Learning Resources

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Lesson

You will learn how to plot polar equations during this Investigation. It is presented in two parts and is perhaps best done in small groups. Contact your on-line teacher regarding this matter.

Carry out Steps A to C of the procedure outlined on pages 291 & 292 of your text, paying particular attention to patterns in various equations. Discuss your observations within your group and draw conclusions as to the effect of different parameters of equations on the shapes of their graphs. Hints and suggestions are provided should you experience difficulty. Use them only as needed.

If you choose to draw the graphs on your calculator, ensure that it is both Degree and Polar MODE. As well, you can use the ZOOM button to fit the complete graph onto the screen.

From your work in Mathematical Modeling, Book 2, you saw that the period for the graph of r = cos, when plotted on rectangular coordinates, is 360°. Why then, if the period is 360°, will the graph of r = cosbe complete on polar paper if only points from are plotted? To answer this, you will need to reflect upon Step A. 

Check your response.

In Mathematical Modeling, Book 2, you did a lot of work with the graphs of 
y = cos and y = sin. Use the knowledge you have gained to answer the question below, before answering the Investigation Questions.

What is the relationship between sin and cos? Specifically, what translation of cos will produce sin? If you are having difficulty answering this question, it might be helpful to sketch y = cos and y = sin on rectangular coordinates. Verify your response.

Answer Investigation Questions #'s 5 to 7. 

Carry out Part 2 of the Investigation. Hints and suggestions are provided should you experience difficulty. Use them only as needed.

Answer the remaining Investigation Questions.

Notebook Entry: Record the definition of a polar equation.

Activity

C.Y.U. page 293 #'s 11 - 13

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

Predict what the following graphs will look like. Your answer should include the shape (circle or number of petals) and where the graph is centred. Assume each equation has domain .

  1. r = 7cos
  2. r = sin
  3. r = cos2
  4. r = 4cos3
  5. r = 6 sin3

Solutions

  1. The graph is a circle with centre (3.5, 0°) and a radius of 3.5. It passes through the origin and is centred on the pole.
  2. The graph is a circle with centre (0.5, 90°) and radius 0.5. It goes through the origin and is centred on the line = 90°. The graph is a 90° counterclockwise rotation of the graph of r = cos.
  3. The graph has four petals, each one unit long. The centre of the graph is at the pole. The petals are centred around the polar axis and the lines
    = 90°, = 180° and = 270°.
  4. The graph has three petals, each four units long. One petal is centred around the polar axis, and the other two are centred around the lines
    = 120° and = 240°.
  5. The graph has three petals, each six units long. It is the same as the graph of r = 6 cos3, rotated by 30°. Thus, the petals will be centred around the lines = 30°, = 150° and = 270°.