Graphing a function detailed algorithm I am trying to create a list of all elements that should be taking into account when trying to graph a function :


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*intersection with X-Y Axis

*Symmetric properties  

*Asymptote

*Critical points

*upward and downward intevals

*Convex

*Inflection point


I have looked online for a detailed algorithm for it but did not found one so I have started to write one.
when finding vertical asymptote I found the definition on wikipedia but did not understand what should be done when the function in not defined in the point   
And when do I know that I have a vertical asymptote and not discontinuity point? I need to check discontinuity by definition?
 A: Most, but not all, vertical asymptotes happen at a boundary point of the domain where the function is undefined. An example is $\frac 1x$ or any rational function where the denominator is zero and the numerator is not. More examples are $\tan x$ and $\sec x$, which can be written as rational functions of trigonometric functions, and again the vertical asymptote is where the denominator is zero and the numerator is not. This and some other ideas mean you should add two ideas to your algorithm:


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*Domain of the function

*Behavior at the boundary of the domain (including vertical asymptotes) (whether or not the function is defined there)


A horizontal asymptote happens when the limit of the function is finite when $x$ approaches positive or negative infinity. An oblique asymptote happens when the difference between the function and a non-horizontal line approaches zero as $x$ approaches positive or negative infinity. In your graph you also want to catch when the function approaches positive or negative infinity in ways other than an oblique asymptote, such as the ends of a polynomial function. Thus you should add these ideas:


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*Range of the function

*End behavior (including horizontal and oblique asymptotes)


There is another idea I often use that you left out. I give it the name "special point" and depends on the function. For example, knowing the vertices of a hyperbola (such as $\frac{ax+b}{cx+d}$) helps me to graph it. You included some special points, such as the $y$-intercept of an exponential function or $x$-intercept of a logarithmic function. But that leaves out others such as the mid-line of a sine or cosine function, vertices, and so on. So probably you should add:


*

*Special points (depending on the kind of function)



When you ask "what to do," do you mean the kinds of things a mathematician or student would do or the kinds of things a computer program would do? Those are not the same. My answers above are slanted to what a mathematician/student would do.
