PART II: WHILE … WEND
Polar Equations & Graphs
|Converting an equation in rectangular coordinates to polar coordinates |
|1) Convert the following equation from rectangular coordinates to polar coordinates. Simplify as much as possible (so that you end up with the fewest terms, and |
|the simplest terms) |
|[pic] |
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|2) Convert the following equation from rectangular coordinates to polar coordinates. Simplify as much as possible (so that you end up with the fewest terms, and |
|the simplest terms). |
|[pic] |
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|3) Convert the following equation from rectangular coordinates to polar coordinates. |
|[pic] |
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|Completing the Square (Review) |
|4) Review: Completing the square (find more examples on p. 115, 4th ed. [p. 120, 3rd ed.]) |
|[pic] |
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|5) Review: Completing the square |
|[pic] |
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|Converting an equation in rectangular coordinates to polar coordinates: circles |
|6) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as much as possible (so that you end up with the fewest terms, and |
|the simplest terms). |
|Once you've done that, graph the resulting rectangular equation on the provided graph |
|[pic] |
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|[pic] |
|7) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as much as possible (so that you end up with the fewest terms, and |
|the simplest terms). |
|Once you've done that, graph the resulting rectangular equation on the provided graph |
|[pic] |
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|[pic] |
|Graphing polar equations by converting to rectangular equations |
|Your goal for the next several sections is to both gain practice in converting equations from polar to rectangular form, and also in graphing the resulting |
|rectangular equations. At the end of this section, you should be able to explain what each of the following equations look like on a graph, as well as being able |
|to get the equation, given a graph with some key points outlined. |
|13) Based on the work you do in the next several sections, make sure that you're able to graph these equations. For each one, be able to describe the center |
|point, and radius, of the circle that is produced. |
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|[pic] [pic] |
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|[pic] [pic] |
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|14) First convert the equation to rectangular form, and then graphing the rectangular equation. Once you've figured out where the center is, and what the radius |
|will be, you can stop. |
|[pic] |[pic] |
| |[pic] |
| |[pic] |
| |etc |
|15) Same activity as the previous exercise. |
|[pic] |[pic] |
| |[pic] |
| |[pic] |
| |etc |
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|16) Same activity as the previous exercises |
|[pic] |[pic] [pic] |
| |[pic] [pic] |
| |[pic] [pic] |
| |etc |
|Figuring out the equation (in polar coordinates) of a given polar equation |
|Given the following pictures, figure out the equation that generated it. Make sure that you're clearly able to identify the center point, and the radius, along |
|with the equation! |
|17) |18) |
|[pic] |[pic] |
|Center Point: Radius: |Center Point: Radius: |
|Equation: |Equation: |
|19) |20) |
|[pic] |[pic] |
|Center Point: Radius: |Center Point: Radius: |
|Equation: |Equation: |
|Converting an equation in rectangular coordinates to polar coordinates: circles & lines |
|8) Convert the following equation from polar coordinates to rectangular coordinates. Simplify as much as possible (so that you end up with the fewest terms, and |
|the simplest terms) |
|Once you've done that, graph the resulting rectangular equation on the provided graph |
|[pic] |
|[pic] |
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|9) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular form, and then graphing the rectangular equation on the |
|provided graph (using the X & Y axes as a rectangular grid) |
|[pic] |[pic] |
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|10) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular form, and then graphing the rectangular equation. |
|[pic] |[pic] |
|11) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular form, and then graphing the rectangular equation. |
|[pic] |[pic] |
|12) Your goal is to graph this polar equation. Do this by first converting the equation to rectangular form, and then graphing the rectangular equation. |
|[pic] |[pic] |
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|Graphing polar equations using a graphing utility |
|Your goal for this section is to get an understanding of how several types of polar equations look when graphed, using your calculator. The first thing you'll need|
|to do is properly configure your calculator. |
|If you're using a TI-83, push the [pic] key, and make sure that your calculator is in Radian mode, and POLar | |
|mode (move the cursor using the arrow keys, and select a mode by pushing |[pic] |
|[pic]) When you're done, your screen should look like this: | |
|Next, push the [pic] button to bring you to the normal screen for entering equations to be graphed. You'll |[pic] |
|notice that much in the same way that rectangular equations define y in terms of x (such as [pic]), these | |
|polar equations define r in terms of θ. So if you want to graph an equation, you'll need to solve it in terms| |
|of r, if it isn't already. Enter the formula listed in the picture to the right (note that the [pic] button | |
|now enters a θ. when pressed). | |
|Next, push the [pic] button, and you'll see a picture of the equation appear. It should look like the one to |[pic] |
|the right | |
|If the screen doesn't look quite right (for example, it's stretched or compressed), then you should press the |[pic] |
|[pic] button, and try selecting the ZTrig setting. | |
|If you're interested in looking at a table of values for a given function (much like the book does), you can |[pic] |
|first press | |
|[pic] [pic] to access the "TBLSET" (or "TaBLe SETup") menu), which should look like the picture to the right. | |
|Typically, you'll want to start your table at 0, and change by [pic], or [pic], etc | |
|When you're done, push the [pic] [pic] buttons to actually see the table. The calculator then shows you the |[pic] |
|values that it plugged in for θ, and the resulting values that it got out of the first equation in the r1 | |
|column. | |
|Graphing polar equations using a graphing utility |
|Graph each of the following using your graphing utility. For each one, you should be able to identify which type of equations you're looking at, given a graph. |
|You do not need to know how to go from the graph back to the equation. |
|(a and b are positive constants) |
|A. Cardioid: [pic], [pic]; try [pic] |
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|B. Limaçon without an inner loop: [pic], [pic], a > b ; try [pic] |
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|C. Limaçon with an inner loop: [pic], [pic], b > a ; try [pic] |
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|D. Lemniscate: [pic], [pic]; try [pic] |
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|E. Rose: [pic], [pic]; try [pic] |
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