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Presented, and Compiled, By

Bryan Grant

Jessie Ross

August 3rd, 2016

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.

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.

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.

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.

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?

5.

How did you graph the first two equations? Did you need the ? / What equation would make the infinity sign symmetric about the axis? / What are your names for these graphs? What are the actual names?

Unit Question: WHY do polar graphs have their unique shapes?

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 , where is the movement on the axis, and on the axis. Polar coordinates are in the form , where is the length of the radius of a circle, and is how many radians you rotate counter-clockwise (think trig) about the origin.

______

Graph the following polar coordinates:

______

Given a right triangle with angle at the origin:
/ Using the triangle, you know that:
so
so
so
Lastly,
Ex. Convert to rectangular coordinates. / Ex. Convert to polar coordinates.

Ex. Convert the following equations to polar form.

(a) (b)

______

Ex. Convert the following equations to rectangular form, and sketch the graph.

(a) / (b) / (c)

______

Work the following problems. Please do NOT use your calculator.

Convert the following equations to polar form.

1. 2. 3.

______

Convert the following equations to rectangular form.

4. 5. 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.

2. Convert the following Cartesian points to Polar.

3. Convert the following polar equations to Cartesian.

4. Convert the following Cartesian equations to Polar.

Day 3 – Understanding Polar Graphs

Days 3 & 4 – Adapted from Alicia Goldner – Butler Senior High School – Pennsylvania

1.)

  1. What do you expect this graph to look like? What is it called?
  1. Fill in the values for in the first table below, and sketch a graph of the equationon the plane below.

/









/ /

/
  1. Why did we choose those specific values for ? If the equation was , what would the first four “nice” values for theta be in your table instead of ?
  1. Notice our table only gave the part of the graph in quadrants 1 and 4. Start the second table with the value . Fill in the “nice” values of up through . Find . Using the first table, how can you get the values of in the rest of the table with very little work? Sketch the rest of the graph.
  1. How many times does the graph touch the origin on the interval ? 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?

2.)

  1. What do you expect this graph to look like? What is it called?
  1. 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.

/ / / /

/ / / /

/

/
  1. How many times will the graph touch the origin on ? Using Algebra, find the exact theta values where this occurs.
  1. What happens to for values of between ? Use the range of cosine to explain your answer.
  1. Fill in the second table above. If your answers are not integers, try to approximate the values without using a calculator. Graph those points.
  1. Fill in the last table by choosing three other values of theta in addition to and finding their respective outputs. Then, complete the graph.
  1. What theta values generate the inner loop of the graph? WHY do they make the inner loop?

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.
  1. Given the equation
  2. Using Algebra, determine how many times the graph goes through the origin on .
  3. What is the largest possible value of generated by this equation? Determine the values of theta where this occurs on .
  4. Graph the equation.
  5. How could you alter the equation so that there are five petals? What about eight?
  1. Given the equation
  2. Using Algebra, determine how many times the graph goes through the origin on .
  3. On what interval of theta is negative between ?
  4. Using a table, graph the equation.
  5. What values of theta generate the inner loop?
  1. Use the range of sine (or cosine) to explain why (or why not) the following graphs have an inner loop. Hint: discuss where is negative. Then, state the type of graph that the equation represents.
  1. Describe how the graph of is created. Why does it look the way it does?

Day 4 – Comparing Polar Graphs through their Auxiliary Equations

Note: Given a polar equation in the form , then the equation created when one replaces with , and with , is known as the auxiliary Cartesian equation of the polar equation.

Part I: Cardioids

Format:

Problem 1:

Polar: / Auxiliary:

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 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 ?

d)On what interval are the values of increasing? What about decreasing? How is this demonstrated on the auxiliary graph?

Problem 2:

Polar: / Auxiliary:

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 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 ?

d)On which interval(s) are the values of increasing? Decreasing? How could you determine these intervals by looking at the auxiliary graph?

On a separate piece of paper, graph the auxiliary equation for each polar equation. Then, match the auxiliary graph to the polar graph.

Part 2: Limaçons with an Inner Loop

Format: where

Problem 1:

Polar: / Auxiliary:

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 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 increasing? Decreasing? How could you determine these intervals by looking at the auxiliary graph?

Part 3: Limaçons without an Inner Loop

Format: where

Problem 1:

Polar: / Auxiliary:

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 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 increasing? Decreasing?

e)Why will the graph never go through the origin when ? Reference both the polar equation AND the rectangular graph for proof.

Part 4: Traditional Rose Petal Graphs

Format: where

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.
Polar Equation / Polar Graph / Graph as many periods of the AUXILIARY equation as will fit on the plane below.

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 with and with . Graph the new auxiliary rectangular equation, and explain how the graph relates to the original polar graph.

1) / 2) / 3) / 4)
5) / 6) / 7) / 8)
9) / 10) / 11) / 12)

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.
The limaçon has the equation
The rose is given by /

1.) Plot the two graphs on your calculator using degrees, simultaneous mode, and a fairly small step so that the graphs plot slowly. Pause the plotting when the graphs reach the intersection point . Approximately what value of do the graphs intersect at ? (Note: if you use a calculator without simultaneous plotting, just trace graph . Record the approximate intersection point, and then either plug that value into OR compare their values of in the table.)

2.) Resume plotting, and then pause it again at the value corresponding to point on the limaçon. Where is the point on the ROSE for this value of ? Explain why is NOT an intersection point of the two graphs. (Note: if you use a calculator without simultaneous plotting, just trace graph . Record the approximate intersection point, and then either plug that value into OR compare their values of in the table.)

3.) Continue the graphing until a complete 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 values on the rose for the points which are not true intersections? (Note: if you use a calculator without simultaneous plotting, just trace graph . Record each approximate intersection point, and then either plug that value into OR compare their values of in the table.)

4.) With your graphing calculator back in function mode, plot the auxiliary Cartesian graphs given below


Then, sketch the result to the right. /

5.) Solve numerically to find the first two positive values of 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 for point 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?