Could you point me out some clear and extensive sources in this regard, please? I haven't found an interesting and extensive document so far.

I have knowledge about parent graphs of functions and their transformations but limited to simple opertations: like stretching/shrinking by multiplication with a costant greater/smaller than 1 and 0, reflection by multiplication by -1, vertical translations by sum and difference with constants and horizontal translations by sum and difference with constants inside products. However, I would like to have more clear in my mind the effects of more complex transformations due to operations between functions (like addition, subraction, multiplication, division, composition between 2 or more functions).

It is for getting more knowledge as I like it but also because I think it helps in order to study the function when there is to determine the right operation for finding its range - or am I wrong and it is useless the knowledge for which I am asking advice in order to find ranges of functions?

PS. You can make some examples of elementary functions and how to graph them, to render more concrete this question. Thus, I can also gain some knowledge from your examples.


Well, when it comes to graphing any sort of function, a very extensive analysis would be that involving its first and second derivatives. Everything pointed out by mvw is great, though perhaps it could be a bit more explained. Anyways, if you really want to get a intuitive feel of how different graphs look like, I recommend you download a graphing calculator, perhaps in your phone, and just play with it; it will do wonders. So, more specifically:

  • Domain: finding out the domain is particular to the different types of functions but you always have to look out for: divisors different than cero, whatever is inside the square root (if there is), domain of the log(x) function.
  • Image is harder to look at before-hand, so I wouldn't worry about it too much at first
  • Intersection with the axis are very useful, but of course the roots are not always easy to find.
  • About derivatives, there are extensive resources online. Still, where the first derivative of a function is positive, the function is increasing; where it is negative it is decreasing.
  • The second derivative, where positive, tells you if the function has a "smiley face" shape, or where negative, that has a "sad face" shape.
  • With the limits, you can evaluate what happens to the function as it approaches its poles, and $\pm \infty$

Using all that, plus the transformations you already know, plus playing with the graphs, I'm sure you'll have a great intuition in no time!

I found these links online for more help: http://faculty.swosu.edu/michael.dougherty/book/chapter05.pdf http://www.teaching.martahidegkuti.com/shared/lnotes/6_calculus/analysis/analysis.pdf

Hope it helps!


In German maths teaching in school, around 10th or 11th year, there is the subject Kurvendiskussion, which should be translated as "Discussion of [the properties of] a Curve". It is a systematic poking of a given function for characteristic properties of its graph.

  • Domain
  • Intersections with $x$- and $y$-axis
  • Symmetries
  • Extrema
  • Inflection points
  • Poles
  • Gaps
  • Limits

This information allows to sketch the graph.

Have a look at the Example, to get an impression.


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