Solve Quadratic Equations by Graphing
|
|
- Marybeth Mosley
- 6 years ago
- Views:
Transcription
1 0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6. Ke Vocabular quadratic equation -intercept, p. 225 roots, p. 575 zero of a function, p. 337 A quadratic equation is an equation that can be written in the standard form a 2 b c 5 0 where a Þ 0. In Chapter 9, ou used factoring to solve a quadratic equation. You can also use graphing to solve a quadratic equation. Notice that the solutions of the equation a 2 b c 5 0 are the -intercepts of the graph of the related function 5 a 2 b c. Solve b Factoring Solve b Graphing ( 2 )( 2 5) or 5 5 To solve , graph From the graph ou can see that the -intercepts are and READING In this course, solutions refers to real-number solutions. To solve a quadratic equation b graphing, first write the equation in standard form, a 2 b c 5 0. Then graph the related function 5 a 2 b c. The -intercepts of the graph are the solutions, or roots, of a 2 b c 5 0. E XAMPLE Solve a quadratic equation having two solutions Solve b graphing. STEP Write the equation in standard form Write original equation Subtract 3 from each side. STEP 2 Graph the function The -intercepts are 2 and 3. c The solutions of the equation are 2 and 3. CHECK You can check 2 and 3 in the original equation Write original equation. (2) 2 2 2(2) 0 3 (3) 2 2 2(3) 0 3 Substitute for Simplif. Each solution checks. 0.3 Solve Quadratic Equations b Graphing 643
2 E XAMPLE 2 Solve a quadratic equation having one solution Solve b graphing. STEP Write the equation in standard form Write original equation Subtract from each side. STEP 2 Graph the function The -intercept is. c The solution of the equation is E XAMPLE 3 Solve a quadratic equation having no solution AVOID ERRORS Do not confuse -intercepts and -intercepts. Although the graph has a -intercept, it does not have an -intercepts. Solve b graphing. STEP Write the equation in standard form Write original equation Subtract 4 from each side. STEP 2 Graph the function The graph has no -intercepts. c The equation has no solution GUIDED PRACTICE for Eamples, 2, and 3 Solve the equation b graphing KEY CONCEPT For Your Notebook Number of s of a Quadratic Equation A quadratic equation has two solutions if the graph of its related function has two -intercepts. A quadratic equation has one solution if the graph of its related function has one -intercept. A quadratic equation has no real solution if the graph of its related function has no -intercepts. 644 Chapter 0 Quadratic Equations and Functions
3 FINDING ZEROS Because a zero of a function is an -intercept of the function s graph, ou can use the function s graph to find the zeros of a function. E XAMPLE 4 Find the zeros of a quadratic function Find the zeros of f() ANOTHER WAY You can find the zeros of a function b factoring: f() ( 7)( 2 ) 527 or 5 Graph the function f() The -intercepts are 27 and. c The zeros of the function are 27 and. CHECK Substitute 27 and in the original function. f(27) 5 (27) 2 6(27) f () f() 5 () 2 6() APPROXIMATING ZEROS The zeros of a function are not necessaril integers. To approimate zeros, look at the signs of the function values. If two function values have opposite signs, then a zero falls between the -values that correspond to the function values. E XAMPLE 5 Approimate the zeros of a quadratic function Approimate the zeros of f() to the nearest tenth. STEP Graph the function f() There are two -intercepts: one between 24 and 23 and another between 2 and 0. STEP 2 Make a table of values for -values between 24 and 23 and between 2 and 0 using an increment of 0.. Look for a change in the signs of the function values. f () INTERPRET FUNCTION VALUES The function value that is closest to 0 indicates the -value that best approimates a zero of the function f() f() c In each table, the function value closest to 0 is 20.. So, the zeros of f() are about 23.7 and about GUIDED PRACTICE for Eamples 4 and 5 4. Find the zeros of f() Approimate the zeros of f() to the nearest tenth. 0.3 Solve Quadratic Equations b Graphing 645
4 E XAMPLE 6 Solve a multi-step problem SPORTS An athlete throws a shot put with an initial vertical velocit of 40 feet per second as shown. a. Write an equation that models the height h (in feet) of the shot put as a function of the time t (in seconds) after it is thrown. b. Use the equation to find the time that the shot put is in the air. 6.5 ft a. Use the initial vertical velocit and the release height to write a vertical motion model. h 526t 2 vt s Vertical motion model h 526t 2 40t 6.5 Substitute 40 for v and 6.5 for s. USE A GRAPHING CALCULATOR When entering h 526t 2 40t 6.5 in a graphing calculator, use instead of h and instead of t. b. The shot put lands when h 5 0. To find the time t when h 5 0, solve 0 526t 2 40t 6.5 for t. To solve the equation, graph the related function h 526t 2 40t 6.5 on a graphing calculator. Use the trace feature to find the t-intercepts. c There is onl one positive t-intercept. The shot put is in the air for about 2.6 seconds. Trace X= Y= GUIDED PRACTICE for Eample 6 6. WHAT IF? In Eample 6, suppose the initial vertical velocit is 30 feet per second. Find the time that the shot put is in the air. CONCEPT SUMMARY For Your Notebook Relating s of Equations, -Intercepts of Graphs, and Zeros of Functions s of an Equation The solutions of the equation are 2 and 6. -Intercepts of a Graph The -intercepts of the graph of occur where 5 0, so the -intercepts are 2 and 6, as shown. Zeros of a Function The zeros of the function f() are the values of for which f() 5 0, so the zeros are 2 and Chapter 0 Quadratic Equations and Functions
5 0.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5 and 5 5 STANDARDIZED TEST PRACTICE Es. 2, 46, 53, and 54. VOCABULARY Write in standard form. 2. WRITING Is a quadratic equation? Eplain. SOLVING EQUATIONS Solve the equation b graphing. EXAMPLES, 2, and 3 on pp for Es } } } ERROR ANALYSIS The graph of the function related to the equation is shown. Describe and correct the error in solving the equation. The onl solution of the equation is 4. 3 EXAMPLE 4 on p. 645 for Es FINDING ZEROS Find the zeros of the function. 22. f() f() f() f() f() f() f() f() f() SOLVING EQUATIONS Solve the equation b graphing } EXAMPLE 5 on p. 645 for Es APPROXIMATING ZEROS Approimate the zeros of the function to the nearest tenth. 37. f() f() f() f() f() f() f() f() f() MULTIPLE CHOICE Which function has a zero between 23 and 22? A f() B f() C f() D f() Solve Quadratic Equations b Graphing 647
6 CHALLENGE Use the given surface area S of the clinder to find the radius r to the nearest tenth. (Use 3.4 for p.) 47. S 5 25 ft S 5 76 m S cm 2 r 6 ft r 3 m r 0 cm PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLE 6 on p. 646 for Es SOCCER The height (in feet) of a soccer ball after it is kicked can be modeled b the graph of the equation where is the horizontal distance (in feet) that the ball travels. The ball is not touched, and it lands on the ground. Find the distance that the ball was kicked. 5. SURVEYING To keep water off a road, the road s surface is shaped like a parabola as in the cross section below. The surface of the road can be modeled b the graph of where and are measured in feet. Find the width of the road to the nearest tenth of a foot DIVING During a cliff diving competition, a diver begins a dive with his center of gravit 70 feet above the water. The initial vertical velocit of his dive is 8 feet per second. a. Write an equation that models the height h (in feet) of the diver s center of gravit as a function of time t (in seconds). b. How long after the diver begins his dive does his center of gravit reach the water? 53. SHORT RESPONSE An arc of water spraed from the nozzle of a fountain can be modeled b the graph of where is the horizontal distance (in feet) from the nozzle and is the vertical distance (in feet). The diameter of the circle formed b the arcs on the surface of the water is called the displa diameter. Find the displa diameter of the fountain. Eplain our reasoning. Displa diameter WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE
7 54. EXTENDED RESPONSE Two softball plaers are practicing catching fl balls. One plaer throws a ball to the other. She throws the ball upward from a height of 5.5 feet with an initial vertical velocit of 40 feet per second for her teammate to catch. a. Write an equation that models the height h (in feet) of the ball as a function of time t (in seconds) after it is thrown. b. If her teammate misses the ball and it lands on the ground, how long was the ball in the air? c. If her teammate catches the ball at a height of 5.5 feet, how long was the ball in the air? Eplain our reasoning. 55. CHALLENGE A stream of water from a fire hose can be modeled b the graph of where and are measured in feet. A firefighter is holding the hose 3 feet above the ground, 37 feet from a building. Will the stream of water pass through a window if the top of the window is 26 feet above the ground? Eplain. MIXED REVIEW PREVIEW Prepare for Lesson 0.4 in Es Evaluate the epression. (p. 0) 56. 2Ï } Ï } Ï } 625 Simplif the epression. (p. 495) (28) } 60. (28) p 9 6 } } Write the number in standard form. (p. 52) QUIZ for Lessons Graph the function. Compare the graph with the graph of 5 2. (p. 628). 52} Graph the function. Label the verte and ais of smmetr (p. 628) (p. 628) (p. 635) (p. 635) 8. 52} (p. 635) (p. 635) Solve the equation b graphing. (p. 643) Find the zeros of the function. (p. 643) 6. f() f() f() EXTRA PRACTICE for Lesson 0.3, p. 947 ONLINE QUIZ at classzone.com 649
2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.
9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept
More information10.7. Interpret the Discriminant. For Your Notebook. x5 2b 6 Ï} b 2 2 4ac E XAMPLE 1. Use the discriminant KEY CONCEPT
10.7 Interpret the Discriminant Before You used the quadratic formula. Now You will use the value of the discriminant. Wh? So ou can solve a problem about gmnastics, as in E. 49. Ke Vocabular discriminant
More informationProperties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a
0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value
More informationGraph Quadratic Functions in Standard Form
TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in
More informationSolving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic
9. Solving Quadratic Equations b Graphing equation in one variable? How can ou use a graph to solve a quadratic Earlier in the book, ou learned that the -intercept of the graph of = a + b variables is
More informationWrite Quadratic Functions and Models
4.0 A..B, A.6.B, A.6.C, A.8.A TEKS Write Quadratic Functions and Models Before You wrote linear functions and models. Now You will write quadratic functions and models. Wh? So ou can model the cross section
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationObjectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
More informationGraph and Write Equations of Parabolas
TEKS 9.2 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Parabolas Before You graphed and wrote equations of parabolas that open up or down. Now You will graph and write equations of parabolas that open
More informationGraph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.
TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationLesson 5.1 Exercises, pages
Lesson 5.1 Eercises, pages 346 352 A 4. Use the given graphs to write the solutions of the corresponding quadratic inequalities. a) 2 2-8 - 10 < 0 The solution is the values of for which y
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s
More informationGraph Square Root and Cube Root Functions
TEKS 6.5 2A.4.B, 2A.9.A, 2A.9.B, 2A.9.F Graph Square Root and Cube Root Functions Before You graphed polnomial functions. Now You will graph square root and cube root functions. Wh? So ou can graph the
More informationDefine General Angles and Use Radian Measure
1.2 a.1, a.4, a.5; P..E TEKS Define General Angles and Use Radian Measure Before You used acute angles measured in degrees. Now You will use general angles that ma be measured in radians. Wh? So ou can
More informationYou studied exponential growth and decay functions.
TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit
More informationChapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square
Chapter Notes Alg. H -A (Lesson -) Solving Quadratic Equations b Finding the Square Root and Completing the Square p. *Calculator Find the Square Root: take the square root of. E: Solve b finding square
More informationSolving Quadratic Equations
9 Solving Quadratic Equations 9. Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square
More informationSquare Root Functions as Inverses. Inverse of a Quadratic Function. y f 1 (x) x
6-1 Square Root Functions as Inverses TEKS FOCUS TEKS ()(C) Describe and analze the relationship between a function and its inverse (quadratic and square root, logarithmic and eponential), including the
More informationAlgebra II Practice Test Quadratic Functions Unit 3 Part II. Period Date NON-CALCULATOR SECTION
Name Period Date Vocabular: Define each word and give an eample.. Quadratic Function NON-CALCULATOR SECTION. Zero (of a function) 3. One-to-One Function Short Answer:. Describe how to find a quadratic
More information7.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI.,
More informationEvaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical
. Georgia Performance Standard(s) MMA2a, MMA2b, MMAd Your Notes Evaluate nth Roots and Use Rational Eponents Goal VOCABULARY nth root of a p Evaluate nth roots and stud rational eponents. Inde of a radical
More informationGraph and Write Equations of Circles
TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges,
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationLesson Master 9-1B. REPRESENTATIONS Objective G. Questions on SPUR Objectives. 1. Let f(x) = 1. a. What are the coordinates of the vertex?
Back to Lesson 9-9-B REPRESENTATIONS Objective G. Let f() =. a. What are the coordinates of the verte? b. Is the verte a minimum or a maimum? c. Complete the table of values below. 3 0 3 f() d. Graph the
More informationSolve Quadratic Equations by Completing the Square
10.5 Solve Quadratic Equations by Completing the Square Before You solved quadratic equations by finding square roots. Now You will solve quadratic equations by completing the square. Why? So you can solve
More informationSkills Practice Skills Practice for Lesson 3.1
Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Define each term in our own words.. quadratic function. vertical motion Problem
More informationAlgebra 1 Unit 9 Quadratic Equations
Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic
More information7.2 Properties of Graphs
7. Properties of Graphs of Quadratic Functions GOAL Identif the characteristics of graphs of quadratic functions, and use the graphs to solve problems. LEARN ABOUT the Math Nicolina plas on her school
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More information7.2 Connecting Intercepts and Linear Factors
Name Class Date 7.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationNonlinear Systems. No solution One solution Two solutions. Solve the system by graphing. Check your answer.
8-10 Nonlinear Sstems CC.9-1.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two
More informationGraph Simple Rational Functions. is a rational function. The graph of this function when a 5 1 is shown below.
TEKS 8.2 2A.0.A, 2A.0.B, 2A.0.C, 2A.0.F Graph Simple Rational Functions Before You graphed polnomial functions. Now You will graph rational functions. Wh? So ou can find average monthl costs, as in E.
More informationUnit 6 Graphing Linear Equations
Unit 6 Graphing Linear Equations NAME: GRADE: TEACHER: Ms. Schmidt _ Lesson 1 Homework Solving a Linear Equation for Solve each equation for and state the slope and -intercept. 1. + = 5. + = 1. 4 = 8 4.
More informationChapter 9 Quadratic Graphs
Chapter 9 Quadratic Graphs Lesson 1: Graphing Quadratic Functions Lesson 2: Vertex Form & Shifts Lesson 3: Quadratic Modeling Lesson 4: Focus and Directrix Lesson 5: Equations of Circles and Systems Lesson
More information6.3 Interpreting Vertex Form and Standard Form
Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationRepresent Relations and Functions
TEKS. a., a., a.5, A..A Represent Relations and Functions Before You solved linear equations. Now You will represent relations and graph linear functions. Wh? So ou can model changes in elevation, as in
More information136 Maths Quest 10 for Victoria
Quadratic graphs 5 Barr is a basketball plaer. He passes the ball to a team mate. When the ball is thrown, the path traced b the ball is a parabola. Barr s throw follows the quadratic equation =.5 +. +.5.
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More information9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
More informationWords Algebra Graph. 5 rise } run. } x2 2 x 1. m 5 y 2 2 y 1. slope. Find slope in real life
TEKS 2.2 a.1, a.4, a.5 Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 46. Ke Vocabular
More informationMAT 1033C -- Martin-Gay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam
MAT 33C -- Martin-Ga Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
More informationCharacteristics of Quadratic Functions
. Characteristics of Quadratic Functions Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete
More informationApply Exponent Properties Involving Quotients. Notice what happens when you divide powers with the same base. p a p a p a p a a
8. Apply Eponent Properties Involving Quotients Before You used properties of eponents involving products. Now You will use properties of eponents involving quotients. Why? So you can compare magnitudes
More information3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
More informationInverse Trigonometric Functions. inverse sine, inverse cosine, and inverse tangent are given below. where tan = a and º π 2 < < π 2 (or º90 < < 90 ).
Page 1 of 7 1. Inverse Trigonometric Functions What ou should learn GOAL 1 Evaluate inverse trigonometric functions. GOAL Use inverse trigonometric functions to solve real-life problems, such as finding
More informationYou evaluated powers. You will simplify expressions involving powers. Consider what happens when you multiply two powers that have the same base:
TEKS.1 a.1, 2A.2.A Before Now Use Properties of Eponents You evaluated powers. You will simplify epressions involving powers. Why? So you can compare the volumes of two stars, as in Eample. Key Vocabulary
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.
More informationStudy Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.
Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +
More informationLesson 4.1 Exercises, pages
Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental
More information1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. (c) ( )
Name: Date: QUADRATIC FUNCTION REVIEW FLUENCY Algebra II 1. Without the use of our calculator, evaluate each of the following quadratic functions for the specified input values. (a) g( x) g g ( 5) ( 3)
More information5.2 Solving Linear-Quadratic Systems
Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker
More informationMathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
More informationShape and Structure. Forms of Quadratic Functions. Lesson 2.1 Assignment
Lesson.1 Assignment Name Date Shape and Structure Forms of Quadratic Functions 1. Analze the graph of the quadratic function. a. The standard form of a quadratic function is f() 5 a 1 b 1 c. What possible
More information5-4. Focus and Directrix of a Parabola. Key Concept Parabola VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
5- Focus and Directri of a Parabola TEKS FOCUS VOCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.
More informationQuadratic Graphs and Their Properties
- Think About a Plan Quadratic Graphs and Their Properties Physics In a physics class demonstration, a ball is dropped from the roof of a building, feet above the ground. The height h (in feet) of the
More informationChapter 10 Answers. Practice (0,0); maximum 2. (0,0); maximum 3. (0,0); minimum y = x 2, y = 3x 2, y = 5x 2 8. y 1
Chapter 0 Answers Practice 0-. (0,0); maimum. (0,0); maimum. (0,0); minimum. (0,0); minimum. (0,0); maimum. (0,0); minimum 7. =, =, =. =, =-, =-. =, =-, = 0. =, =, =-. =, =, =-7. =, =, =........ 7. 0 7...
More informationThe Quadratic Formula
- The Quadratic Formula Content Standard Reviews A.REI..b Solve quadratic equations by... the quadratic formula... Objectives To solve quadratic equations using the Quadratic Formula To determine the number
More informationUnit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More informationStudy Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
More informationName Class Date. Solving by Graphing and Algebraically
Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem
More information20.3 Applying the Zero Product Property to Solve Equations
20.3 Applying the Zero Product Property to Solve Equations Essential Question: How can you use the Zero Product Property to solve quadratic equations in factored form? Resource Locker Explore Understanding
More informationAdditional Factoring Examples:
Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic
More informationMath 141 Review for Midterm
Math 141 Review for Midterm There will be two parts to this test. Part 1 will have graph sketching, and no calculator is allowed. Part will have everthing else, and a calculator and/or graphing calculator
More informationNAME DATE PERIOD. Study Guide and Intervention. Solving Quadratic Equations by Graphing. 2a = -
NAME DATE PERID - Study Guide and Intervention Solving Quadratic Equations by Graphing Solve Quadratic Equations Quadratic Equation A quadratic equation has the form a + b + c = 0, where a 0. Roots of
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationName Class Date. Identify the vertex of each graph. Tell whether it is a minimum or a maximum.
Practice Quadratic Graphs and Their Properties Identify the verte of each graph. Tell whether it is a minimum or a maimum. 1. y 2. y 3. 2 4 2 4 2 2 y 4 2 2 2 4 Graph each function. 4. f () = 3 2 5. f ()
More informationQuadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry
Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function = - -1 0 1 1 Eample 1: Make a table,
More informationPRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.
MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for
More informationSolve Trigonometric Equations. Solve a trigonometric equation
14.4 a.5, a.6, A..A; P.3.D TEKS Before Now Solve Trigonometric Equations You verified trigonometric identities. You will solve trigonometric equations. Why? So you can solve surface area problems, as in
More informationName Class Date. Deriving the Standard-Form Equation of a Parabola
Name Class Date 1. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of
More informationHow can you determine the number of solutions of a quadratic equation of the form ax 2 + c = 0? ACTIVITY: The Number of Solutions of ax 2 + c = 0
9. Solving Quadratic Equations Using Square Roots How can ou determine the number of solutions of a quadratic equation of the form a + c = 0? ACTIVITY: The Number of Solutions of a + c = 0 Work with a
More informationComparing Linear and Nonlinear Functions 5.5. ACTIVITY: Finding Patterns for Similar Figures. How can you recognize when a pattern
5.5 Comparing Linear and Nonlinear Functions in real life is linear or nonlinear? How can ou recognize when a pattern ACTIVITY: Finding Patterns for Similar Figures Work with a partner. Cop and complete
More information3.1. Shape and Structure Forms of Quadratic Functions ESSENTIAL IDEAS TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS 169A
Shape and Structure Forms of Quadratic Functions.1 LEARNING GOALS KEY TERMS In this lesson, ou will: Match a quadratic function with its corresponding graph. Identif ke characteristics of quadratic functions
More informationEssential Question How can you use a quadratic function to model a real-life situation?
3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.
More information9-4. Quadratics and Projectiles. Vocabulary. Equations for the Paths of Projectiles. Activity. Lesson
Chapter 9 Lesson 9-4 Quadratics and Projectiles Vocabulary force of gravity initial upward velocity initial height BIG IDEA Assuming constant gravity, both the path of a projectile and the height of a
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 2 Stud Guide-Chapters 8 and 9 Name Date: Time: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all square roots of the number. ) 600 9,
More informationModel Inverse Variation
. Model Inverse Variation Rational Equations and Functions. Graph Rational Functions.3 Divide Polynomials.4 Simplify Rational Epressions. Multiply and Divide Rational Epressions.6 Add and Subtract Rational
More information3.2. Properties of Graphs of Quadratic Relations. LEARN ABOUT the Math. Reasoning from a table of values and a graph of a quadratic model
3. Properties of Graphs of Quadratic Relations YOU WILL NEED grid paper ruler graphing calculator GOAL Describe the ke features of the graphs of quadratic relations, and use the graphs to solve problems.
More information4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2
COMMON CORE. d Locker d LESSON Parabolas Common Core Math Standards The student is epected to: COMMON CORE A-CED.A. Create equations in two or more variables to represent relationships between quantities;
More informationFinal Exam Review Part 2 #1 Page 1 / 21
Final Eam Review Part #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: v 1. Solve for, where is a real number. v v + 1 + =. Solve for, where is a real number. +
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationCompleting the Square
5-7 Completing the Square TEKS FOCUS TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS
More informationMath 2 1. Lesson 4-5: Completing the Square. When a=1 in a perfect square trinomial, then. On your own: a. x 2 18x + = b.
Math 1 Lesson 4-5: Completing the Square Targets: I can identify and complete perfect square trinomials. I can solve quadratic equations by Completing the Square. When a=1 in a perfect square trinomial,
More informationLearning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of
More informationAnswers (Anticipation Guide and Lesson 9-1)
Answers (Anticipation Guide and Lesson 9-) Chapter Resources NAME DATE PERID 9 Anticipation Guide Quadratic and Eponential Functions Step Before ou begin Chapter 9 Read each statement. Decide whether ou
More informationOverview QUADRATIC FUNCTIONS PATTERNS IN CHANCE
Overview UNIT 7 UNIT 8 QUADRATIC FUNCTIONS Lesson 1 Quadratic Patterns....................... 462 1 Pumpkins in Flight............................... 463 2 Golden Gate Quadratics............................
More informationSelf- assessment 1010 (Intermediate Algebra)
Self- assessment (Intermediate Algebra) If ou can work these problems using a scientific calculator, ou should have sufficient knowledge to demonstrate master of Intermediate Algebra and to succeed in
More information1-D and 2-D Motion Test Friday 9/8
1-D and -D Motion Test Frida 9/8 3-1 Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocit, force, momentum A scalar has onl a magnitude. Some scalar
More informationApply Properties of Logarithms. Let b, m, and n be positive numbers such that b Þ 1. m 1 log b. mn 5 log b. m }n 5 log b. log b.
TEKS 7.5 a.2, 2A.2.A, 2A.11.C Apply Properties of Logarithms Before You evaluated logarithms. Now You will rewrite logarithmic epressions. Why? So you can model the loudness of sounds, as in E. 63. Key
More informationChapter 8 Vocabulary Check
28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator
More informationAdd, Subtract, and Multiply Polynomials
TEKS 5.3 a.2, 2A.2.A; P.3.A, P.3.B Add, Subtract, and Multiply Polynomials Before You evaluated and graphed polynomial functions. Now You will add, subtract, and multiply polynomials. Why? So you can model
More information6.1 Solving Quadratic Equations by Graphing Algebra 2
10.1 Solving Quadratic Equations b Graphing Algebra Goal 1: Write functions in quadratic form Goal : Graph quadratic functions Goal 3: Solve quadratic equations b graphing. Quadratic Function: Eample 1:
More informationCompleting the Square
6.3 Completing the Square GOAL Write the equation of a parabola in verte form by completing the square. LEARN ABOUT the Math The automated hose on an aerial ladder sprays water on a forest fire. The height
More informationf(x) = 2x 2 + 2x - 4
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
More informationEvaluate and Graph Polynomial Functions
5.2 Evaluate and Graph Polynomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polynomial functions. Why? So you can model skateboarding
More information