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My teacher first sketched a graph r $\theta$ which is simply graph of $1+\cos\theta$
we all know. And then transformed it into strange graph like this.
What steps I go through in order to obtain it?

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This second graph is in what are known as polar coordinates. Instead of graphing in the usual $x,y$ sense, it's generated using $r,\theta$. There are tons of video tutorials on polar coordinates and graphic polar graphs that are available on YouTube. It's the kind of thing probably best explained in a video where you can watch them draw the graph. Searching for "polar coordinates tutorial" brings up many such videos.

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    I found a video [link](https://www.youtube.com/watch?v=8QnQxAMfbWM) – 2017-01-12
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    @BaimyrzaShamyr great! Do you have any further questions about the process or concept of polar graphing? – 2017-01-12
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The right graph has $r$ on the vertical axis and $\theta$ on the horizontal axis. The graph of $r = 1 + 2 \cos \theta$ repeats every $2\pi$, as you probably see.

The left graph treats the two variables $r,\theta$ as polar coordinates in the two-dimensional plane, where $r$ is the distance from the origin $(0,0)$ and $\theta$ is the angle the point makes with respect to the origin, going counterclockwise starting with the positive horizontal axis.

To see how the points map, start at the point $(\theta, r) = (0,3)$ on the right graph (this is the intersection with the vertical axis), and at the point on the horizontal axis of the left graph (it would be $(3,0)$ in the $x-y$ plane).

Now, as you trace the right graph to the right, you go counterclockwise around the graph on the left (including "going around" the little loop). When you reach the point $(\theta, r) = (2 \pi, 3)$ on the right graph, you're back to where you started on the left.