What is the connection between trigonometric graphs and polar graphs? (Day 2 of 2)
Lesson 5 of 7
Objective: SWBAT use the rectangular graph of a trigonometric equation to create a polar graph of the function.
I begin today's class with a review of the concepts covered in yesterday's lesson. The bell work (page 1) gives the students a polar graph. By working with students it appears that the challenging concept is an equation with negative values. The students begin by creating a sinusoidal graph.
I ask the students questions like:
- How many times will the function touch the pole? (see student work)
- Why does it cross the pole twice?
- How can we find the zeros of the function?
- Where does the function have negative values?
As we work through the problem, I use different colors to highlight the features of the graph so that students can see how the polar graph relates to the sinusoidal graph.
Next, I ask the students to explain the graph's symmetry. We talk about the graph's symmetry with the polar axis and the pi/2 axis. I question my students, "How can I use the equation to predict symmetry?". Students usually notice that looking at the trigonometric function helps determine symmetry.
I also ask, "Why are cosine graphs symmetric with the polar axis and sine graphs symmetric with the pi/2 axis?" This questions reinforces the concept that sine is y/r and cosine is x/r. Many students see the polar axis as the x-axis and the pi/2 axis as the y-axis. This helps students to relate the unit circle with the polar coordinate system.
The trouble with graphing negative values is that the graph reflects across the pole. It may still be difficult for students to completely grasp the structure of a polar coordinate system, making it possible to shift up or to the right when there is a negative number.
After discussing the first equation I give the students another problem to graph to practice graphing with negative numbers. The problems is on page 2 of the bell work.
I now present more complicated graphs to the class, including the graphs of polar equations involving r squared. The students have not yet analyzed the graph for the y=sqrt(sin x). My goal is for students to see how y=sin x is connected to y=sqrt(sin x).
Students sketch the graph y=sin x by hand. We then use the graphing calculator to graph y=sqrt sin x. After looking at the calculator, the students determine that the graph is and I put a sketch on the board.
"Why does the graph of y= sqrt (sin x) only graph from 0 to pi, 2pi to 3pi, etc?"
Students begin to realize that the only where y=sin x will be graphed in y=sqr(sin x)?
"Why is only this graphed in y=sqrt (sin x)?" This is a great time to remind students of how a composition works. Once students see this connection, we can use this idea to graph the r-squared equation.
Next, the students are given a polar equation with r-squared (other polar graphs page 1).
This type of problem can be confusing to start the students graph y=4sin(2theta)( see lemniscate). I then ask my students, "How do we get r instead of r squared?". By looking at the sinusoidal graph, you can see that the students only found the positive sqrt(4sin(2theta)). Together, we simplified the equation and determined that the difficult equation has a amplitude of 2 and thus would only graph under the positive arc of r=4sin(2theta) (graphed in red).
After sketching the sinusoidal graph, we used this to sketch the lemniscate. I now remind the students that we have not written all the values for r squared. At the moment, someone in class will usually state that we need the positive or negative sqrt(4sin (2theta)). The class sketches the negative portion of the graph (page 3 green graph).
Lastly, I ask: "What happens when we graph the negative values?". This allows students to see how the negative values will trace the positive values.
Other polar graphs.
I believe that it's important during this lesson for students to see and practice with other graphs as well. On the worksheet, other polar graphs page 2, there is a problem where students can sketch a spiral graph, exposing them to ways of making the sinusoidal graph more interesting. However, one challenging aspect of this graph is determining where to put pi on the r axis.
For page 3 of the slides I allow the student use their calculator. I ask students:
- How is the graph of tangent different than sine and cosine?
- What happens to the polar graph as theta gets close to pi/2?
Seeing how the asymptote of tangent affects the polar graph is interesting for students. Some students quickly realize what is happening while others need to see the sinusoidal graph first and then discuss the positive and negative parts of the graph.
As class comes to an end, students will continue working on the polar graphing worksheet.
I ask the students what the graph looks like if you have both sine and cosine in an equation, such as r=2 sin (theta)+3 cos (theta). You can ask your students, "Does the graph have symmetry?" and "Why does the graph have the shape that it does?", to reinforce these important concepts.