Unit+5

=Unit 5 Quadratic Equations and Functions=

Vocabulary

 * absolute value of a complex number (p. 271) || greatest common factor (GCF) of an expression (p. 255) || standard form of a quadratic equation (p. 263) ||
 * axis of symmetry (p. 235) || //i// (p. 270) || standard form of a quadratic function (p. 234) ||
 * completing the square (p. 278) || imaginary number (p. 270) || vertex form of a quadratic function (p. 248) ||
 * complex number (p. 271) || parabola (pp. 235, 543) || vertex of a parabola (p. 235) ||
 * complex number plane (p. 271) || perfect square trinomial (p. 258) || zero of a function (pp. 264, 309) ||
 * difference of two squares (p. 259) || Quadratic Formula (p. 285) || Zero Product Property (p. 263) ||
 * discriminant (p. 287) || quadratic function (p. 234) ||  ||
 * factoring (p. 255) ||

5-1 Objectives
The **standard form of a quadratic function** is //f//(//x//) = //ax//2 + //bx// + //c//, where //a// �� 0. The quadratic term is //ax//2. The graph of a **quadratic function** is a **parabola.** The **axis of symmetry** is a line that divides a parabola into two mirror images. **The vertex of a parabola** is the point at the intersection of the parabola and its axis of symmetry. Corresponding points on the parabola are the same distance from the axis of symmetry. You can find a quadratic model for a set of data by solving a system of three equations for //a//, //b//, and //c//, or by using the quadratic regression feature of a graphing calculator.
 * 1) To identify quadratic functions and graphs (p. 234)
 * 2) To model data with quadratic functions (p. 236)

5-2 and 5-3 Objectives
The constants //a//, //b//, and //c// characterize the graph of //y// = //ax//2 + //bx// + //c//. The axis of symmetry is //x// = −, the vertex is at , and //f//  is the maximum or minimum value. The **vertex form of a quadratic function** is //y// = //a//(//x// − //h//)2 + //k//. The vertex is (//h//, //k//), the maximum or minimum value is //k// , and the axis of symmetry is the line //x// = //h//. If //a// > 0, the parabola opens up. If //a// < 0, it opens down.
 * 1) To graph quadratic functions (p. 241)
 * 2) To find maximum and minimum values of quadratic functions (p. 243)
 * 3) To use the vertex form of a quadratic function (p. 248)

5-4 and 5-5 Objectives
You can solve some quadratic equations by finding the square root of each side or by finding the zeros of the related function. You can solve some quadratic equations in the **standard form of a quadratic equation** //ax//2 + //bx// + //c// = 0 by **factoring** if you can find two factors with product //ac// and sum //b//. Then use the **Zero Product Property.** For a **perfect square trinomial,** //ax//2 ± 2//abx// + //b//2 = (//a// ± //b//)2. For the **difference of two squares,** //a//2 − //b//2 = (//a// + //b//)(//a// − //b//). In all cases, first factor out the **greatest common factor (GCF) of the expression.**
 * 1) To find common and binomial factors of quadratic expressions (p. 255)
 * 2) To factor special quadratic expressions (p. 258)
 * 3) To solve quadratic equations by factoring and by finding square roots (p. 263)
 * 4) To solve quadratic equations by graphing (p. 264)

5-6 Objectives
An **imaginary number** has the form //a// + //bi//, where //b// �� 0. The imaginary number //**i**// is defined as //i//2 = − 1. A **complex number** has the form //a// + //bi//, where //a// and //b// are any real numbers. The **absolute value of a complex number** is its distance from the origin in the **complex number plane.** You graph //a// + //bi// in the complex plane just as you graphed (//a//, //b//) in the coordinate plane. Complex numbers follow rules of operation like those of real numbers. Some quadratic equations have imaginary numbers as roots. Functions of complex numbers may be used to generate fractals.
 * 1) To identify and graph complex numbers (p. 270)
 * 2) To add, subtract, and multiply complex numbers (p. 272)

5-7 and 5-8 Objectives
You can solve any quadratic equation by using the Quadratic Formula. If //ax//2 + //bx// + //c// = 0, then //x// =. The discriminant //b//2 − 4//ac// determines the number and type of solutions of the equation. If //b//2 − 4//ac// > 0, the equation has two real solutions. If //b//2 − 4//ac// = 0, the equation has one real solution. If //b//2 − 4//ac// < 0, the equation has no real solutions and two imaginary solutions.
 * 1) To solve equations by completing the square (p. 278)
 * 2) To rewrite functions in vertex form by completing the square (p. 280)
 * 3) To solve quadratic equations by using the Quadratic Formula (p. 285)
 * 4) To determine types of solutions by using the discriminant (p. 287)
 * Completing the square** is based on the relationship //x//2 + //bx// + [[image:http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037881-X/Ch05/05EM/images/Fparen_bover2_sq.gif width="30" height="33"]] = [[image:http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037881-X/Ch05/05EM/images/x_plus_bover2_sq.gif width="52" height="32"]] . You can use it to write a quadratic function in vertex form. If the coefficient of the quadratic term is not 1, you must factor out the coefficient from the variable terms.