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 + cos θ

Accepted Answer

The polar equation for symmetry and sketch is graph where we have R equals nine plus nine halves, cosine data. And we have a graph here. We can also plot points on. Now to check this graph, we need to check three lines of symmetry. Our first will be polar symmetry for polar symmetry. We want a check if our data when equals negative data gives us the same answer as our original function. In other words, if I take R equals nine plus nine halves, cosine negative data, do we get the original function back out? Now we have cosine negative data. We know that cosine is an even function because cosines an even function we know cosine negative data will equal cosine data itself, meaning we have R equals nine plus nine halves, cosine data which does pass the polar symmetry check. Now let's check our next symmetry. Our next symmetry we wanna check is over the line theta equals pi divided by two. Now the check data equals pi divided by two symmetry. We need to see if, when we change our, our data to negative R negative data, do we get the same function back out So we will check negative R equals nine plus nine halves, cosine negative data. We get negative R equals nine plus nine FS cosign data based on our even function rule. And we can divide both sides by negative one to get R by itself R equals negative nine minus nine halves cos I data, this is not the same as the original function. So we can't confirm whether this has symmetry over the line data equals pi divided by two. So we will go ahead and move to our final symmetry check. Our final symmetry check will be over the pole itself. So now we want Paul Symmetry to check our pulse symmetry, we will set our R equals to negative R and see if we get the same function back out. We get negative R equals nine plus nine halves plus nine data which solving for R by itself by dividing everything by negative one, get R equals negative nine minus nine halves data which similarly to the previous problem, her previous check the same. We do not know for sure if this is symmetrical. So we will say it's not for now now that we have our symmetry checks, we know it only has polar symmetry. We will go ahead and find points on our graph. Now let's say we check a few points here. We'll have data values on the top and our values on the bottom zero pi over six pi divided by three divided by two, two pi divided by three pi pi divided by six and pi itself. Now, if we were to plug this into a calculator, we would get the following values 13.5 12. 11.25 nine, 6.75 5.1 and 4.5 which we can now graph all of these points on our plane at angle zero. We have an R length of 13.5 which is roughly around here on the graph power 6, 12.9 pi over three, it's 11.25. So we go up to 11 hour radius, draw a point. And now we have pi divided by two which we have nine, two pi divided by three, which gives us 6. five pi divided by six, which gives us 5.1. And finally pi in the direction of 4.5 as a radius, I should say right there. Now that is, this is symmetrical um polar or polar symmetry is the same. So we will draw our graph going through the points and then the same thing reflected over the axis. So I'll make this a little better and dry it again like so, oh, so one more time, there we go. That is how a graph should look roughly. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

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