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Student Text and Homework Helper UNIT 3 QUADRATIC FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles • Allan E. Bellman • Basia Hall William G. Handlin, Sr. • Dan Kennedy Stuart J. Murphy • Grant Wiggins Boston, Massachusetts • Chandler, Arizona • Glenview, Illinois • Hoboken, New Jersey Copyright © 2016 Pearson Education, Inc., or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, write to Rights Management & Contracts, Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030. Pearson, Prentice Hall, Pearson Prentice Hall, and MathXL are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc., or its affiliates. Apple, the Apple logo, and iTunes are trademarks of Apple Inc., registered in the U.S. and other countries. Google Play is a trademark of Google Inc. ISBN-13: 978-0-13-330063-5 ISBN-10: 0-13-330063-3 1 2 3 4 5 6 7 8 9 10 V0YJ 20 19 18 17 16 15 14 UNIT 3 QUADRATIC FUNCTIONS AND EQUATIONS TOPIC 8 Quadratic Functions and Equations TOPIC 8 Overview ............................................324 8-1 Quadratic Graphs and Their Properties ...........................326 8-2 Quadratic Functions ...........................................333 8-3 Transformations of Quadratic Functions ..........................340 8-4 Vertex Form of a Quadratic Function .............................346 8-5 Solving Quadratic Equations ....................................350 8-6 Factoring to Solve Quadratic Equations ...........................356 8-7 Writing Quadratic Functions ....................................360 8-8 Completing the Square ........................................367 8-9 The Quadratic Formula and the Discriminant ......................372 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS (TEKS) FOCUS (6)(A) Determine the domain and range of quadratic functions and represent the domain and range using inequalities. (6)(B) Write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f (x) = a(x - h)2 + k), and rewrite the equation from vertex form to standard form (f (x) = ax2 + bx + c). (6)(C) Write quadratic functions when given real solutions and graphs of their related equations. (7)(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry. (7)(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions. (7)(C) Determine the effects on the graph of the parent function f (x) = x2 when f (x) is replaced by af (x), f (x) + d, f (x - c), f (bx) for specific values of a, b, c, and d. (8)(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula. (8)(B) Write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems. (12)(B) Evaluate functions, expressed in function notation, given one or more elements in their domains. PearsonTEXAS.com xix Topic 8 Quadratic Functions and Equations TOPIC OVERVIEW VOCABULARY 8-1 Quadratic Graphs and Their English/Spanish Vocabulary Audio Online: Properties English Spanish 8-2 Quadratic Functions axis of symmetry, p. 327 eje de simetría 8-3 Transformations of Quadratic completing the square, p. 367 completar el cuadrado Functions discriminant, p. 372 discriminante 8-4 Vertex Form of a Quadratic Function maximum, p. 327 máximo 8-5 Solving Quadratic Equations minimum, p. 327 mínimo parabola, p. 327 parábola 8-6 Factoring to Solve Quadratic quadratic equation, p. 350 ecuación cuadrática Equations quadratic formula, p. 372 fórmula cuadrática 8-7 Writing Quadratic Functions quadratic function, p. 326 función cuadrática 8-8 Completing the Square root of an equation, p. 350 raíz de una ecuación 8-9 The Quadratic Formula and vertex, p. 327 vértice the Discriminant zero of a function, p. 350 cero de una función DIGITAL APPS PRINT and eBook Access Your Homework . ONLINE HOMEWORK You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. HOMEWORK TUTOR APP YOUR DIGITAL Do your homework anywhere! You can access the Practice and RESOURCES Application Exercises, as well as Virtual Nerd tutorials, with this PearsonTEXAS.com Homework Tutor app, available on any mobile device. STUDENT TEXT AND HOMEWORK HELPER Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 324 Topic 8 Quadratic Functions and Equations The Long Shot Have you ever been to a basketball game where they hold contests at halftime? A Math popular contest is one where the contestant needs to make a basket from half court to win Act Math Act Act - - the prize. Contestants often 3 3 shoot the ball in different ways. They might take a regular basketball shot, a hook shot, or an underhand toss. What’s the best way to shoot the basketball to make a basket? In this video you’ll see various shots and you decide! Scan page to see a video for this 3-Act Math Task. If You Need Help . VOCABULARY ONLINE LEARNING ANIMATIONS INTERACTIVE MATH TOOLS You’ll find definitions of math You can also access all of the These interactive math tools terms in both English and stepped-out learning animations give you opportunities to Spanish. All of the terms have that you studied in class. explore in greater depth audio support. key concepts to help build understanding. INTERACTIVE EXPLORATION STUDENT COMPANION VIRTUAL NERD You’ll have access to a robust Refer to your notes and Not sure how to do some of assortment of interactive solutions in your Student the practice exercises? Check explorations, including Companion. Remember that out the Virtual Nerd videos interactive concept explorations, your Student Companion is also for stepped-out, multi-level dynamic activitites, and topic- available as an ACTIVebook instructional support. level exploration activities. accessible on any digital device. PearsonTEXAS.com 325 Quadratic Graphs and 8-1 Their Properties TEKS FOCUS VOCABULARY TEKS (6)(A) Determine the domain and ĚAxis of symmetry – the line ĚQuadratic function – a function range of quadratic functions and represent that divides a parabola into two that can be written in the form the domain and range using inequalities. matching halves y = ax2 + bx + c, where a ≠ 0 TEKS (1)(D) Communicate mathematical ĚFalling object model – The ĚQuadratic parent function – the 2 ideas, reasoning, and their implications function h =-16t + c models simplest quadratic function 2 2 using multiple representations, including the height of a falling object, f(x) = x or y = x symbols, diagrams, graphs, and language as where h is the object’s height in ĚStandard form of a quadratic feet, t is the time in seconds since appropriate. function – f(x) = ax2 + bx + c, the object began to fall, and c is Additional TEKS (1)(B), (7)(A) where a ≠ 0 the object’s initial height. ĚVertex – the highest or lowest ĚMaximum – The vertex of a point on a parabola parabola that opens downward is a maximum point. The ĚImplication – a conclusion that y-coordinate of the vertex is the follows from previously stated maximum value of the function. ideas or reasoning without ĚMinimum – The vertex of a being explicitly stated parabola that opens upward ĚRepresentation – a way to is a minimum point. The display or describe information. y-coordinate of the vertex is the You can use a representation to minimum value of the function. present mathematical ideas and ĚParabola – the graph of a data. quadratic function ESSENTIAL UNDERSTANDING A quadratic function is a type of nonlinear function that models certain situations where the rate of change is not constant. The graph of a quadratic function is a symmetric curve with a highest or lowest point corresponding to a maximum or minimum value. Key Concept Standard Form of a Quadratic Function A quadratic function is a function that can be written in the form y = ax 2 + bx + c, where a ≠ 0. This form is called the standard form of a quadratic function. Examples y = 3x 2 y = x 2 + 9 y = x 2 - x - 2 326 Lesson 8-1 Quadratic Graphs and Their Properties Key Concept Parabolas 2 2 The simplest quadratic function f (x) = x or y = x is the quadratic parent function. y The graph of a quadratic function is a U-shaped curve called a parabola. 2 The parabola with equation y = x 2 is shown at the right. x Ϫ2 O 2 You can fold a parabola so that the two sides match exactly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry. The highest or lowest point of a parabola is its vertex, which is on the axis of symmetry. For graphs of functions of the form y = ax 2, the vertex is at the origin. The axis of symmetry is the y-axis, or x = 0. If a 7 0 in y = ax 2 + bx + c, If a 6 0 in y = ax 2 + bx + c, the parabola opens upward. the parabola opens downward. v v The vertex is the minimum point, The vertex is the maximum point, or lowest point, of the parabola. or highest point, of the parabola. Problem 1 Identifying Key Attributes of a Quadratic Function Graph each function.
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