# chapter 6: quadratic functions and inequalities ?· quadratic functions and inequalities chapter...

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Pacing suggestions for the entire year can be found on pages T20T21.

Quadratic Functions and InequalitiesChapter Overview and Pacing

Quadratic Functions and InequalitiesChapter Overview and Pacing

PACING (days)Regular Block

Basic/ Basic/ Average Advanced Average Advanced

Graphing Quadratic Functions (pp. 286293) 1 1 0.5 0.5 Graph quadratic functions. Find and interpret the maximum and minimum values of a quadratic function.

Solving Quadratic Equations by Graphing (pp. 294300) 1 2 0.5 1 Solve quadratic equations by graphing. (with 6-2 Estimate solutions of quadratic equations by graphing. Follow-Up)Follow-Up: Modeling Real-World Data

Solving Quadratic Equations by Factoring (pp. 301305) 1 1 0.5 0.5 Solve quadratic equations by factoring. Write a quadratic equation with given roots.

Completing the Square (pp. 306312) 2 1 1 0.5 Solve quadratic equations by using the Square Root Property. Solve quadratic equations by completing the square.

The Quadratic Formula and the Discriminant (pp. 313319) 1 1 0.5 0.5 Solve quadratic equations by using the Quadratic Formula. Use the discriminant to determine the number and type of roots of a quadratic equation.

Analyzing Graphs of Quadratic Functions (pp. 320328) 2 1 1.5 0.5Preview: Families of Parabolas (with 6-6 (with 6-6 Analyze quadratic functions of the form y a(x h)2 k. Preview) Preview) Write a quadratic function in the form y a(x h)2 k.

Graphing and Solving Quadratic Inequalities (pp. 329335) 1 1 0.5 0.5 Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable.

Study Guide and Practice Test (pp. 336341) 1 1 0.5 0.5Standardized Test Practice (pp. 342343)

Chapter Assessment 1 1 0.5 0.5

TOTAL 11 10 6 5

LESSON OBJECTIVES

284A Chapter 6 Quadratic Functions and Inequalities

*Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters,SC School-to-Career Masters, SM Science and Mathematics Lab Manual

Chapter 6 Quadratic Functions and Inequalities 284B

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See pages T12T13.

284C Chapter 6 Quadratic Functions and Inequalities

Mathematical Connections and BackgroundMathematical Connections and Background

Graphing Quadratic FunctionsWhen graphing a quadratic function, use all the

available information to produce the most accurate pos-sible graph. This includes a table of values containing thevertex and the y-intercept. Be sure the points on thegraph are connected with a smooth curve and that thegraph has a U-shape and not a V-shape at the vertex ofthe graph. Unlike the letter U, however, the graphshould become progressively wider and have arrowsindicating that the graph continues to infinity.

Solving Quadratic Equations by GraphingIt is important to distinguish between finding

solutions or roots of an equation and finding zeros of itsrelated function. An equation of the form ax2 bx c 0has a related function, f(x) ax2 bx c. The zeros off(x) are the x-coordinates of the points where the graphcrosses the x-axis. These x values are the solutions of therelated quadratic equation. Without the use of a graph-ing calculator, this method of solving quadratic equa-tions will usually provide only an estimate of solutions.Solutions that appear to be integers should be verifiedby substituting them into the original equation.

Solving Quadratic Equations by FactoringWhen solving a quadratic equation by factoring,

it is important to review factoring techniques. Theseinclude the techniques for factoring general trinomials,perfect square trinomials, and a difference of squares.Also remember to look for a greatest common factor(GCF) that might be factored out or the possibility of fac-toring by grouping. Before factoring, the equation shouldbe rewritten so that one side of the equation is 0. If theGCF of the terms of the polynomial being factored is avariable or the product of a number and a variable, suchas 3x, one solution to the equation is 0.

Complete the SquareCompleting the square is a technique most often

used to solve quadratic equations that are not factorable.To use this technique it is desirable to rewrite the equationso that it is equal to a constant. Then divide the coefficientof the linear term by 2 and square the result. Add thisvalue to both sides of the equation. One side of theequation will now be a perfect square trinomial that canbe rewritten as the square of a binomial. To help isolate thevariable on one side of the equation, take the square root

Prior KnowledgePrior KnowledgeIn the previous chapter students exploredhow to factor quadratic expressions, how towork with radicals, and how to performoperations on complex numbers. They havealso solved linear equations and inequalities,and they are familiar with using formulas.

This Chapter

Future ConnectionsFuture ConnectionsStudents will continue to explore roots andzeros of equations, examining higher-orderpolynomial equations in Chapter 7. Studentswill learn to recognize other types of equa-tions that can be solved using the QuadraticFormula. They will explore systems of quad-ratic inequalities in Chapter 8.

Continuity of InstructionContinuity of Instruction

This ChapterStudents solve quadratic equations by graph-ing, by factoring, by completing the square,and by using the Quadratic Formula. Theyexplore how values of a quadratic equationare reflected in the parabola that representsit, and use equations and graphs to explorequadratic equations that have 0, 1, or 2 roots.They relate the value of the discriminant tothe number of roots and to whether the roots

are rational, irrational, or complex.

Chapter 6 Quadratic Functions and Inequalities 284D

of both sides, remembering that the square root of theconstant on one side of the equation will result in twovalues, one positive and one negative, represented by a sign. Finally, isolate the variable using the Addition,Subtraction, Multiplication, and/or Division Propertiesof Equality. Simplify any solution that involves thesymbol by simplifying two separate expressions, oneusing the plus sign and the other using the minus sign.When using the technique of completing the square,be sure that the coefficient of the quadratic term is 1. If it is not 1, divide both sides of the equation by thatcoefficient. You can then solve the equation by com-pleting the square.

The Quadratic Formula and the DiscriminantWhile the technique of completing the square

can be used to solve any quadratic equation, imple-menting this technique can lead to operations involvingunwieldy fractions. The Quadratic Formula can alsobe used to solve any quadratic equation. Using thisformula, the variable is isolated in the very first stepand the other steps involve simplifying the solutionsby simplifying radicals and fractions. The discrimi-nant is simply the portion of the Quadratic Formulathat appears underneath the radical, b2 4ac. Thisvalue alone will determine the number and type ofroots (solutions) of the equation. This is because thesquare root of this value could result in a rational number, such as 6, an irrational number, such as 2,the value 0, or an imaginary number, such as 3i. Byfinding and examining just the value of the discrimi-nant, you can tell very quickly what type of solutions aquadratic equation will have. This can serve as acheck when solving the equation.

Analyzing Graphs ofQuadratic FunctionsTo write a quadratic equation in the form

y a(x h)2 k, called vertex form, it is important toremember that an equation is a statement of equality.When an equation is rewritten in a different form, thisequality must be maintained. In previous lessons, if avalue was added to one side of an equation, it was alsoadded to the other side of the equation in order to main-tain equality. Another way to maintain equality is toadd a value to one side and then subtract that samevalue from that side. For example, if an addition of 5 isshown on the right side of an equation, a subtractionof 5 would also be shown on the right side. So in

essence this would be an overall addition of 5 5 or 0,which does not affect the equality of the statement.For example, y 3x is equivalent to the statementy 3x 5 5. Be especially careful when adding avalue inside a set of parentheses, since the use ofparentheses often involves multiplication. For exampley 3(x) is not equivalent to y 3(x 1) 1, butinstead to y 3(x 1) 3(1).

Graphing and SolvingQuadratic InequalitiesOne way of solving a linear inequality not dis-

cussed in Chapter 1 is to first solve its related linearequation and the

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