That's an interesting question, have you tried to graph the functions on a graphing calculator ( or any other graphing utility) and then graph their products or sums in the same viewing window and then compare the graphs?
I just finished answering a thread on graphing and I realized that I am lacking in that area as well.
I'm sure I should know this stuff already, but I don't. So I know how to graph trig and exponential functions when they are shifted and things like that, but how do you graph them when you have a combination? For instance, let's say I wanted to graph:
y = e^(2x) + e^(5x) ...to keep it simple
so i can graph e^2x and e^5x, piece of cake, but graphing their sum, I don't know.
or how about products:
e^x * sinx
or e^x * sinx + e^(3x)cosx
or sin(x + 2)*cos(x - 3)
I was also curious about the applications of graphs in biology, when they talk about "steady state solutions" and their graphs, but i guess i'll leave that for another thread another time.
i have seen graphs of functions like these, but i was unable to, at first glance, recognize any helpful similarities between them and the functions i do know how to graph.
Find the derivatives, critical points, asymptotes, turning points, infection points, all that stuff.
Find the graph for e^(2x), easy.y = e^(2x) + e^(5x) ...to keep it simple
FInd the graph for e^(5x), easy.
The new graph is the sum of the coordinates.
The only smart about I see is through Calculus.e^x * sinx
But I happen to know how it looks even before doing any Calculus on it nor graphing it .
That is a famous curve which appears in unstable vibrational systems.
(Have you done that in Differencial Equations?)