Graphing the binomial coefficients: appearances? Is it appropriate to describe the function
$$f(x) = \binom{x}{2}$$ as parabolic?
How does it vary over $c$:
$$f(x) = \binom{x}{c}?$$
And, is there any intuition behind these graphs? For example, their resemblance to the polynomials of degree $c?$

I tried inputting one into wolfram alpha and it spat out a continuous function, which confused me. This is what prompted me to ask the question.
Yes, you can generalise the factorial, choose function, etc. to not whole numbers too.
$$\binom{x}{k}=\frac{x!}{k!(x-k)!}$$
$$=\frac{x(x-1)\dots(x-k)!}{k!(x-k)!}$$
$$=\frac{x(x-1)\dots(x-k+1)}{k!}$$
So for $k=2$ you will get:
$$\binom x2=\frac{x(x-1)}{2!}=\frac{x^2-x}{2}$$
And for $k=3$:
$$\binom x3=\frac{x(x-1)(x-2)}{3!}=\frac{x^3-3x^2+2x}{6}$$
And so on.
I have usually used the binomial formula with discrete whole numbers. If you want to plot this, you get points on a grid.

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$