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I attended an interview at Cambridge several months ago and during this, the interviewer referred to the graph of $y = x^3$ as a parabola (Maths Interview, Maths Professor). The exact dialogue went as follows:

  • "We'll start with something simple, can you draw the graph of $x^2$" - I could
  • "What is the word for this type of graph" - Parabola
  • "Great, now can you mark on the point $(1,1)$ and draw the other parabola, $x^3$"

If this had not been right at the start of the first interview, I would have questioned him on this out of curiosity but since I hadn't shown any skill yet I thought it was best not saying anything in case I was wrong.

Since getting home I have searched for any reference to $y=x^3$ being a parabola, I have asked teachers and tried to bend any definitions I know of a parabola to make sense for $y=x^3$. My thoughts being:

  • Intersection of horizontal plane and a cone
  • All incoming vertical lines will reflect to a common focus
  • The loci of points equidistant from the focus and directrix

None of these seem to make sense. I wonder if the mention of it was simply a test to see if I would notice. Turns out that wasn't of too much importance since I was offered a place anyway but out of peace of mind, I would love an explanation for this.

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    Maybe it was a slip of the tongue; did he continue to refer to it as a parabola, or was it just the one time? The really breathtaking part of it, to me, is the wording "the *other* parabola" (emphasis added). – 2017-02-03
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    It didn't sound like it. There very much seemed to be a sense of distinction in - "yes, that is a parabola, I want to draw the _other_ one too". I spoke to one of the 2nd year history students after who said that the Professor is a well-known eccentric and pedantic so maybe he's developed his own definition that he swears by. – 2017-02-03
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    It probably was a slip of tongue, or he might have wanted to test you on whether you're able to catch the error. – 2017-02-03
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    That's saying something.. – 2017-02-03
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    So this was with a Cambridge's mathematics professor (Cambridge England or, at least, Cambridge USA, right?) , but then you asked...a 2nd year **history** student about this? – 2017-02-03
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    Cambridge, England. As in 'the university of'. Between interviews we were just waiting in a lobby with some 1st/2nd years to chat to us to pass time. There wasn't any maths or physics students but he asked how things went and I mentioned this and he said yes, he's got a reputation, clearly one that extends as far as other subjects – 2017-02-03
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    Yes, it can be called a "cubical parabola": http://mathworld.wolfram.com/CubicalParabola.html – 2017-02-03
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    Oh, that's very interesting. What exactly makes it parabolic? – 2017-02-03
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    Generally I consider a parabola to be a conic section, and so it definitely isn't the graph of a cubic polynomial. – 2017-02-03

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It is not completely unheard of to refer to any curve of the form $y=px^k$ as a "higher parabola", simply because of the formal resemblance to $y=px^2$.

This terminology seems to have been introduced by some of the pioneers of analytic geometry in the 1600s (Fermat, Pascal, Cavalieri), who had figured out methods to calculate the area under an ordinary parabola, and were generalizing those to $px^k$, but did not yet have the machinery to handle general polynomials.

There were a bunch of other generalizations in the same vein, such as calling $x^ky^m=p$ a "higher hyperbola" by analogy with $xy=p$.

As your experience shows, the terminology has not gone entirely away, but it is certainly not a standard usage in the modern day. Usually a "parabola" does refer specifically to a conic section.