4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms at the left. Lesson 4-1 quadratic function quadratic term linear term constant term f() = 2 2 + 2-4 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. parabola ais of smmetr verte maimum value minimum value Chapter 4 59 Glencoe Algebra 2
Lesson 4-1 (continued) Main Idea Details Graph Quadratic Functions Graph f() = 2 2 + 2 + 8. Fill in the missing verbal and mathematical steps. Write the function in standard form. Identif the coefficients. Identif the -intercept. Make a table. Put the verte in the center. Maimium and Minimum Values Graph the ordered pairs. Check the ais of smmetr. Find the domain and range for the function f() = -2 2 + 12-5. Find -coordinate of verte. Find -coordinate of verte. Domain = Range = Helping You Remember How can ou remember the wa to use the 2 term of a quadratic function to tell whether it has a maimum or a minimum value? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 60 Glencoe Algebra 2
4-2 Solving Quadratic Equations b Graphing What You ll Learn Scan the tet in the lesson. Write two facts ou learned about solving quadratic equations b graphing as ou scanned the tet. 1. 2. Active Vocabular Review Vocabular Graph the linear function and solve the related linear equation. (Lesson 1 3) Lesson 4-2 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. = 2-5 2-5 = 0 How is the graph related to the solution of the equation? New Vocabular Write the correct term beside each definition. the solution of a quadratic equation quadratic functions that are set equal to zero the -intercepts of the graph of a function a 2 + b + c = 0, where a 0, and a, b, and c are integers Chapter 4 61 Glencoe Algebra 2
Lesson 4-2 (continued) Main Idea Details Solve Quadratic Functions Show that the zeros of the function f() = 2 2 + 5-3 are the roots of the equation 2 2 + 5-3 = 0. 2 2 + 5-3 = 0 = and = 2(-3) 2 + 5(-3) - 3 = 0 2(0.5) 2 + 5(0.5) - 3 = 0 Estimate Solutions Graph the parabola. Locate zeros on the graph. Helping You Remember Finding zeros in a table. Quadratics on a Calculator Solve quadratic equations. Think of a memor aid that can help ou recall what is meant b the zeros of a quadratic function. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 62 Glencoe Algebra 2
4-3 Solving Quadratic Equations b Factoring What You ll Learn Scan the lesson. List two headings ou would use to make an outline of this lesson. 1. 2. Active Vocabular factored form New Vocabular Fill in each blank with the correct term or phrase. A form of a quadratic equation written as = where p and q represent the of the of the equation. FOIL method A method for changing a quadratic equation from Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. form to form. The foil method uses the Propert to multipl. Vocabular Link Make a table of values for = 2 - - 6 and = ( - 3)( + 2). Graph the equations. -5-2 0 3 5-5 -2 0 3 5 What can ou determine about the two equations? Lesson 4-3 Chapter 4 63 Glencoe Algebra 2
Lesson 4-3 (continued) Main Idea Details Factored Form Factor each trinomial using the steps listed. Given 2-35+ 2 8+3 2-10 Write in standard form. Identif the product and sum. Find two integers to satisf the product and sum. Epand the middle term and find GCF of first two terms and second two terms. Solve Equations b Factoring Eplain the error made in the solution to the quadratic equation. Solutions: 2-4 - 12 = 5 ( - 6)( + 2) = 5-6 = 5 or + 2 = 5 Error: = 11 = 3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 64 Glencoe Algebra 2
4-4 Comple Numbers What You ll Learn Skim the lesson. Predict two things that ou epect to learn based on the headings and the Ke Concept bo. 1. 2. Active Vocabular New Vocabular Place each number in a bo. All numbers should be used once: 4, 0, 5, 1 2, π, 2, 0.5. (Lesson 1-2) Integers Real Numbers Rational Numbers Whole Numbers Irrational Numbers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. square root propert comple conjugates imaginar unit pure imaginar number Natural Numbers Vocabular Link Match the term with its definition b drawing a line to connect the two. square root of a negative real number i, which is defined as i 2 = -1 a propert which sas that if 2 = a, then = ± a an number which can be written in the form a + bi, where a and b are real numbers and i is the imaginar unit Lesson 4-4 comple number two comple numbers of the form a + bi and a - bi Chapter 4 65 Glencoe Algebra 2
Lesson 4-4 (continued) Main Idea Details Pure Imaginar Numbers Simplif the epression b completing each empt bo............... Operations with Comple Numbers Write each listed number under each categor that applies. 7, 12i, 3 + 4i, -12, 0, 2 + i, i 5, 5, 2 3, 1 2 + 3 2 i, - 1 3 i Helping You Remember Comple Real Imaginar How can ou use what ou know about the factors of a polnomial that is a difference of two squares to help ou remember how to simplif fractions with imaginar numbers in the denominator? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 66 Glencoe Algebra 2
4-5 Completing the Square What You ll Learn Skim the Eamples in the lesson. Predict two things ou think ou will learn about completing the square. 1. 2. Active Vocabular Review Vocabular Solve the equation using each method. (Lessons 4-2, 4-3, and 4-4) Factoring 2-9 = 0 Graphing Square Root Propert 2-9 = 0 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. completing the square Which method do ou prefer. Eplain our answer. New Vocabular Fill in the blanks with the correct terms. a method used to manipulate a so that one side is a perfect ; Once one side is a perfect square, the Propert can be used to solve the equation. Lesson 4-5 Chapter 4 67 Glencoe Algebra 2
Lesson 4-5 (continued) Main Idea Details Square Root Propert Solve each equation using the Square Root Propert, if possible. If not possible, eplain wh. 2-25 = 36 2 + 6 + 36 = 100 Complete the Square Solve the equation b completing the square. 2-8 - 25 = 0 Does (- 2) 8 2 = - 25? No. Add to each side. Helping You Remember (- 2) 8 =. Add to each side. Factor the left and simplif the right. Solve using the Square Root Propert. Add to each side. How can ou use the rules for squaring a binomial to help ou remember the procedure for changing a binomial into a perfect square trinomial? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 68 Glencoe Algebra 2
4-6 The Quadratic Formula and the Discriminant What You ll Learn Skim the lesson. Write two things ou alread know about the quadratic formula and the discriminant. 1. Active Vocabular 2. Review Vocabular Complete the square to solve each equation. (Lesson 4-5) Lesson 4-6 2-2 - 15 = 0 2 2 - - 3 = 0 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. New Vocabular Complete the Quadratic Formula and discriminant b filling in each bo with the missing constant, variable or operation. Quadratic Formula = -b 2-4a a Discriminant b 2 4 Vocabular Link Eplain how the non-mathematical meaning of the word discriminate can help ou to remember the mathematical meaning of this word. Chapter 4 69 Glencoe Algebra 2
Lesson 4-6 (continued) Main Idea Details Quadratic Formula Solve the quadratic equation b completing the square and b using the Quadratic Formula. Completing the Square 2-4 + 12 = 0 Quadratic Formula 2-4 + 12 = 0 a = b = c = Roots and the Discriminant Complete the chart about discriminants below in our own words. Helping You Remember positive perfect square negative number What does the discriminant tell ou? positive non-perfect square Based on what ou know about the discriminant, eplain wh it is not possible to have onl one comple root. zero Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 70 Glencoe Algebra 2
4-7 Transformations with Quadratic Functions What You ll Learn Skim the Eamples in the lesson. Predict two things ou think ou will learn about transformations with quadratic functions. 1. 2. Active Vocabular Review Vocabular Label each graph with a function. (Lesson 2-7) f() = - 4 f() = 2 Lesson 4-7 f() = Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. f() = - 2 verte form New Vocabular Write the definition net to the term. Chapter 4 71 Glencoe Algebra 2
Lesson 4-7 (continued) Main Idea Details Write Quadratic Functions in Verte Form Write the equation of the parabola shown to the right. Verte Form = a( - h) 2 + k Fill in known information. = a( - ) 2 + Solve for a: (1, 4) Write in verte form. = 2( - 3) 2-4 (3, 4) Transformations of Quadratic Functions Describe how each characteristic of the quadratic function affects the graph of the quadratic function. Graph the function. Helping You Remember = -3(-4) 2 + 3 How can ou remember that = ( + 4) 2 is a translation to the left and that = ( - 5) 2 is a translation to the right? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 72 Glencoe Algebra 2
4-8 Quadratic Inequalities What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. 2. Active Vocabular Review Vocabular Graph the inequalities in two variables. (Lesson 2-8) > -2 + 5 2-4 6 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. quadratic inequalities New Vocabular Fill in each blank with the correct term or phrase. A quadratic inequalit in two can be graphed in the same wa that ou graph linear. A quadratic inequalit in two variables that is graphed consists of a and. A quadratic inequalit in one variable Lesson 4-8 can be solved using the of the related function. The solution set is given in notation. Chapter 4 73 Glencoe Algebra 2
Lesson 4-8 (continued) Main Idea Details Graph Quadratic Inequalities Graph the quadratic inequalit < 2 + 6 + 7. Use the boes on the left to show and eplain an work. Decide if the line is dotted or shaded. Graph the parabola b writing it in verte form. Test the point (0, 0) and shade accordingl. Solve Quadratic Inequalities Graph the parabola = -( - 2) 2 + 4. Circle the section(s) of the parabola in which the -value is greater than zero. Write the -values in the section(s) of the parabola using set-builder notation. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 74 Glencoe Algebra 2