Graphing Quadratics Using the TI-83 - Buffalo State Collegemath.buffalostate.edu/~it/projects/renzoni.pdf · Graphing Quadratics Using the TI-83 9th Grade Algebra ... 606. The students

Graphing QuadraticsUsing the TI-83

9th Grade Algebra

Paul Renzoni12/01/02

I2T2 Project

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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

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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 = ax2 + 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 realnumbers and algebraic expressions.

7A Represent and analyze functions using verbal descriptions, tables, equations, andgraphs.

7B Apply linear and quadratic functions in the solution of problems.

7C Translate among the verbal descriptions, tables, equations, and graphic forms offunctions.

7D Model real-world situations with appropriate functions.

NCTM Standards:

• Numbers and Operations

• Algebra

• Communication

• Representations

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Resources

Scott Foresman Addison Wesley. UCSMP Algebra. Chapter 9, Sections 9-1 – 9-5,including Lesson Master 9-3B, pgs546-577. (1998)

www.exploremath.com/activities/Activity_page.cfm?ActivityID=13, QuadraticsPolynomial Form, No author given.

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Materials and Equipment Needed

• UCSMP Algebra Text

• Class Set of TI-83 Calculators

• Overhead with Calculator unit

• Computers with internet access

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Overview:

Day1:

Students will be assigned group projects that will be one assessment. Students will useTI-83 graphing calculators to explore variations of y = ax2.

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 AutomaticGrapher. (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 Appon the TI-83.

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Day 1

Lesson Plan:

Objectives:

1. Students will be able to graph and interpret equations of the form y = ax2.2. 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 ax2 = 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. Therewill be four of each card. These represent project numbers on pgs. 605 – 606. Thestudents will be given 10 minutes to meet with their group members to discuss theproject, 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 = ax2. Students will then be selected to present their solutions to the classeither 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 toaddress any concerns they had with the lesson.

What are the two most important pieces of information that are determined by the a in theequation y = ax2?

Homework:

Read pgs. 548 – 551, complete pgs. 551-553 # 5 – 9, 12, 13.

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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 toassess the students understanding of the lesson covered.

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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 = 0 12. 144 ft.9. up, down 13. 100 ft.

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Exploring y = ax2

Name: ____________________

Period: _____

Directions: With your partner complete the following questions using your graphingcalculator to create graphs.

1. Graph the following equations in the y = window and use zoom standard to view thegraphs.

a. y = x2

b. y = 2x2

c. y = .5x2

d. y = 4x2

e. y = .25x2

2. Sketch the graphs on the grid below.

3. What happens to the graph of y = ax2 when the value of a gets larger? gets smaller?

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4. Graph the following equations in the y = window and use zoom standard to view thegraphs.

a. y = x2

b. y = -x2

c. y = 2x2

d. y = -2x2

5. Sketch the graphs in the grid below.

6. What happens to the graph of y = ax2 when a changes positive to negative?

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Day 2

Lesson Plan:

Objectives:

1. Students will be able to interpret the graphs of equations of the form y = ax2 + 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 aspreviously instructed. They will given a worksheet as they enter, it will have the websitethey have to find.


Students will use the website to answer the questions on the worksheet regarding thegraph of y = ax2 + bx + c.

Ticket Out:


What happens to the graph of y = ax2 + bx + c when the value of c changes?

Homework:

Read pgs. 554 – 557, complete pgs. 558 – 559 #5, 7, 9, 12

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Teacher’s Notes:


1.Equation of parabola Vertex x-intercept(s) y-intercept Axis of symmetryy = x2 (0, 0) (0, 0) (0, 0) x = 0y = x2 + 2 (0, 2) None (0, 2) x = 0y = x2 – 2 (0, -2) 1.41 and –1.41 (0, -2) x = 0y = x2 + x (-.5, -.25) 0 and –1 (0, 0) x = -.5y = x2 + 5x (-2.5, -6.25) 0 and –5 (0, 0) x = -2.5y = x2 – 5x (2.5, -6.25) 5 and 0 (0, 0) x = 2.5y = x2 – 3x +2 (1.5, -.25 2 and 1 (0, 2) x = 1.5y = x2 – 4x (-2,-4) 0 and –4 (0, 0) x = -2y = x2 – 3x – 4 (1.5, -6.25) 4 and –1 (0, -4) x = 1.5

2. Answers will vary.

3. a: Changing a will make the parabola more narrow or wide. Changing the sign of awill 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 toassess the students understanding of the lesson covered.

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5. True7b.

7c. (1, -4); minimum

7d. –3

9a. (-3,1)

9b. x = -3

12a. (-4, 35)

12b. x = -4

12c. 19, 5, -13

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Exploring y = ax2 + bx + c

Name: ____________________

Period: _____

Directions: Log on to the computer and go to the following website:

http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=13

Set the graph tool so that a = 1, b = 0, and c = 0. Click in the box that says showvertex/intercept data.

1. Complete the table below:

Equation of parabola Vertex x-intercept y-intercept Axis of symmetryy = x2

y = x2 + 2y = x2 – 2y = x2 + xy = x2 + 5xy = x2 – 5xy = x2 – 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 onlythat value.

a:

b:

c:

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Day 3

Lesson Plan:

Objectives:

1. Students will be able to graph and interpret equations of the form y = ax2 + 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 = ax2 + bx + c if1. its vertex is its minimum point2. the y-axis is the axis of symmetry of the graph3. The point (0, 6) is on the graph.


The students will work with a partner to complete Lesson Master 9-3B. The students willthen be selected to present their answers to the class using the overhead calculator and theboard.

Ticket Out:


When graphing on a TI-83 how important is it to select the correct window size?

Homework:

Read pgs. 562 – 564, complete pg. 565 # 7 –10.

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Teacher’s Notes:

Solutions to Opening Activity:

1. The value of a must be greater than 0.

2. b = 0

3. c = 6


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. Theirvertices 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). Itis quite narrow.

3. 5 < x < 15, -40 < y < 10 4. 5 < x < 20, -10 < y < 50

5. 0 < x < 3, 0 < y < 5

6a. (-10, -144) b. x = -10 c. –14.7 and –5.3

7b. (-1, 2), (3, 10)

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7. –10 < x < 10, -10 < y < 30 8. –5 < x < 10, -25 < y < 10

9a.

9b. (-9, -12) 9c. x = -9 9d. –12 and –7

10a. 3 10b. 16

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20

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Day 4

Lesson Plan:

Objectives:

1. Students will be able to use quadratic equations to solve problems about paths ofprojectiles.

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 theteacher will circulate about the room to answer questions.


The students will remain in their groups to discuss the following question. List as manyobjects as you can that are either in the shape of a parabola or travel in the path of aparabola. For example the water that comes out of the drinking fountain. The groupswill have time to discuss and then the teacher will pass around the blank sheet oftransparency and have the groups add to the list as it gets to them. We will then discussas a class.

Ticket Out:


Are there any items on the list that need to be further discussed to prove to you that theirpaths a parabolas?

Homework:

Read pgs. 567 – 569, complete pg. 569 # 12 –15.

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Teacher’s Notes:


This activity is designed to make the students relate parabolas to their everyday life. Itwill also help them to better understand the reading and the homework problems.


12. 45 meters

13. 25 meters

14. after 2 and 4 seconds

15. 35 meters

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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.


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:


What happens when b2 – 4ac is a negative number?

Homework:

Read pgs. 573 – 576 , complete pg. 577 # 5 – 8, 15.

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Teacher’s Notes:

Solutions to Opening Activity:

1. 5 2. 9 3. 3

4. 8 5. 2 6. 1

7. _


1. x = 1.5 or - 2 2. x = -2

3. x = 5 or – 5 4. x = 2 or 5

5. x = - 2/3 or 7/4 6. no real solutions


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 5.015b.

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Solving Quadratics

Name: ____________________

Period: _____

Directions: Solve each of the following quadratic equations using the QuadraticFormula listed below. Show all work.

1. 2x2 + x – 6 = 0 2. x2 + 4x + 4 = 0

3. x2 – 25 = 0 4. x2 – 7x + 10 = 0

5. 12x2 – 13x – 14 = 0 6. x2 + 2x + 2 = 0

Directions: Using the same six equations above check your answers using the PolySmltApp on your calculator. Choose Poly Root Finder with degree 2. Remember a2 = a, a1 =b, and a0 = c.

1. 2.

3. 4.

5. 6.

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Unit TestGraphing and Solving Quadratics

Name: ____________________

Period: _____

Directions: Answer all questions on this test. You may use your graphing calculator forany problem on this test. You must show work on the Quadratic Formula questions butyou may check your answers with the PolySmlt App.

Graph the following equations. Also draw the axis of symmetry.

1. y = 3x2 2. y = 2x2 – 4x + 2

3. y = -x2 + 9 4. y = 4x2 – 4x – 8

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For questions 5 and 6 identify the vertex, the axis of symmetry, and the y-intercept.

5. y = x2 – 25 6. y = 2x2 – 4x + 5

vertex: vertex:

axis of symmetry: axis of symmetry:

y-intercept: 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 byh = 0.8t2 + 10t.

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. 3x2 + 6x – 9 = 0 9. 2x2 – 5x – 12 = 0 10. 56x2 +70x + 14 = 0

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Answer Key to Unit Test

1. 2.

3. 4.

5. y = x2 – 25 6. y = 2x2 – 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)

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7a.

7b. 2.5 and 10 seconds

8. 3x2 + 6x – 9 = 0 9. 2x2 – 5x – 12 = 0 10. 56x2 +70x + 14 = 0

x = 1 or –3 x = -3/2 or 4 x = -1/4 or –1

Documents

Graphing Quadratics Using the TI-83 - Buffalo State Collegemath.buffalostate.edu/~it/projects/renzoni.pdf · Graphing Quadratics Using the TI-83 9th Grade Algebra ... 606. The students