Graphing Quadratics Using the TI-83

Transcription

1 Graphing Quadratics Using the TI-83 9 th Grade Algebra Paul Renzoni 12/01/02 I2T2 Project

2 Table of Contents: Unit Objectives, NYS Standards, NCTM Standards 3 Resources 4 Materials and Equipment 5 Overview 6 Day 1 7 Day 2 12 Day 3 16 Day 4 21 Day 5 23 Unit Test 26 2

3 Unit Objectives 1. Students will be able to solve quadratic equations using the quadratic formula. 2. Students will be able to identify and use the properties of quadratic equations. 3. Students will be able to use quadratic equations to solve problems about paths of projectiles. 4. Students will be able to graph equations of the form y = ax 2 + bx + c. New York State Standards: (Math A) 2A Understand and use rational and irrational numbers. 3A Use addition, subtraction, multiplication, division and exponentiation with real numbers and algebraic expressions. 7A Represent and analyze functions using verbal descriptions, tables, equations, and graphs. 7B Apply linear and quadratic functions in the solution of problems. 7C Translate among the verbal descriptions, tables, equations, and graphic forms of functions. 7D Model real-world situations with appropriate functions. NCTM Standards: Numbers and Operations Algebra Communication Representations 3

4 Resources Scott Foresman Addison Wesley. UCSMP Algebra. Chapter 9, Sections , including Lesson Master 9-3B, pgs (1998) Quadratics Polynomial Form, No author given. 4

5 Materials and Equipment Needed UCSMP Algebra Text Class Set of TI-83 Calculators Overhead with Calculator unit Computers with internet access 5

6 Overview: Day1: Students will be assigned group projects that will be one assessment. Students will use TI-83 graphing calculators to explore variations of y = ax 2. Day 2: Students will explore graphing y = ax2 + bx + c on the website Exploremath.com. Day 3: Students will work with partners on Lesson Master 9-3B, Graphing with an Automatic Grapher. (TI-83) Day 4: Student will explore real-world examples of parabolas. Day 5: Students will solve quadratic equations both with Quadratic Formula and PolySmlt App on the TI-83. 6

7 Day 1 Lesson Plan: Objectives: 1. Students will be able to graph and interpret equations of the form y = ax Students will be able to recognize axis of symmetry from a table of values and from a graph. 3. Students will be able to solve equations of the form ax 2 = k. Standards: NCTM Standards covered: Algebra, Representation NYS Standards covered: 3A, 7A, 7C Materials: Graphing calculators Student worksheet and overhead transparency of worksheet Overhead with calculator unit Opening Activity: Students will be given a card with a number 1 7 on it when they enter the room. There will be four of each card. These represent project numbers on pgs The students will be given 10 minutes to meet with their group members to discuss the project, which will be due the day after the unit test. This will be one of the assessments. Developmental Activity: Students will return to their seats and work with their partners to complete the worksheet, Exploring y = ax 2. Students will then be selected to present their solutions to the class either at the board or on the overhead TI-83 unit. Ticket Out: Students will have the last 5 minutes of class to respond to the following question, also to address any concerns they had with the lesson. What are the two most important pieces of information that are determined by the a in the equation y = ax 2? Homework: Read pgs , complete pgs # 5 9, 12, 13. 7

8 Teacher s Notes: Solutions to Developmental Activity: #1 and #2: #3: Answers will vary #4 and #5: #6: Answers will vary Ticket Out: Answers will be collected as the students exit the room. The information will be used to assess the students understanding of the lesson covered. 8

9 Solutions to Homework: 5b. 5c. A parabola which opens up whose axis of symmetry is x = 0 and whose vertex is (0,0). 6b. 6c. A parabola which opens down whose axis of symmetry is x = 0 and whose vertex is (0,0). 7. (0,0) 8. x = ft. 9. up, down ft. 9

10 Exploring y = ax 2 Name: Period: Directions: With your partner complete the following questions using your graphing calculator to create graphs. 1. Graph the following equations in the y = window and use zoom standard to view the graphs. a. y = x 2 b. y = 2x 2 c. y =.5x 2 d. y = 4x 2 e. y =.25x 2 2. Sketch the graphs on the grid below. 3. What happens to the graph of y = ax 2 when the value of a gets larger? gets smaller? 10

11 4. Graph the following equations in the y = window and use zoom standard to view the graphs. a. y = x 2 b. y = -x 2 c. y = 2x 2 d. y = -2x 2 5. Sketch the graphs in the grid below. 6. What happens to the graph of y = ax 2 when a changes positive to negative? 11

12 Day 2 Lesson Plan: Objectives: 1. Students will be able to interpret the graphs of equations of the form y = ax 2 + bx + c. 2. Students will be able to identify the vertex, axis of symmetry, y-intercept, and x- intercept(s) if they exist. Standards: NCTM Standards covered: Algebra, Representation NYS Standards covered: 7A, 7C Materials: Computers with internet access Worksheets for students Opening Activity: Students will enter computer lab, take their seats, and log in to the computers as previously instructed. They will given a worksheet as they enter, it will have the website they have to find. Developmental Activity: Students will use the website to answer the questions on the worksheet regarding the graph of y = ax 2 + bx + c. Ticket Out: Students will have the last 5 minutes of class to respond to the following question, also to address any concerns they had with the lesson. What happens to the graph of y = ax 2 + bx + c when the value of c changes? Homework: Read pgs , complete pgs #5, 7, 9, 12 12

13 Teacher s Notes: Solutions to Developmental Activity: 1. Equation of parabola Vertex x-intercept(s) y-intercept Axis of symmetry y = x 2 (0, 0) (0, 0) (0, 0) x = 0 y = x (0, 2) None (0, 2) x = 0 y = x 2 2 (0, -2) 1.41 and 1.41 (0, -2) x = 0 y = x 2 + x (-.5, -.25) 0 and 1 (0, 0) x = -.5 y = x 2 + 5x (-2.5, -6.25) 0 and 5 (0, 0) x = -2.5 y = x 2 5x (2.5, -6.25) 5 and 0 (0, 0) x = 2.5 y = x 2 3x +2 (1.5, and 1 (0, 2) x = 1.5 y = x 2 4x (-2,-4) 0 and 4 (0, 0) x = -2 y = x 2 3x 4 (1.5, -6.25) 4 and 1 (0, -4) x = Answers will vary. 3. a: Changing a will make the parabola more narrow or wide. Changing the sign of a will make the graph open up or down. b: Changing b will move the vertex of the parabola. c: Changing c will change the y-intercept. Ticket Out: Answers will be collected as the students exit the room. The information will be used to assess the students understanding of the lesson covered. 13

14 Solutions to Homework: 5. True 7b. 7c. (1, -4); minimum 7d. 3 9a. (-3,1) 9b. x = -3 12a. (-4, 35) 12b. x = -4 12c. 19, 5,

15 Name: Period: Exploring y = ax 2 + bx + c Directions: Log on to the computer and go to the following website: Set the graph tool so that a = 1, b = 0, and c = 0. Click in the box that says show vertex/intercept data. 1. Complete the table below: Equation of parabola Vertex x-intercept y-intercept Axis of symmetry y = x 2 y = x y = x 2 2 y = x 2 + x y = x 2 + 5x y = x 2 5x y = x 2 3x +2 (-2,-4) 0 and 4 (0, 0) (1.5, -6.25) 4 and 1 (0, -4) 2. Was it difficult to graph the parabola given only the vertex and the intercepts? 3. For each letter a, b, and c give a general rule for what happens when you change only that value. a: b: c: 15

16 Day 3 Lesson Plan: Objectives: 1. Students will be able to graph and interpret equations of the form y = ax 2 + bx + c. Standards: NCTM Standards covered: Algebra, Representation NYS Standards covered: 7A, 7C Materials: Graphing calculators Lesson Master 9-3B and overhead transparency of worksheet Overhead with calculator unit Opening Activity: The students will answer the following questions upon entering the room. Tell what you know about a, b, or c in the equation y = ax 2 + bx + c if 1. its vertex is its minimum point 2. the y-axis is the axis of symmetry of the graph 3. The point (0, 6) is on the graph. Developmental Activity: The students will work with a partner to complete Lesson Master 9-3B. The students will then be selected to present their answers to the class using the overhead calculator and the board. Ticket Out: Students will have the last 5 minutes of class to respond to the following question, also to address any concerns they had with the lesson. When graphing on a TI-83 how important is it to select the correct window size? Homework: Read pgs , complete pg. 565 #

17 Teacher s Notes: Solutions to Opening Activity: 1. The value of a must be greater than b = 0 3. c = 6 Solutions to Developmental Activity: 1. a. 5 b. 4 c. x = 5 2a. 2b. They will all open down and have the y-axis as their axis of symmetry. Their vertices are at different points, and their graphs appear to get narrower. 2c. It opens down and has the y-axis as its axis of symmetry. The vertex is at (0, -10). It is quite narrow < x < 15, -40 < y < < x < 20, -10 < y < < x < 3, 0 < y < 5 6a. (-10, -144) b. x = -10 c and 5.3 7b. (-1, 2), (3, 10) 17

18 Solutions to Homework: < x < 10, -10 < y < < x < 10, -25 < y < 10 9a. 9b. (-9, -12) 9c. x = -9 9d. 12 and 7 10a. 3 10b

19 19

20 20

21 Day 4 Lesson Plan: Objectives: 1. Students will be able to use quadratic equations to solve problems about paths of projectiles. Standards: NCTM Standards covered: Algebra, Representation NYS Standards covered: 7A, 7C, 7D Materials: Blank sheet of overhead transparency Overhead Opening Activity: As students enter the room they will be divided into their groups to discuss their projects. The students will be given the first 10 minutes of class to discuss project progress and the teacher will circulate about the room to answer questions. Developmental Activity: The students will remain in their groups to discuss the following question. List as many objects as you can that are either in the shape of a parabola or travel in the path of a parabola. For example the water that comes out of the drinking fountain. The groups will have time to discuss and then the teacher will pass around the blank sheet of transparency and have the groups add to the list as it gets to them. We will then discuss as a class. Ticket Out: Students will have the last 5 minutes of class to respond to the following question, also to address any concerns they had with the lesson. Are there any items on the list that need to be further discussed to prove to you that their paths a parabolas? Homework: Read pgs , complete pg. 569 #

22 Teacher s Notes: Developmental Activity: This activity is designed to make the students relate parabolas to their everyday life. It will also help them to better understand the reading and the homework problems. Solutions to Homework: meters meters 14. after 2 and 4 seconds meters 22

23 Day 5 Lesson Plan: Objectives: 1. Students will be able to solve quadratic equations using the quadratic formula. 2. Students will be able to solve quadratic equations using the PolySmlt App. Standards: NCTM Standards covered: Algebra, Representation NYS Standards covered: 2A, 3A, 7A, 7C Materials: Graphing calculators and overhead unit Worksheet for students with overhead transparency Opening Activity: The students will answer these questions when they enter the room. Evaluate each expression when a = a, b = -5, and c = 1. Developmental Activity: The students will work with a partner to complete the Quadratic Formula worksheet. Students will be called to the board to show solutions. Ticket Out: Students will have the last 5 minutes of class to respond to the following question, also to address any concerns they had with the lesson. What happens when b 2 4ac is a negative number? Homework: Read pgs , complete pg. 577 # 5 8,

24 Teacher s Notes: Solutions to Opening Activity: _ Solutions to Developmental Activity: 1. x = 1.5 or x = x = 5 or 5 4. x = 2 or 5 5. x = - 2/3 or 7/4 6. no real solutions Solutions to Homework: 5a. a = 12, b = 7, c = 1 b. x = - _ or x = 1/3 6a. a = 3, b = 1, c = -2 b. x = 2/3 or x = -1 7a. a = 1, b = 6, c = 9 b. x = -3 8a. a = -1, b = 0, c = 4 b. x = -2 or x = 2 15a. 2.5 and b. 24

25 Solving Quadratics Name: Period: Directions: Solve each of the following quadratic equations using the Quadratic Formula listed below. Show all work. 1. 2x 2 + x 6 = 0 2. x 2 + 4x + 4 = 0 3. x 2 25 = 0 4. x 2 7x + 10 = x 2 13x 14 = 0 6. x 2 + 2x + 2 = 0 Directions: Using the same six equations above check your answers using the PolySmlt App on your calculator. Choose Poly Root Finder with degree 2. Remember a 2 = a, a 1 = b, and a 0 = c

26 Name: Period: Unit Test Graphing and Solving Quadratics Directions: Answer all questions on this test. You may use your graphing calculator for any problem on this test. You must show work on the Quadratic Formula questions but you may check your answers with the PolySmlt App. Graph the following equations. Also draw the axis of symmetry. 1. y = 3x 2 2. y = 2x 2 4x y = -x y = 4x 2 4x 8 26

27 For questions 5 and 6 identify the vertex, the axis of symmetry, and the y-intercept. 5. y = x y = 2x 2 4x + 5 vertex: axis of symmetry: y-intercept: vertex: axis of symmetry: y-intercept: 7. One of the first astronauts who traveled to the moon hit a golf ball on the moon. Suppose that the height h in meters of a ball t seconds after it is hit is described by h = 0.8t t. a. Graph the equation. b. Find the times at which the ball is at a height of 20 meters. Solve for x using the Quadratic Formula. Show all work! 8. 3x 2 + 6x 9 = x 2 5x 12 = x 2 +70x + 14 = 0 27

28 Answer Key to Unit Test y = x y = 2x 2 4x + 5 vertex: (0, -25) vertex: (1, 3) axis of symmetry: x = -25 axis of symmetry: x = 1 y-intercept: (0, -25) y-intercept: (0, 5) 28

29 7a. 7b. 2.5 and 10 seconds 8. 3x 2 + 6x 9 = x 2 5x 12 = x 2 +70x + 14 = 0 x = 1 or 3 x = -3/2 or 4 x = -1/4 or 1 29