ࡱ>  ;=89:_E uUbjbjO O C_-a-aK###TD$D$D$D$,p'D$l,< 9999_aaaaaa$ #L99&~ 9#9__J \nS9BhcgBK0 x S #S"1I]&~  : Math Analysis CP, 2013 Due Date 12/17/2013 Polar Project 100 points The purpose of the Polar Project is to familiarize students with polar coordinates and polar equations. You will do graphing calculator investigations, solve simultaneous polar equations, create your own polar design, and research historical polar curves. You will complete this project in groups of 3 or 4 students. This project is worth 100 points and is due on Tuesday, December 17 at the beginning of the period. Ten points will be deducted for each day late. The polar project consists of the ten assignments listed below. Each assignment is worth 10 points. You will be graded on completeness, accuracy, neatness, and format. The assignments marked with a * must be completed by each individual student. The remaining assignments are to be completed as a group. Assignments 1, 6, and 7 must be typed. You will turn in: A " rigid (non-flexible) binder with all assignments in the order listed A title page (This may be put inside a "view binder", but is not required) A table of contents matching the order of the assignments Each individual assignment * clearly labeled with students name DO NOT put any pages into sheet protectors Assignments for Polar Project * Math Analysis Letter Write a letter to a new student describing this course. Be specific with your comments and suggestions. This should be about 1 full page and it must be typed. What kinds of things have you learned? Is it similar to previous math courses? Is it easier or harder than other math classes? What advice would you give to a student at the beginning of the course? * Textbook assignments All graphs must be on drawn on polar graph paper (Found on Q). P. 558 (1-3, 5 13, 15) P. 565 (1, 2, 11 19) Include a table of values (from the calculator) for each graph. P. 571 (4 12) Graphing Investigation I Print and complete the Changing Values of n Worksheet from Q. Graphing Investigation II Print and complete the Polar Curves Investigation from Q. Systems of Polar Equations Solve the followingGroups of three students must complete any six systems; groups of four students must do all eight. Solve each system algebraically and show your work. Write your answer as an ordered pair. Sketch a graph of each system. Plot and label the points of intersection on your graph. i.  EMBED Equation.DSMT4 ii.  EMBED Equation.DSMT4 iii.  EMBED Equation.DSMT4 iv.  EMBED Equation.DSMT4 v.  EMBED Equation.DSMT4 vi.  EMBED Equation.DSMT4  vii.  EMBED Equation.DSMT4 viii.  EMBED Equation.DSMT4  Spiral of Archimedes Write a brief (two or three paragraph) report on the Spiral of Archimedes. Include historical facts about Archimedes, the spiral, and why a polar curve is named after him. Sketch a graph of his curve (by hand) on polar graph paper and include its general equation. You must use at least two resources for your research and reference them using MLA format. This report must be typed and printed. Other Curves Choose two polar curves (three curves if your group has four members) from the list below. Write a brief report (2-3 paragraphs) on each curve. Sketch a graph of each curve (by hand) on graph paper and include its general equation. You must use at least two resources for your research and reference them using MLA format. This report must be typed and printed. * Original Design Create an original design using at least two polar equations on your graphing calculator. You must print the graph itself, the equations used, and your window settings. You may color or embellish your design after you have printed it. * Polar Multiple Choice WS You must print and complete Worksheet from Q. Your answers must be written on the printed worksheet in the space provided. You must show your work on the back or separate sheet. The worksheet will be graded for accuracy. Can You Study Guide and Evaluation Write a "Can You" Study Guide for this unit. List the topics a student should be able to do after completing the polar project. Include a question and an answer for each topic. Be sure each member of your group keeps a copy. Groups with four members must write a brief evaluation of the project, describing what you liked and disliked about the process. This should be at least half of a page. List of Polar Curves Astroid Catenary Conchoid Double Folium Epitrochoid Freeth's Nephroid Kampyle of Eudoxus Lituus Pearls Rhodonea Curves Talbot's Curve Trifolium Bicorn Cayley's Sextic Conchoid of de Sluze Durer's Shell Curves Equiangular Spiral Hyperbolic Spiral Kappa Curve Neile's Parabola Plateau Curves Right Strophoid Tractrix Trisectrix of Maxlaurin Cartesian Oval Cissoid of Diocles Cycloid Eight Curve Fermat's Spiral Hypocycloid Lame Curves Nephroid Pursuit Curve Serpentine Tricuspoid Tschirnhaus' Cubic Cassinian Ovals Cochleoid Devil's Curve Epicycloid Folium of Descartes Hypotrochoid Lissajous Curves Pear-shaped Quartic Quadratix of Hippias Sinusoidal Spirals Trident of Newton Watt's Curve  Note: We will be in the library on 12/11, 12/12 and 12/13. You should plan on using this time to research your Polar Curves and print your original Polar Design. Resources posted on Q (You should print these as soon as possible): Detailed grading rubric Notes Polar graph paper Graphing Investigation I Graphing Investigation II Polar Review Worksheet POLAR PROJECT RUBRIC AssignmentPoints PossibleStudent 1 Student 2Student 3Student 4 0. Binder Title Page Table of Contents/Pages Labeled 51. Letter to future student Reflective Typed, double spaced 5*2. Textbook Assignments Complete Accurate 10*3. Graphing Investigation I Neat and accurate graphs on polar grid Changes to n clearly described 104. Graphing Investigation II Complete Neat and Accurate Graphs Questions correctly answered  105. Systems of Equations Complete (6 or 8) Algebraic solutions accurate Graphs on polar grid, neat POI clearly labeled 106. Spiral of Archimedes Report Typed, double spaced Historical facts provided Graph and equation provided 2 references, MLA format 107. Other Polar Curves Reports Typed, double spaced Historical facts provided Graph and equation provided 2 references, MLA format 108. Original Design At least 2 equations Graph, Eqn, and Window Printed Title 10*9. Review Worksheet Complete Accurate 10*10. Study Guide/Group Evaluation Complete Typed, double spaced 10TOTAL  100 WS- Graphing Investigation I Changing Values of n Graph the following equations on the polar grid provided using different colored pens or pencils. Graph  EMBED Equation.DSMT4  in blue. Graph  EMBED Equation.DSMT4  in green. Graph  EMBED Equation.DSMT4  in red. Graph  EMBED Equation.DSMT4  in yellow. Verbally describe the changes to each curve as n changes: Graph the following equations on the polar grid provided using different colored pens or pencils. Graph  EMBED Equation.DSMT4  in blue. Graph  EMBED Equation.DSMT4  in green. Graph  EMBED Equation.DSMT4  in red. Graph  EMBED Equation.DSMT4  in yellow. Verbally describe the changes to each curve as n changes: Graph the following equations on the polar grid provided using different colored pens or pencils. Graph  EMBED Equation.DSMT4  in blue. Graph  EMBED Equation.DSMT4  in green. Graph  EMBED Equation.DSMT4  in red. Verbally describe the changes to each curve as n changes: Graph the following equations on the polar grid provided using different colored pens or pencils. Graph  EMBED Equation.DSMT4  in blue. Graph  EMBED Equation.DSMT4  in green. Graph  EMBED Equation.DSMT4  in red. Verbally describe the changes to each curve as n changes: WS- Graphing Investigation II Polar Curves Investigation In this investigation, you will look at different equations and their graphs. Many of the pictures will look neat! Later, you will explore a design using your own equations. It will be helpful to have a grasp of the terms on your calculator. Switch your calculator mode to POL (Polar Mode). Pressing Y= now takes you to a menu containing EMBED Equation.DSMT4. Further, your calculator will no longer display an "X" when you press the  button. Instead, it will display EMBED Equation.DSMT4. So you will input an angle measurement and will get an output that is the length of a point to the origin. Thus, you are graphing the ordered pairs EMBED Equation.DSMT4. Finally, your window will be extremely important. In general, your window settings should be as follows. Although you may use radians, it is recommended that you use degrees. EMBED Equation.DSMT4 Min: 0 or EMBED Equation.DSMT4X Min: -6Y Min: -4EMBED Equation.DSMT4 Max: 360 or EMBED Equation.DSMT4X Max: 6Y Max: 4EMBED Equation.DSMT4 Step: 2 or 0.03 RadiansX Scl: 1Y Scl: 1For different problems, however, you may wish to change the max/min on the two axes to better see the graph. The Rose General Form: EMBED Equation.DSMT4 or EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 0.25 Dist. Between Circles: 0.5 Dist. Between Circles: 1 Complete the table for EMBED Equation.DSMT4 Complete the table for EMBED Equation.DSMT4 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 So, what is the general shape of the Rose graph? _______________________________________________________________ In general, what does the a in EMBED Equation.DSMT4 do? ______ Graph: EMBED Equation.DSMT4 (Use the table below) Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 So in general, what does the b in EMBED Equation.DSMT4 do? How does the graph change when the b value is even vs. odd? Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 (Use the table) Dist. Between Circles: 0.25 Dist. Between Circles: 0.5 Dist. Between Circles: 1 How do the cosine Rose graphs differ from the sine graphs? ____________ Complete the table for EMBED Equation.DSMT4 Complete the table for EMBED Equation.DSMT4 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 The Leminscate General Form: EMBED Equation.DSMT4 or EMBED Equation.DSMT4 NOTE: You must take the square root of both sides since it is EMBED Equation.DSMT4. Graph: EMBED Equation.DSMT4 (Use the table) Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 0.25 Dist. Between Circles: 1 Dist. Between Circles: 1 What is the general shape of the Leminscate? _________________________________________________________________ How do the sine Leminscates differ from the cosine graphs? ______________________________________________________ _______________________________________________________________________________________________________ Now lets see what happens when a is imaginary Graph: EMBED Equation.DSMT4 (Use the table) Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 0.25 Dist. Between Circles: 1 Dist. Between Circles: 1 So what does the a do in EMBED Equation.DSMT4 or EMBED Equation.DSMT4? Be sure to discuss both real and imaginary values. _____________________________________________________________________________ Complete the table for EMBED Equation.DSMT4 Complete the table for EMBED Equation.DSMT4 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 The Limacon General Form: EMBED Equation.DSMT4 or EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 So, what is the basic shape (s) of the Limacon graph? ___________________________________________________________ _______________________________________________________________________________________________________ How do the sine and cosine graphs differ? ____________________________________________________________________ _______________________________________________________________________________________________________ In general, what do a and b in EMBED Equation.DSMT4 or EMBED Equation.DSMT4 do? ______ What is/are the requirements to have a loop in the middle versus just an indent? Starting with EMBED Equation.DSMT4 can you find a value for a that does not make a loop or an indentation? ______________________ _______________________________________________________________________________________________________ Create a cosine equation where there is a loop in the middle: Create a sine equation where there is only an indentation: The Cardioid (A Special Limacon) General Form: EMBED Equation.DSMT4 or EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 (Use the table) Graph: EMBED Equation.DSMT4 Graph: EMBED Equation.DSMT4 (Use the table) Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 Why do you suppose we call this graph a Cardoid? ____________ In general, what does the value of a in EMBED Equation.DSMT4 or EMBED Equation.DSMT4 do? _______________________________________ _______________________________________________________________________________________________________ What, if anything, changes when the "+" is changed to a "-"? Complete the table for EMBED Equation.DSMT4 Complete the table for EMBED Equation.DSMT4 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 EMBED Equation.DSMT4rEMBED Equation.DSMT4r021030225452406027090300120315135330150360180 Worksheet #01- Polar Project Review Write the letter of the correct answer on the blank. YOU MUST INCLUDE YOUR WORK!!! Which is a value for EMBED Equation.DSMT4 if the points EMBED Equation.DSMT4 and EMBED Equation.DSMT4 are equivalent? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the graph of EMBED Equation.DSMT4? a pointa vertical linea horizontal line Which is the polar equation for a circle with center at the pole and radius EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 Which of the following is an equation of a spiral of Archimedes? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 How many petals does the rose given by EMBED Equation.DSMT4 have? 28164 What is the name of the classical curve represented by  EMBED Equation.DSMT4  ? rosecardiodleminscatelimacon What is the solution of this system of equations? EMBED Equation.DSMT4 EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 & EMBED Equation.DSMT4EMBED Equation.DSMT4 & EMBED Equation.DSMT4 What are the polar coordinates of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What are the rectangular coordinates of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the polar form of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 Which of the following is equivalent to EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the simplest form of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the polar form of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the rectangular form of EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is the product of EMBED Equation.DSMT4 and EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is EMBED Equation.DSMT4 EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is EMBED Equation.DSMT4? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 What is EMBED Equation.DSMT4? 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