In Exercises 64–70, graph each polar equation. Be sure to test...

Question

In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ

Accepted Answer

That the polar equation for symmetry and sketch its graph R equals plus 12 sine data. And we're given a polar coordinate system here to graph our equation on now to solve this, we first need to test our symmetry. We have three types of symmetry. We can check our first is our polar symmetry. For polar symmetry. We just wanna take our data and replace it with negative data. If we get our original function back out, we have polar symmetry, you will get R equals 12 plus 12 sign of negative data. Now sign is an odd function. This means if we have a sign of a negative angle, it's the same as saying negative sign of a positive angle like so we can then say R equals 12 plus 12 multiplied by negative sine data. We then the R equals 12 minus 12 science data. This is not the same as our original. So we cannot confirm if we have Polo symmetry or not. We now want to check symmetry over the line beta equals pi divided by two. To do this, we will take both our radius and our data trend them to be negative and see if we get the same answer back out. When substituting we get negative R equals 12 plus 12. Not by our odd function, we get 12 minus 12 sign the. And finally, we can divide negative one to get R by itself negative 12 plus 12 2nd. The this is also not the same as the original function. So we can't confirm symmetry over the line data equals pi divided by two. We finally had to check our last symmetry which is Pole symmetry. For Paul Symmetry. Let us to replace our radius with a negative radius and check to see if we get the original function. Negative part equals 12 plus 12 S data where we get R people say the 12 minus 12 data, which is also not the same as our original. So we can't confirm if we have any of these three symmetries now that we test our symmetry. Let's go ahead and graph our function. We can graph our function by finding values to plug in for data and we will go from 0 to 2 pi see if data has our input radius as our output. Let say we have zero and divided by six and divided by three pi and divided by two, two pi divided by three, five P divided by six in pi. And we will also check until two data. So we have seven pi divided by six or pi divided by three, three pi divided by two pi pi divided by 11, pi divided by six and finally two pi now we can throw all of these into the calculator can find our values which I've done here. We have 12 in 22. 24 22.4 18, 12 for a top row, six, 1.601 0.66 and 12 for our bottom row. Now, we can grab all of our points in direction of angle zero degrees or zero radiance. Our radius is 12. So we will put a point at pi divided by six. We have 18 pi divided by 3 22. divided by 2 two pi divided by 3 22.4 five pi divided by 6 18 pi and or our other values 75 divided by six and six four pi divided by 31. 35, led by 20 55, divided by 31.6 11 pi divided by 66. And finally two pi 12, we can now draw a curve that connects all of our points. And we notice here, draw curve. This graph is in the shape of a cardioid. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ

Key Concepts

Polar Coordinates and Equations

Graphing Polar Equations

Symmetry in Polar Graphs

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