I have trouble grasping parabolas, and mainly the cartesian equations describing the,. In my mind, there are 4 possible parabolas, a parabola shaped like a mountain ($\cap$), a parabola shaped like a valley ($\cup$), a parabola shaped like the greater than sign ($\supset$) and a parabola shaped like the smaller than sign ($\subset$), of course without the sharp edges like they have in my examples. I know one standard equation:
$y-b = \dfrac{1}{4c}(x-a)^2$, with the vertex at $T(a,b)$ and the focus $F(a; b+c)$.
However, I don't know for which of the 4 this one is, so my questions are:
1. What is the equation of the $\cap$ shaped parabola?
2. What is the equation of the $\cup$ shaped parabola?
3. What is the equation of the $\subset$ shaped parabola?
4. What is the equation of the $\supset$ shaped parabola?
To clarify: I would like them in the same form as the one I noted above in the example, thanks in advance.
Best Answer
This general equation you gave is a parabola that opens either up or down (U or ^ shaped, in your words) depending on the signs of $c$: if $c > 0$ it opens up, if $c < 0$ it opens down. You can get the formulas for the right/left ones (C and >) by switching $x$ and $y$, and again whether it opens right or left depending on whether $c$ is positive or negative, respectively. There many other sorts of parabolas too, though--for example, you might have a parabola at a 45-degree angle to the axes.