![]() ![]() While we do have students practice graphing limacons by hand we also give plenty of short answer questions or have students describe key features of the graph without actually drawing it.\right)\), as shown below. The area inside the region bounded by the rays a and band the curve r f(. The area of a region in polar coordinates can be found by adding up areas of innitesimal circular sectors as in Figure 2(b). Our main goal in this lesson is not that students draw perfect limacon graphs without a calculator/graphing tool but that they would be able to connect key features of a polar graph with its equation. specied in polar coordinates, it is often helpful to convert to Cartesian coordinates and proceed as in Sections 9.2 and 9.3. Of the three types of limacons, which one never touches the pole? Why? How can you determine the length and location of a loop from the equation? What would happen if the value of a was zero? What about if b was zero? Be ready to hear some things you may not have thought about yourself! My students always surprise me with their observations! Some additional questions to enrich the discussion are: Each group shares one thing and we keep circling back to the groups until groups have no more unique contributions. In the debrief we use the round-robin protocol to collect these ideas on the whiteboard or poster paper. Assign a reporter in each group that will share out the main take-aways or patterns observed from their group. Today’s lesson lends itself well to a whole-class discussion. Because the experience portion of the lesson is meant to spark curiosity and help students use inductive reasoning to form conjectures, it is okay to save some of these conversations for the debrief. Similarly, students saw that the a-value represented the y-intercept for cosine graphs and the x-intercept for sine graphs, but needed more prompting to explain what this had to do with the behavior of sine and cosine at the angles of 0, π/2, π, 3π/2, and 2π. Many students noticed that the sum of the a and b values is the furthest point away from the pole, but were not able to explain why. Throughout the activity, the idea of maximum distance away, or maximum radius will become important. They notice patterns about when the graphs make dimples, heart shapes, and loops, though their vocabulary to describe these might be different. ![]() Example 1 Graphing a Polar Equation by Point Plotting. They first graph equations with cosine in them and later equations with sine in them. This section approaches curve sketching in the polar coordinate system similarly. Today students finish yesterday’s exploration of polar graphs by looking specifically at limacons. Day 4: Calculating Instantaneous Rate of Change.Day 3: Calculating Instantaneous Rate of Change.Day 2: Average versus Instantaneous Rates of Change.Day 3: Evaluating Limits with Direct Substitution.Day 7: Infinite Geometric Sequences and Series.Day 6: Geometric Sequences and Finite Series.Day 2: Using Sequences and Series to Describe Patterns.Day 3: Solving Systems with Elimination.Day 2: Solving Systems with Substitution.Day 15: Parametric Equations (With Trig).Day 9: Equations in Polar and Cartesian Form.Day 10: Transformations of Sine and Cosine Graphs.Day 11: Exponential and Logarithmic Modeling.Day 9: Solving Exponential and Logarithmic Equations.Day 3: Compound Interest and an Introduction to "e".Unit 3: Exponential and Logarithmic Functions. ![]()
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