`f(x) = x^2 +ce^(-x)` Describe how the graph of `f`varies as `c` varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when `c` changes. You should also identify any transitional values of `c` at which the basic shape of the curve changes.
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Although this question looks tough, it is really simple if you break it down into familiar component parts.
f(x) is composed of the component "basis" functions which include x^2 and e^-x. I would first graph these familiar functions and study how they behave over their respective domains. f(x) will have an overall behavior which is determined by the behavior of each of these familiar component functions.
It is important to realize that c is a real constant. Thus, f(x) actually represents a "family of functions" where the MAGNITUDE and SIGN of c will determine which member of the family of functions we are dealing with.
I would recommend that you graph f(x) when c = 0, 1, and -1. These members of the f(x) family will give you a quick sense of how the family of functions will behave for different values of c over the domain of f(x).
Lastly, I would recommend that you look at the "end behavior" of f(x) as x approaches positive infinity as well as negative infinity.
This is the type of logic that must be employed when one attempts to graph functions like f(x). Build upon what you already know to better understand f(x).
If you go to https://mathway.com/graph , you will find a graphing calculator that will allow you to perform the aforementioned explorations.
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