Both.
I also want to know how, is it simply by differentiation?
But differentiation seems to have a lot of steps before correct shape can be plotted by finding first and second derivative.
You do it by finding zeros, intercepts, critical points, inflections points, etc. but typically in Physics we just graph the thing. That means for the simpler equations it just comes down to recognition and the nice thing is that there aren't that many to recognize.
Graphs in Physics tend to come in two categories: directly proportional and inversely proportional. The direct proportions are of the form:
\(\displaystyle y \propto x\)
which implies that
\(\displaystyle y = mx\)
where m is some constant. These graphs are just lines with either positive or negative slopes. (Notice that I'm lumping lines with non-zero intecepts into this category so the general equation here is \(\displaystyle y = mx + b\).)
The inverse proportions are of the form
\(\displaystyle y \propto \frac{1}{x}\)
or
\(\displaystyle y = \frac{k}{x}\)
where k is a constant. These graphs are hyperbolas (usually only part of the hyperbola is physically satisfactory.) That is the blue graph in my original post.
If the equation is more complicated then we can usually change the form of the equation. For example say we have
\(\displaystyle y = Ae^{mx}\)
Take the natural log of both sides:
\(\displaystyle ln(y) = mx + ln(A)\)
This is a line graph if we plot ln(y) vs. x. (The slope is m and the intercept is ln(A).)
Other problems have you interpret equations. For example, say that the power output through a resistor is constant. How does the voltage drop across the resistor vary with respect to the current? Well
\(\displaystyle P = IV\)
\(\displaystyle V = \frac{P}{I}\)
where P is a constant. If we graph V vs. I (this is typically the form "y vs. x" but always check the order) then this is a inverse proportion. That is to say as we increase I we decrease V. (Look at the blue graph above.)
There are obviously graphs where this will not work (just look at the distance vs. time graph for a particle with a constant non-zero acceleration: it's a parabola.) But typically when you need to discuss a relationship on a test it will be in one of the proportion formats.
-Dan