In Exercises 13–34, test for symmetry and then graph each polar e...

Question

In Exercises 13–34, test for symmetry and then graph each polar equation.r = 2 + 2 cos θ

Accepted Answer

Test the polar equation for symmetry and sketch its graph where we have R equals eight plus eight cosine data. And we also have a graph here, the graph our equation on. Now our first thing is to check our symmetry. We have three checks of symmetry. We need to do our first check will be the polar symmetry test or flex over our axis heading in the direction of zero. Other words, we want to check over this line of symmetry. No, to check polar symmetry, we need to set theta equals to negative the. Now if we get the same function as our original back out, we do our substitution, we confirm polar symmetry. So let's check, we have R equals eight plus eight cosine negative data. Now cosine we know isn't even function, which means if we have cosine of a negative angle, it is the same as cosine of a positive angle. So we can then write that this equation is the same as R equals eight plus eight cosine data, which is the same as our original. So we do confirm we have polar symmetry. Now let's check our next line of symmetry which is theta equals pi divided by two. To check this symmetry, we check if we have our data and see if we take negative R negative data. Do we get the same equation back out? We will get negative R equals eight plus eight to assign negative data which we can simplify. We get negative R equals eight plus eight cosine data by R even function dividing everything by negative one gives us R equals negative eight minus eight cosine data which this is not the same as the original function. So we cannot be for sure if there is symmetry over this line. So now we will check our final symmetry which is Pole symmetry. For pole symmetry. We see the check if we replace our R with negative R and simplify do we get the same function back out? Pr equals A plus eight cosine theta and we'll put a negative in front of the R now is solved for art by dividing both sides by negative one to get R equals negative eight minus eight cos data which is not the same as the original. So we cannot confirm if this has Pole Symmetry, we can only confirm that this has polar symmetry which means will reflect our graph over this line heading in the angle of zero in the direction of angle is zero. I should say now we see the graph, our function we can play in some values into our graph. But if we were to do that, we would end up with a graph that goes way out towards the angle that equals zero and slowly starts to spiral inwards as it approaches the angle pie. So you draw points on our graph, we can find these values with the calculator. But if we were to do that, we should eventually have a graph that looks like this. Now, this is reflected based on our symmetry. So we will also make a reflection going in the opposite direction just like that. It's a little rough, but the rough idea is a function that heads the direction of zero degrees spirals inwards until it gets the pie and then reflected over the polar axis. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 13–34, test for symmetry and then graph each polar equation.r = 2 + 2 cos θ

Key Concepts

Polar Coordinates

Symmetry in Polar Graphs

Graphing Polar Equations

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