In the book of _Classical Mechanics_ by Goldstein, at page 88, it is given that

[![enter image description here][1]][1]

[![enter image description here][2]][2]

However, the orbit might not be bounded, so there might not be two turning point; just one. In such a case, how can we argue that the orbit equation always has two turning points and it is symmetric about both axes?

  [1]: https://i.sstatic.net/OAo9p.png
  [2]: https://i.sstatic.net/tm8uK.png