Why are there two standard forms for parabolas? I am helping my girlfriend do her math homework.  They have been calling
$$ (x-h)^{2} = 4a(y-k) $$
the standard form.  Suddenly in her homework, a different equation has been referred to as standard form:
$$ y = ax^{2} + bx + c, $$
which is familiar to me from high school. It's been frustrating for her to see both of these equations called the standard form, especially since questions are asking to convert a freeform equation into 'standard form' when standard form seems ambiguous.
Both equations represent parabolas in different ways, but does anyone have an idea why they are both referred to as standard form?
 A: "Standard form" is context dependent. If you are interested in the coefficients then $ax^2 +bx +c$ is your friend. If you want to be able to see the symmetry and find the roots then you want the other form.
If someone asks you to convert something to "standard form" then  they are required to tell you what "standard form" means.
The actual algebra is independent of the labels.
A: If a book author has two different descriptions of the same phenomenon, and calls both "the standard form", the author has, in my opinion, made a mistake.  If this is what is happening, I would suggest that you find an alternative text.
That being said, a possible explanation for this ambiguity is as follows:

*

*In many texts, a polynomial is said to be in standard form if that polynomial is written with its terms in decreasing order of degree.  For example,
$$ 9x^4 + 3x^2 - 8x + 10 $$
is in "standard form", while
$$ 10 - 8x + 9x^4 + 3x^2 $$
is not.  Thus a quadratic polynomial has been written in standard form when it is written as
$$ ax^2 + bx + c,$$
where $a$, $b$, and $c$ are real (or complex, or whatever) coefficients.


*Other texts which focus more on conic sections might describe the standard form of a parabola as
$$ (x-h)^2 = 4a(y-k), $$
where $a$, $h$, and $k$ are real numbers which have geometric meaning (specifically, $(h,k)$ is the vertex, and $a$ describes a scaling ($a$ is the distance from the vertex to the focus (in one direction) and the directrix (in the other direction); the sign of $a$ indicates orientation)).
Hence it is possible for "the" standard form of a parabola to be one of (at least) two different things:  the standard form of a degree two polynomial, or the standard form of a conic section.  They way in which you are supposed to interpret what is meant by "standard form" is going to depend on what you are trying to describe:  is the salient idea that the curve is the graph of a polynomial?  or is it that the curve is a conic section?
Finally, as a bit of advice for tackling the homework:  if you are given a problem which is phrased like "Write the following in 'standard form': ...", and the meaning of "standard form" is ambiguous, write it in every form which could be "standard.  If nothing else, you get familiar with the symbol manipulation.
