P a g e 1 Presented, and Compiled, By Bryan Grant Jessie Ross August 3 rd, 2016
P a g e 2 Day 1 Discovering Polar Graphs Days 1 & 2 Adapted from Nancy Stephenson - Clements High School, Sugar Land, Texas Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equations on your calculator (or DESMOS), sketch the graphs on this sheet, and answer the questions. 1. r 2cos r 3cos r 3cos r 2sin r 3sin r 3sin Describe the graphs in part 1. Make up a generic name for the graphs in part 1. What is the actual name of this type of graph? (TOGETHER!) 2. r 2 2cos r 1 2cos r 2 cos r 2 2sin r 1 2sin r 2 sin What determines whether the graph goes through the origin? Which graphs have an inner loop? Why? What is your name for these graphs? What is the actual name?
3. r 2cos3 r 3cos5 r 2sin 3 r 3sin 5 P a g e 3 What determines the number of petals? How can you tell the difference between sine and cosine petals? What is your name for these graphs? What is the actual name? 4. r 3cos2 r 2cos 4 r 3sin 2 r 2sin 4 What s different between the graphs in part 3 and part 4? How can you tell the difference between sine and cosine petals? What is your name for these graphs? What is the actual name? 2 2 5. r 4cos 2 r 4sin 2 r How did you graph the first two equations? Did you need the ±? What equation would make the infinity sign symmetric about the y axis? What are your names for these graphs? What are the actual names? Unit Question: WHY do polar graphs have their unique shapes?
P a g e 4 Day 2 Notes - Polar Coordinates and Polar Graphs Days 1 & 2 Adapted from Nancy Stephenson - Clements High School, Sugar Land, Texas Goal: Develop fluency moving between rectangular and polar coordinates and equations. Rectangular coordinates are in the form (x, y), where x is the movement on the x axis, and y on the y axis. Polar coordinates are in the form (r, θ), where r is the length of the radius of a circle, and θ is how many radians you rotate counter-clockwise (think trig) about the origin. Graph the following polar coordinates: A4, 2 3 B 3, 7 6 5 C 2, 4 D 2, 6 Given a right triangle with angle θ at the origin: Using the triangle, you know that: cos θ = so x = sin θ = so y = x 2 + y 2 = so r = Lastly, tan(θ) = 5 Ex. Convert 2, 6 to rectangular coordinates. Ex. Convert 3, 3 to polar coordinates.
P a g e 5 Ex. Convert the following equations to polar form. (a) y = 4 (b) x 2 2 y 25 Ex. Convert the following equations to rectangular form, and sketch the graph. (a) rsin 3 (b) r 2cos 2 (c) 3 Work the following problems. Please do NOT use your calculator. Convert the following equations to polar form. 1. y = 3 2. 3x 5y 2 0 3. x 2 2 y 25 Convert the following equations to rectangular form. 5 4. r 3sec 5. r 2sin 6. 6
P a g e 6 Day 2 Homework: Polar Practice 1. Graph each point below. Then, convert each ordered pair to Cartesian (rectangular) coordinates. (no decimals; use your special triangles if you forget the formulas!). Make sure the point in rectangular coordinates matches the location of the point in polar form. a. (2, π 4 ) b. ( 2, 7π 4 ) c. (3, π 6 ) d. ( 4, π 2 ) e. (5, 2π 3 ) 2. Convert the following Cartesian points to Polar. a. (1, 1) b. (0, 4) c. ( 3, 1) d. (3, 4) e. (0, 2) f. ( 3, 3)
P a g e 7 3. Convert the following polar equations to Cartesian. a. ) r sin θ = 0 b. ) r = 4 csc θ c. ) r cos θ + r sin θ = 1 d. ) r 2 = 1 e. ) r = 5 sin θ 2 cos θ f. ) r = cot θ csc θ 4. Convert the following Cartesian equations to Polar. a. ) x = 7 b. ) y 2 = 4x c. ) x = y d. ) x 2 + (y 2) 2 = 4 e. ) x 2 + y 2 = 4 f. ) (x + 3) 2 + y 2 = 9
P a g e 8 Day 3 Understanding Polar Graphs Days 3 & 4 Adapted from Alicia Goldner Butler Senior High School Pennsylvania 1.) r = 4 sin(2θ) a. What do you expect this graph to look like? What is it called? b. Fill in the values for r in the first table below, and sketch a graph of the equation on the plane below. θ 0 r θ r π 12 π 4 5π 12 π 2 1 2 3 4 5 r 7π 12 3π 4 11π 12 2π π c. Why did we choose those specific values for θ? If the equation was r = 4 sin(3θ), what would the first four nice values for theta be in your table instead of {0, π 12, π 4, 5π 12 }? d. Notice our table only gave the part of the graph in quadrants 1 and 4. Start the second table with the value θ = 13π. Fill in the nice values of θ up through 2π. Find r (13π). Using the first table, how can you get 12 12 the values of r in the rest of the table with very little work? Sketch the rest of the graph. e. How many times does the graph touch the origin on the interval [0,2π]? Using Algebra, find the exact theta values where this occurs. How could those points be useful in graphing the rose? What other values might be useful?
P a g e 9 2.) r = 1 + 2 cos(θ) a. What do you expect this graph to look like? What is it called? b. There are three tables below. Fill in ONLY the top table (the first four values of theta given). Then, graph only the part of the equation represented by the first table. θ 0 π 3 r π 2 2π 3 θ r 3π 4 π 7π 6 4π 3 1 2 3 4 5 r θ 2π r c. How many times will the graph touch the origin on [0,2π]? Using Algebra, find the exact theta values where this occurs. d. What happens to r for values of θ between ( 2π, 4π )? Use the range of cosine to explain your answer. 3 3 e. Fill in the second table above. If your answers are not integers, try to approximate the values without using a calculator. Graph those points. f. Fill in the last table by choosing three other values of theta in addition to 2π and finding their respective outputs. Then, complete the graph. g. What theta values generate the inner loop of the graph? WHY do they make the inner loop?
P a g e 10 Day 3 Practice Problems (Separate Piece of Paper) 1. For each Rose Petal Curve, identify (a) the equation and (b) the interval of theta which will create only one petal. A. B. C. D. E. F. 2. Given the equation r = 2 cos(3θ) a. Using Algebra, determine how many times the graph goes through the origin on [0,2π]. b. What is the largest possible value of r generated by this equation? Determine the values of theta where this occurs on [0,2π]. c. Graph the equation. d. How could you alter the equation so that there are five petals? What about eight? 3. Given the equation r = 2 + 4 sin(θ) e. Using Algebra, determine how many times the graph goes through the origin on [0,2π]. f. On what interval of theta is r negative between [0,2π]? g. Using a table, graph the equation. h. What values of theta generate the inner loop? 4. Use the range of sine (or cosine) to explain why (or why not) the following graphs have an inner loop. Hint: discuss where r is negative. Then, state the type of graph that the equation represents. i. r = 1 + 3 sin(θ) j. r = 2 + 3 sin(θ) k. r = 3 + 3 sin(θ) l. r = 4 + 3 sin(θ) m. r = 5 + 3 sin(θ) 5. Describe how the graph of r = θ is created. Why does it look the way it does?
Day 4 Comparing Polar Graphs through their Auxiliary Equations P a g e 11 Note: Given a polar equation in the form r = f(θ), then the equation created when one replaces r with y, and θ with x, is known as the auxiliary Cartesian equation of the polar equation. Part I: Cardioids Format: r = a ± a cos θ and r = a ± a sin θ Problem 1: Polar: r = 4 + 4 cos θ Auxiliary: y = 4 + 4 cos(x) a) Match each smiley face on the Cartesian graph with their corresponding points on the polar curve. Label the coordinates in polar form on the polar graph. b) How are the y values of the smiley faces on the auxiliary graph represented on the polar graph? c) What theta values are necessary to generate the entire polar graph? How do those theta values relate to the graph of y = 4 + 4 cos (x)? d) On what interval are the values of r increasing? What about decreasing? How is this demonstrated on the auxiliary graph?
P a g e 12 Problem 2: Polar: r = 3 + 3 sin θ Auxiliary: y = 3 + 3 sin(x) a) Match each smiley face on the Cartesian graph with their corresponding points on the polar curve. Label the coordinates in polar form on the polar graph. b) How are the y values of the smiley faces on the auxiliary graph represented on the polar graph? c) What theta values are necessary to generate the entire polar graph? How do those theta values relate to the graph of y = 3 + 3 sin (x)? d) On which interval(s) are the values of r increasing? Decreasing? How could you determine these intervals by looking at the auxiliary graph?
P a g e 13 On a separate piece of paper, graph the auxiliary equation for each polar equation. Then, match the auxiliary graph to the polar graph. 1. r = 2 + 2 cos θ 2. r = 2 2 cos θ 3. r = 2 + 2 sin θ 4. r = 2 2 sin θ 5. r = 3 + 3 cos θ 6. r = 3 3 sin θ 7. r = 5 + 5 cos θ 8. r = 4 + 4 sin θ A. B. C. D. E. F. G. H.
P a g e 14 Part 2: Limaçons with an Inner Loop Format: r = a ± b cos θ and r = a ± b sin θ where a < b Problem 1: Polar: r = 2 + 4 sin θ Auxiliary: y = 2 + 4 sin(x) a) Match each smiley face on the Cartesian graph with their corresponding points on the polar curve. Label the coordinates in polar form on the polar graph. b) How are the y values of the smiley faces on the auxiliary graph represented on the polar graph? c) What theta values generate the inner loop? How is the inner loop of the polar graph related to the rectangular graph provided? d) On which interval(s) are the values of r increasing? Decreasing? How could you determine these intervals by looking at the auxiliary graph?
P a g e 15 Part 3: Limaçons without an Inner Loop Format: r = a ± b cos θ and r = a ± b sin θ where a > b Problem 1: Polar: r = 5 + 4 cos θ Auxiliary: y = 5 + 4 cos(x) a) Match each smiley face on the Cartesian graph with their corresponding points on the polar curve. Label the coordinates in polar form on the polar graph. b) What are the y values of the smiley faces? How do they relate to the polar coordinates on the polar graph? c) What theta values generate the inner loop? How is the inner loop of the polar graph related to the rectangular graph provided? d) On which interval(s) are the values of r increasing? Decreasing? e) Why will the graph never go through the origin when a > b? Reference both the polar equation AND the rectangular graph for proof.
P a g e 16 Part 4: Traditional Rose Petal Graphs Format: r = b cos(kθ) and r = b sin(kθ) where b > 0 Fill in the following table, graphing when necessary Polar Equation Polar Graph Graph as many periods of the AUXILIARY equation as will fit on the plane below. r = 3 cos(θ) 1 2 3 4 5 r r = 3 cos(2θ) 1 2 3 4 5 r r = 3 cos(3θ) 1 2 3 4 5 r r = 3 cos(4θ) 1 2 3 4 5 r
Polar Equation Polar Graph P a g e 17 Graph as many periods of the AUXILIARY equation as will fit on the plane below. r = 3 sin(θ) 1 2 3 4 5 r r = 3 sin(2θ) 1 2 3 4 5 r r = 3 sin(3θ) 1 2 3 4 5 r r = 3 sin(4θ) 1 2 3 4 5 r Conclusion: How does the period of the rectangular equation relate to the polar equation, specifically the number and location of the petals?
Directions: Match each graph to its corresponding equation. Then, replace r with y and θ with x. Graph the new auxiliary rectangular equation, and explain how the graph relates to the original polar graph. A. B. C. P a g e 18 D. E. F. G. H. I. J. K. L. 1) r = 2 + 4sin θ 2) r = 6 cos 4θ 3) r = 3 + 2 sin θ 4) r = 4 3 cos θ 5) r = 3 + 3 cos θ 6) r = 2 + 2 sin θ 7) r = 6 sin 9θ 8) r = 4 + 3 cos θ 9) r = 3 4 cos θ 10) r = 2 4 sin θ 11) r = 3 + 5cos θ 12) r = 4 sin 2θ
P a g e 19 Day 5 Extra Practice Day 5 Extra Practice Adapted from Precalculus & Trigonometry Explorations by Paul Foerster Goal: Plot polar curves on your graphing calculator, and find intersection points on polar curves. Introduction: The figure to the right shows the intersection of two polar curves. P 2 P 1 The limaçon has the equation r 1 = 3 + 2 cos(θ) The rose is given by r 2 = 5 sin (2θ) 1.) Plot the two graphs on your calculator using degrees, simultaneous mode, and a fairly small θ step so that the graphs plot slowly. Pause the plotting when the graphs reach the intersection point P 1. Approximately what value of θ do the graphs intersect at P 1? (Note: if you use a calculator without simultaneous plotting, just trace graph r 1. Record the approximate intersection point, and then either plug that θ value into r 2 OR compare their values of r in the table.) 2.) Resume plotting, and then pause it again at the θ value corresponding to point P 2 on the limaçon. Where is the point on the ROSE for this value of θ? Explain why P 2 is NOT an intersection point of the two graphs. (Note: if you use a calculator without simultaneous plotting, just trace graph r 1. Record the approximate intersection point, and then either plug that θ value into r 2 OR compare their values of r in the table.) 3.) Continue the graphing until a complete 360 has been plotted. Which of the 8 apparent intersections in the figure are true intersections, and which are not? What do you notice about the r values on the rose for the points which are not true intersections? (Note: if you use a calculator without simultaneous plotting, just trace graph r 1. Record each approximate intersection point, and then either plug that θ value into r 2 OR compare their values of r in the table.)
P a g e 20 4.) With your graphing calculator back in function mode, plot the auxiliary Cartesian graphs given below y 1 = 3 + 2 cos(θ) y 2 = 5 sin(2θ) Then, sketch the result to the right. 5.) Solve numerically to find the first two positive values of θ where the graphs in Problem 4 intersect. Show that these correspond to the two points where the polar graphs intersect. 6.) Show on the auxiliary graphs in Problem 4 that the second-quadrant angle θ for point P 2 corresponds to a point on the limaçon, but not to a point on the rose. 7.) What did you learn as a result of doing this Exploration that you did not know before?