ࡱ>  rGfz] Kbjbj:/:/ !XEbXEbyA+ HH4'4'4'Dx'x'x''<4+x'(P/< <<<<   :<<<<<<$`4'L `HH<<u666<H~ <4'<:6:66f2 p"n< j4ei(D&07774'p "+6CW   ``<   7          B : Math Analysis CP, 2017 Due Date 12/11/2017 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 2 or 3 students. This project is worth 100 points and is due on Monday, December 11 at the beginning of the period. Ten points will be deducted for each day late. The polar project consists of the nine 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. You will turn in: A " rigid (non-flexible) binder with all assignments in the order listed A creative, original title page (This may be put inside a "view binder", but is not required) A table of contents matching the outline below. Assignments should be in this order. Each individual assignment * clearly labeled with students name DO NOT put any pages into sheet protectors; include one copy of the rubric Assignments for Polar Project A description of the polar coordinate system with a short paragraph about its history. Be sure to cite your sources. A description of conversions between polar and rectangular points. Include several examples of each, showing conversions in both directions. Transitions Worksheet Distance in Polar Coordinate System Worksheet Graphing Investigation I Print and complete the Changing Values of n Worksheet from Website. Graphing Investigation II Print and complete the Polar Curves Investigation from Website. Systems of Polar Equations Solve the following. . . Choose 6 of the 8 to solve. 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 Other Curves Choose three polar curves 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 three polar equations on Desmos. You must print the graph itself and the equations used. You may color or embellish your design after you have printed it. * Polar Multiple Choice WS You must print and complete Worksheet from Website. 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. 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  Resources posted on Website (You should print these as soon as possible): Detailed grading rubric Polar graph paper Graphing Investigation I (Included in this document) Graphing Investigation II (Included in this document) Polar Multiple Choice Worksheet (Included in this document) Transitions WS Distance WS 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? EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4EMBED Equation.DSMT4 Which of the following is a root of the equation EMBED Equation.DSMT4? 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