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Common Core Math 2 Unit 1A Modeling with Quadratics

Common Core Math 2 Unit 2 Modeling with QuadraticsMain ConceptsStudy GuideVocabularyFactoringWord Problems with factoringProperties of parabolasVertex formulaComparing QuadraticsQuadratic formulaDiscriminantNon-linear systemsQuadratic inequalitiesQuadratic applicationsTest reviewAppendix: Using the calculatorHw AnswersPage 4546-4950-5152-5455-583

Common Core Math 2 Unit 1A Modeling with QuadraticsCommon Core StandardsA.SSE.1Interpret expressions that represent a quantity in terms of its context.«a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. Forexample, interpret (1 𝑟)! as the product of P and a factor not depending on P.A.SSE.2Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 –(y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 y2).A.APR.1Understand that polynomials form a system analogous to the integers, namely, they are closedunder the operations of addition, subtraction, and multiplication; add, subtract, and multiplypolynomials.A.APR.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros toconstruct a rough graph of the function defined by the polynomial.A.CED.1Create equations and inequalities in one variable and use them to solve problems. Includeequations arising from linear and quadratic functions, and simple rational and exponentialfunctions.A.REI.4Solve Equations in One Variable b. Solve quadratic equations by inspection (e.g., for x2 49),taking square roots, completing the square, the quadratic formula and factoring, as appropriateto the initial form of the equation. Recognize when the quadratic formula gives complex solutionsand write them as a bi for real numbers a and b.A.REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted inthe coordinate plane, often forming a curve (which could be a line).A.REI.11Explain why the x--‐ coordinates of the points where the graphs of the equations y f(x) and y g(x) intersect are the solutions of the equation f(x) g(x); find the solutions approximately, e.g.,using technology to graph the functions, make tables of values, or find successiveapproximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolutevalue, exponential, and logarithmic functions.F.IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statementsthat use function notation in terms of a context.F.IF.4For a function that models a relationship between two quantities, interpret key features ofgraphs and tables in terms of the quantities, and sketch graphs showing key features given averbal description of the relationship. Key features include: intercepts; intervals where thefunction is increasing, decreasing, positive, or negative; relative maximums and minimums;symmetries; end behavior; and periodicity.F.IF.5Relate the domain of a function to its graph and, where applicable, to the quantitativerelationship it describes.F.IF.8Write a function defined by an expression in different but equivalent forms to reveal and explaindifferent properties of the function. a. Use the process of factoring and completing the square ina quadratic function to show zeros, extreme values, and symmetry of the graph, and interpretthese in terms of a context.F.IF.9Compare properties of two functions each represented in a different way (algebraically,graphically, numerically in tables, or by verbal descriptions).4

Common Core Math 2 Unit 2 Modeling with QuadraticsUnit DescriptionIn this unit, students will continue their exploration of quadratic functions begun in Math I. Students willlearn about the three different solution types for quadratic equations and how to determine what type ofsolution a quadratic equation has by analyzing the equation and the graph of the related function. Studentswill continue solving quadratic equations using inspection, square roots, and factoring. Students will beintroduced to the quadratic formula and the concept of non-real solutions. Students will use the discriminantof the quadratic formula to determine the number of real solutions. Students will comprehend equivalentstructures of quadratics through factoring in order to find the zeros of the graph. Students will continue touse the zeros and symmetry to find the value of the vertex of the graph, but also learn algebraic methods forfinding the vertex. Students will also use quadratic functions and equations to solve real world problems,interpreting values in context.Essential QuestionsBy the end of this unit, I will be able to answer the following questions: Evaluate which representations of a function are most useful for solving problems in differentmathematical and real world settings. Note: This statement does not have the same meaning in questionformat, so we left it the way it is. How are the key features identified, described, and interpreted from different representations ofquadratic functions? How are geometric transformations (translations, reflections, rotations, and dilations) of figures related totransformations of functions?Why is symmetry the key feature of a quadratic function and how is it revealed in the different forms ofthis function (graph, table, equation, and verbal)?I can . . .Operate with polynomials Determine whether an expression is a polynomial. (A-APR.1)Add and subtract polynomials.* (A-APR.1)Multiply up to three linear expressions.**Solve quadratic equations and systems involving quadratic equations Create quadratic equations in one variable and use them to solve problems.Solve quadratic equations by inspection (e.g., 𝑥 2 49), taking square roots, the quadratic formula,and factoring.o Justify each step in solving a quadratic equation by factoring.o Use the discriminant to determine the number of real solutions of a quadratic equation andwhen a quadratic equation has non-real solutions.o Choose an appropriate solution method based on the initial form of the quadratic equation.o Construct a viable argument to justify a solution method.Solve a simple system consisting of a linear equation and a quadratic equation in two variablesalgebraically and graphically.Explain why the x-coordinates of the points where the graphs of the equations y f(x) and y g(x)intersect are the solutions of the equation f(x) g(x). Find the solutions approximately, e.g., usingtechnology to graph the functions, make tables of values, or find successive approximations. (Note:Limit f(x) and g(x) to linear or quadratic functions.)Represent constraints by systems of linear-quadratic equations and/or inequalities based on amodeling context.Interpret solutions of linear-quadratic systems as viable or non-viable options in a modelingcontext.5

Common Core Math 2 Unit 1A Modeling with QuadraticsAnalyze quadratic functions using different representations Given a function in verbal, algebraic, table, or graph form, determine whether it is a quadraticfunction. (F-BF.1)Recognize equivalent forms of quadratic functions. For example, standard form 𝑎𝑥 2 𝑏𝑥 𝑐 0,and factored form 𝑦 𝑎(𝑥 𝑟1 )(𝑥 𝑟2 ). (A-SSE.2)Identify the coefficients and constants of a quadratic function and interpret them in a contextualsituation. (A-SSE.1a)Use the process of factoring to find the zeros of a quadratic function. (A-SSE.2; A-SSE.3)Find the vertex of a quadratic function algebraically and using technology.For a quadratic function that models a relationship between two quantities, I can interpret keyfeatures of the graph and table forms of the function in context. Key features include: intercepts;intervals where the function is increasing, decreasing, positive or negative; maximum or minimum,and symmetry.Construct a rough graph of a quadratic function using zeros, intercepts, the vertex and symmetry.Sketch the graph of a quadratic function that was graphed using technology, showing key features.Determine the appropriate viewing window and scale to reveal the key features of the graph of aquadratic function. (N-Q.1)Relate the domain of a quadratic function to its graph and, when given a context, to thequantitative relationship it describes.Identify the effect on the graph of a quadratic function f(x) in vertex form of replacing f(x) by f(x) k, f(x k) and k f(x) for specific values of k (both positive and negative). Experiment with cases andillustrate an explanation of the effects on the graph using technology.Find the value of k given the graphs of a parent quadratic function and its transformation.Use function notation Evaluate quadratic functions for inputs in their domains.Interpret statements that use function notation in terms of a context.Build quadratic functionsWrite a quadratic function that describes a relationship between two quantities. (F-BF.1)6

Common Core Math 2 Unit 2 Modeling with QuadraticsVocabulary: Define each word and give examples and notes that will help you remember the word/phrase.Axis of SymmetryExample and Notes to help YOU remember:DiscriminantExample and Notes to help YOU remember:ExtremaExample and Notes to help YOU remember:Intercept Form of aQuadraticExample and Notes to help YOU remember:MaximumExample and Notes to help YOU remember:MinimumExample and Notes to help YOU remember:7

Common Core Math 2 Unit 1A Modeling with QuadraticsParabolaExample and Notes to help YOU remember:Quadratic FormulaExample and Notes to help YOU remember:Quadratic FunctionExample and Notes to help YOU remember:Root of an EquationExample and Notes to help YOU remember:Standard Form of aQuadraticExample and Notes to help YOU remember:VertexExample and Notes to help YOU remember:8

Common Core Math 2 Unit 2 Modeling with QuadraticsVertex Form of aQuadraticExample and Notes to help YOU remember:X-InterceptExample and Notes to help YOU remember:Zeros of a FunctionExample and Notes to help YOU remember:Zero ProductPropertyExample and Notes to help YOU remember:Example and Notes to help YOU remember:Example and Notes to help YOU remember:9

Common Core Math 2 Unit 1A Modeling with QuadraticsSolving Quadratics Algebraically Investigation : Factoring ReviewInstructions: Today we will find the relationship between 2 linear binomials and their product which is aquadratic expression represented by the formpatterns.ax2 bx c . First we will generate data and the look forPart I. Generate DataUse the distributive property to multiply and then simplify the following binomials.1.( x 3)( x 5)2.( x 4)( x 2)2. Where do you expect each of the above equations to “hit the ground”?Chart II.Organize DataFill in the following chart by multiplying the factorsFACTORSPRODUCTax2 bx ca( x 3)( x 5)( x 4)( x 2)Part III. Analyze DataAnswer the following questions given the chart you filled in above1. Initially, what patterns do you see?2. How is the value of “a” related to the factors you see in each problem?3. How is the value of “b” related to the factors you see in each problem?4. How is the value of “c” related to the factors you see in each problem?10bc

Common Core Math 2 Unit 2 Modeling with QuadraticsPart IV: ApplicationKnowing this, fill out the values for a, b, and c in the following chart. Work backwards using your rules frompart III to find 2 binomial factors for each product. Put these in the first column.PRODUCTFACTORS( x 4)( x ax2 bx c)abcHint: Listfactors of“c”x2 6x 8x 2 7 x 12x 2 13 x 12x 2 3x 10x 2 3x 10x 2 15 x 54For each of the quadratics above, use your graphing calculator to inspect where the quadratic “hits the ground”,or touches the x-axis.1. What do you notice about the relationship between the factors and the x-intercepts?2. Why is factoring a useful skill to learn?3. Choose one of the quadratics above and create arough sketch of the graph using all the informationyou know about quadratic equations.11

Common Core Math 2 Unit 1A Modeling with QuadraticsPART V: Factoring Quadratics where a 1What if the problem has “a” value that is not equal to 1?For example,4x 2 8x 3 0:How can we algebraically find where this graph 0?The concept of un-distributing is still the same!! 4x 2 8x 3 0In this case we need to find out what multiplies to give usa c but adds to give us b .Let’s list all the factors of (4 3) or 12:Which one of those sets of factors of 12 also add to give us the b value, 8? Rewrite the original equation using an equivalent structure:4 x 2 8x 3 0 Remember!! It doesn’t matter whichorder you write the factors in!4x 2 6x 2x 3 0(4 x 2 6 x)( 2 x 3) 0Group the first two and last two! (4x 2 6x)( 2x 3) 0Undistribute what is common to bothterms2x(2x 3) 1(2x 3) 0Create factors out of the repeatedfactor, and the undistributed factors(2x 1)(2x 3) 0Check through multiplication (box,distribution, or FOIL) that it is equal!Reverse Box Method of Factoring by Grouping 12ax21stfactor2ndfactorc

Common Core Math 2 Unit 2 Modeling with Quadratics13

Common Core Math 2 Unit 1A Modeling with QuadraticsHW Factoring ( Kuta Software – Infinite Algebra 2)14

Common Core Math 2 Unit 2 Modeling with Quadratics15

Common Core Math 2 Unit 1A Modeling with Quadratics16

Common Core Math 2 Unit 2 Modeling with QuadraticsQuadratic Word Problems using factoring to solveAnnotating Math Word Problems - CUBESJust like in Language Arts, we sometimes need to annotate problems to better understand them.C – Circle important numbersU – Underline important wordsB – Box what the problem is asking you to solveE – EquationS – SolveThere are several standard types: problems where the formula is given, falling object problems, problemsinvolving geometric shapes. Just to name a few. There are many other types of application problems that usequadratic equations, however, we will concentrate on these types to simplify the matter.We must be very careful when solving these problems since sometimes we want the maximum or minimum ofthe quadratic, and sometimes we simply want to solve or evaluate the quadratic.ZERO PRODUCT PROPERTYIf the of two expressions is zero, then or of theexpressions equals zero.AlgebraIf A and B are expressions and AB , then A or B .ExampleIf (x 5)(x 2) 0, then x 5 0 or x 2 0. That is,x or x .Example:X2 - 2x - 8 0xy-3-20145Word Problem Examples1. Eight more than the square of a number is the same as 6 times the number. Find the number.17

Common Core Math 2 Unit 1A Modeling with Quadratics2. A 4 m by 6 m rug covers half of the floor area of a room and leaves a uniform strip of bare floor around theedges. What are the dimensions of the room?3. One hundred feet of fencing is available to enclose a rectangular yard along side of the St. John’s River,which is one side of the rectangle as shown. What dimensions will produce an area of 800 ft2 ?4. The perimeter of a rectangle is 50 ft. The area is 100 ft2. What are the dimensions of the rectangle?5. A company sells team photos for 10 each, and the coaches find that they sell on average 30 photographsper team. The coaches do a survey and find out for each reduction in price of 0.50, an additional twophotographs will be sold. At what price will the revenue from the photographs be 150.6. The current price of an amateur theater ticket is 10, and the venue typically sells 50 tickets. A survey foundthat for each 1 increase in ticket price, 2 fewer tickets are sold? When will the revenue equal 300.18

Common Core Math 2 Unit 2 Modeling with QuadraticsHW Factoring Word Problems1. The altitude of a triangle is 5 less than its base. The area of the triangle is 42 square inches. Find its base andaltitude.2. The length of a rectangle is 7 units more than its width. If the width is doubled and the length is increased by2, the area is increased by 42 square units. Find the dimensions of the original rectangle.3. The ages of three family children can be expressed as consecutive integers. The square of the age of theyoungest child is 4 more than eight times the age of the oldest child. Find the ages of the three children.4. In a trapezoid, the smaller base is 3 more than the height, the larger base is 5 less than 3 times the height,and the area of the trapezoid is 45 square centimeters. Find, in centimeters, the height of the trapezoid.5. A rectangular garden has a perimeter of 140m and an area of 1200m2. Find the dimensions of the garden.6. Paul wants to build a dog run in his backyard using the side of the house as one side of the run. He has 80feet of fencing. How big will the run be if the area is 600 square feet?19

Common Core Math 2 Unit 1A Modeling with Quadratics7. A museum has a café with a rectangular patio. The museum wants to add 464square feet to the area of the patio by expanding the existing patio as shown.a.) Find the area of the existing patio.b.) Write an equation that you can use to find the value of x.c.) Solve the equation. By what distance x should the length and thewidth of the patio be expanded?8. At last year’s school fair, an 18 foot by 15 foot rectangular section of land was roped off for a dunking booth.The length and width of the section will each by increased by x feet for this year’s fair in order to triple theoriginal area. Write and solve an equation to find the value of x. What is the length of rope needed toenclose the new section?9. A rectangular deck for a recreation center is 21 feet long by 20 feet wide. Its area is to be halved bysubtracting the same distance x from the length and the width. Write and solve an equation to find the valueof x. What are the deck’s new dimensions?10. A square garden has sides that are 10 feet long. A gardener wants to double the area of the garden by addingthe same distance x to the length and the width. Write an equation that x must satisfy. Can you solve theequation you wrote by factoring? Explain why or why not.20

Common Core Math 2 Unit 2 Modeling with QuadraticsProperties of Parabolay ax 2 bx c A is a function that can be written in the Standard Form of where a, b, and c are real numbers and a 0. Ex: y 5 x 2 y 2 x 2 7The domain of a quadratic function is . The graph of a quadratic function is a U-shaped curve called a . All parabolas have a , the lowest or highest point on the graph (depending upon whethery x2 x 3it opens up or down). The is an imaginary line which goes through the vertexand about which the parabola is symmetric.Axis of SymmetryRootsor xinterceptsY-interceptVertexCharacteristics of the Graph of a Quadratic Function:y ax2 bx c Direction of Opening: When a 0 , the parabola opens :When a 0 , the parabola opens : Stretch: When a 1 , the parabola is vertically .When a 1 , the parabola is vertically . Axis of symmetry: This is a vertical line passing through the vertex. Its equation is . Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of symmetry.To find the vertex, plug in x band solve for y. This yields a point ( , )2a x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y 0 and solve for x. y-intercept: is the point where the parabola crosses the y-axis. Plug in x 0 and solve for y: y c.21

Common Core Math 2 Unit 1A Modeling with QuadraticsWithout graphing the quadratic functions, complete the reque

Common Core Math 2 Unit 1A Modeling with Quadratics 4 Common Core Standards A.SSE.1 Interpret expressions that represent a quantity in terms of its context.« a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

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