solving Systems by Graphing. In consistent systems, Independent systems consist of. Three Cases: A. consistent and independent
3.1 Solving Systems by Graphing In consistent systems, 2. y Independent systems consist of Three Cases: x A. consistent and independent B. inconsistent and independent 3. y C. consistent and dependent
More informationPre-AP Algebra 2 Unit 8 - Lesson 2 Graphing rational functions by plugging in numbers; feature analysis
Pre-AP Algebra 2 Unit 8 - Lesson 2 Graphing rational functions by plugging in numbers; feature analysis Objectives: Students will be able to: Analyze the features of a rational function: determine domain,
More informationMTH 1825 Sample Exam 4 Fall 2014
Name (print) Section Signature PID Instructions: Please check to make sure your exam has all 8 pages (including cover) before you begin. Please read the following instructions carefully. 1. DO NOT OPEN
More informationLesson 6.1 Linear Equation Review
Name: Lesson 6.1 Linear Equation Review Vocabulary Equation: a math sentence that contains Linear: makes a straight line (no Variables: quantities represented by (often x and y) Function: equations can
More informationPre-Calc Conics
Slide 1 / 160 Slide 2 / 160 Pre-Calc Conics 2015-03-24 www.njctl.org Slide 3 / 160 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections
More informationThe Picture Tells the Linear Story
The Picture Tells the Linear Story Students investigate the relationship between constants and coefficients in a linear equation and the resulting slopes and y-intercepts on the graphs. This activity also
More informationHonors Algebra 2 Assignment Sheet - Chapter 1
Assignment Sheet - Chapter 1 #01: Read the text and the examples in your book for the following sections: 1.1, 1., and 1.4. Be sure you read and understand the handshake problem. Also make sure you copy
More informationMath 165 Section 3.1 Linear Functions
Math 165 Section 3.1 Linear Functions - complete this page Read the book or the power point presentations for this section. Complete all questions on this page Also complete all questions on page 6 1)
More informationGraphing - Slope-Intercept Form
2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,
More informationMATH Exam 2 Solutions November 16, 2015
MATH 1.54 Exam Solutions November 16, 15 1. Suppose f(x, y) is a differentiable function such that it and its derivatives take on the following values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y)
More informationProlegomena. Chapter Using Interval Notation 1
Chapter 1 Prolegomena 1.1 Using Interval Notation 1 Interval notation is another method for writing domain and range. In set builder notation braces (curly parentheses {} ) and variables are used to express
More informationMath 259 Winter Recitation Handout 6: Limits in Two Dimensions
Math 259 Winter 2009 Recitation Handout 6: its in Two Dimensions As we have discussed in lecture, investigating the behavior of functions with two variables, f(x, y), can be more difficult than functions
More informationThe Chain Rule, Higher Partial Derivatives & Opti- mization
The Chain Rule, Higher Partial Derivatives & Opti- Unit #21 : mization Goals: We will study the chain rule for functions of several variables. We will compute and study the meaning of higher partial derivatives.
More informationLesson 24: Finding x-intercepts Again?
Opening Discussion The quadratic function, y = x 2 6x + 8, can be written as y = (x 2)(x 4) and as y = (x 3) 2 1. Deshi and Ame wanted to find the x-intercepts of this function. Their work is shown below.
More informationM.I. Transformations of Functions
M.I. Transformations of Functions Do Now: A parabola with equation y = (x 3) 2 + 8 is translated. The image of the parabola after the translation has an equation of y = (x + 5) 2 4. Describe the movement.
More informationSection 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions.
Section 8.4: The Equations of Sinusoidal Functions Stop Sine 1 In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation.
More informationReview for Mastery. Identifying Linear Functions
Identifying Linear Functions You can determine if a function is linear by its graph, ordered pairs, or equation. Identify whether the graph represents a linear function. Step 1: Determine whether the graph
More informationTHE DOMAIN AND RANGE OF A FUNCTION Basically, all functions do is convert inputs into outputs.
THE DOMAIN AND RANGE OF A FUNCTION Basically, all functions do is convert inputs into outputs. Exercise #1: Consider the function y = f (x) shown on the graph below. (a) Evaluate each of the following:
More informationANNOUNCEMENTS. GOOD MORNING or GOOD AFTERNOON AGENDA FOR TODAY. Quickly Review Absolute Values Graphing Quadratics. Vertex Form Calculator Activity
ANNOUNCEMENTS GOOD MORNING or GOOD AFTERNOON AGENDA FOR TODAY Quickly Review Absolute Values Graphing Quadratics Vertex Form Calculator Activity M314 Algebra II Section 9-4 and 9-5: Quadratics Presented
More informationExam: Friday 4 th May How to Revise. What to use to revise:
National 5 Mathematics Exam Revision Questions Exam: Friday 4 th May 2018 How to Revise Use this booklet for homework Come to after school revision classes Come to the Easter holiday revision class There
More information5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs
Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2
More informationAlgebra & Trig. 1. , then the slope of the line is given by
Algebra & Trig. 1 1.4 and 1.5 Linear Functions and Slope Slope is a measure of the steepness of a line and is denoted by the letter m. If a nonvertical line passes through two distinct points x, y 1 1
More informationKeystone Exams: Algebra I Assessment Anchors and Eligible Content. Pennsylvania Department of Education
Keystone Exams: Algebra I Assessment Anchors and Pennsylvania Department of Education www.education.state.pa.us 2010 PENNSYLVANIA DEPARTMENT OF EDUCATION General Introduction to the Keystone Exam Assessment
More informationSection 5.2 Graphs of the Sine and Cosine Functions
A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in
More informationStudy Guide and Review - Chapter 3. Find the x-intercept and y-intercept of the graph of each linear function.
Find the x-intercept and y-intercept of the graph of each linear function. 11. The x-intercept is the point at which the y-coordinate is 0, or the line crosses the x-axis. So, the x-intercept is 8. The
More informationAim #35.1: How do we graph using a table?
A) Take out last night's homework Worksheet - Aim 34.2 B) Copy down tonight's homework Finish aim 35.1 Aim #35.1: How do we graph using a table? C) Plot the following points... a) (-3, 5) b) (4, -2) c)
More information5.1, 5.2, 5.3 Properites of Exponents last revised 12/4/2010
48 5.1, 5.2, 5.3 Properites of Exponents last revised 12/4/2010 Properites of Exponents 1. *Simplify each of the following: a. b. 2. c. d. 3. e. 4. f. g. 5. h. i. j. Negative exponents are NOT considered
More informationNOTES: SIGNED INTEGERS DAY 1
NOTES: SIGNED INTEGERS DAY 1 MULTIPLYING and DIVIDING: Same Signs (POSITIVE) + + = + positive x positive = positive = + negative x negative = positive Different Signs (NEGATIVE) + = positive x negative
More informationSection 3.5. Equations of Lines
Section 3.5 Equations of Lines Learning objectives Use slope-intercept form to write an equation of a line Use slope-intercept form to graph a linear equation Use the point-slope form to find an equation
More information2.3: The Human Cannonball
2.3: The Human Cannonball Parabola Equations and Graphs As a human cannonball Rosa is shot from a special cannon. She is launched into the air by a spring. Rosa lands in a horizontal net 150 ft. from the
More informationYou analyzed graphs of functions. (Lesson 1-5)
You analyzed graphs of functions. (Lesson 1-5) LEQ: How do we graph transformations of the sine and cosine functions & use sinusoidal functions to solve problems? sinusoid amplitude frequency phase shift
More informationMathematics 205 HWK 19b Solutions Section 16.2 p750. (x 2 y) dy dx. 2x 2 3
Mathematics 5 HWK 9b Solutions Section 6. p75 Problem, 6., p75. Evaluate (x y) dy dx. Solution. (x y) dy dx x ( ) y dy dx [ x x dx ] [ ] y x dx Problem 9, 6., p75. For the region as shown, write f da as
More informationSection 2.3 Task List
Summer 2017 Math 108 Section 2.3 67 Section 2.3 Task List Work through each of the following tasks, carefully filling in the following pages in your notebook. Section 2.3 Function Notation and Applications
More informationActivity 1 A D V A N C E D H O M E W O R K 1
Activity 1 A D V A N C E D H O M E W O R K 1 A D V A N C E D H O M E W O R K 2 Activity 2 Research Required: Recursive Functions Activity 3 A D V A N C E D H O M E W O R K 3 A D V A N C E D H O M E W O
More informationAlgebra 1 Online:
Dear Algebra 2 Students, Within this packet you will find mathematical concepts and skills learned in Algebra 1 that are the foundation from which Algebra 2 is built. These concepts need to be reviewed
More informationSection 7.2 Logarithmic Functions
Math 150 c Lynch 1 of 6 Section 7.2 Logarithmic Functions Definition. Let a be any positive number not equal to 1. The logarithm of x to the base a is y if and only if a y = x. The number y is denoted
More informationDeveloping Algebraic Thinking
Developing Algebraic Thinking DEVELOPING ALGEBRAIC THINKING Algebra is an important branch of mathematics, both historically and presently. algebra has been too often misunderstood and misrepresented as
More information4.4 Slope and Graphs of Linear Equations. Copyright Cengage Learning. All rights reserved.
4.4 Slope and Graphs of Linear Equations Copyright Cengage Learning. All rights reserved. 1 What You Will Learn Determine the slope of a line through two points Write linear equations in slope-intercept
More informationLesson 18: Solving Quadratic Equations
Opening Exercise 1. The area of a rectangle can be represented by the expression xx 2 + 2xx 3. A. If the dimensions of the rectangle are known to be the linear factors of the expression, write each dimension
More information6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line.
6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line. Toolkit: - Rate of change - Simplifying fractions Main Ideas: Definitions Rise: the vertical distance between two
More informationC.2 Equations and Graphs of Conic Sections
0 section C C. Equations and Graphs of Conic Sections In this section, we give an overview of the main properties of the curves called conic sections. Geometrically, these curves can be defined as intersections
More informationTrigonometric Equations
Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric
More informationSolving Equations and Graphing
Solving Equations and Graphing Question 1: How do you solve a linear equation? Answer 1: 1. Remove any parentheses or other grouping symbols (if necessary). 2. If the equation contains a fraction, multiply
More informationS56 (5.1) Logs and Exponentials.notebook October 14, 2016
1. Daily Practice 21.9.2016 Exponential Functions Today we will be learning about exponential functions. A function of the form y = a x is called an exponential function with the base 'a' where a 0. y
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y
More informationGeneral Functions and Graphs
General Functions and Graphs Section 7 Functions Graphs and Symmetry Functions can be represented both as algebraic expressions and as graphs. So far we have concentrated on algebraic operations related
More informationLogs and Exponentials Higher.notebook February 26, Daily Practice
Daily Practice 2.2.2015 Daily Practice 3.2.2015 Today we will be learning about exponential functions and logs. Homework due! Need to know for Unit Test 2: Expressions and Functions Adding and subtracng
More informationAlgebra EOC Practice Test #3
Class: Date: Algebra EOC Practice Test #3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Write the monomial 4x 2 y 3y 3 without the use of negative exponents.
More informationALGEBRA LOGS AND INDICES (NON REAL WORLD)
ALGEBRA LOGS AND INDICES (NON REAL WORLD) Algebra Logs and Indices LCHL New Course 206 Paper Q4 (b) 204S Paper Q2 (b) LCOL New Course 204S Paper Q (a) 204S Paper Q (c) 204S Paper Q (d) 203 Paper Q3 (c)
More informationPolynomials - Special Products
Polynomials - Special Products There are a few shortcuts that we can take when multiplying polynomials. If we can recognize them the shortcuts can help us arrive at the solution much quicker. These shortcuts
More informationSection 1.3. Slope formula: If the coordinates of two points on the line are known then we can use the slope formula to find the slope of the line.
MATH 11009: Linear Functions Section 1.3 Linear Function: A linear function is a function that can be written in the form f(x) = ax + b or y = ax + b where a and b are constants. The graph of a linear
More informationFinal Exam Study Guide High School 2014
Teacher : M. Grant Classes : Algebra I The following are samples of the types of problems that will be on your final exam. They are the same types that are on the EOC test. 1. Write the monomial 4x² y
More informationThis early Greek study was largely concerned with the geometric properties of conics.
4.3. Conics Objectives Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations
More informationIn this section, we find equations for straight lines lying in a coordinate plane.
2.4 Lines Lines In this section, we find equations for straight lines lying in a coordinate plane. The equations will depend on how the line is inclined. So, we begin by discussing the concept of slope.
More information