When analyzing the functions

$f(x) = \sqrt{x-5}$

$g(x) = \frac{1}{x-5}$

$h(x) = 2^{x-5}$

we know that it is useful to think about what happens at $x = 5$.

For the function $f$, this logic will yield the vertex of a graph.

For the function $g$, this logic will yield the vertical asymptote of a graph.

For the function $h$, this logic doesn't really yield and particularly special point, but still helps us draw a graph.

I have been telling my algebra students that in all three cases,

$x = 5$ is an important $x$-value, or

all the action is happening near $x=5$.

However, my textbook does not talk this way or make this connection across different types of functions. Is there any better way to talk about this idea? Does the idea have a better name than "important x-values?" The word "critical" is already taken. The word "important" is imprecise. What should I be saying to my students?

  • 4
    $\begingroup$ These are all roots of some part of the expression (the radicand, the denominator and the exponent). I might invent a name for this kind of value (e.g. a "key number"), but I think I would also want to take the time to use the word for which part of the expression my eye is drawn toward (e.g. the "denominator"). I don't have a good name for this kind of value, so I don't think this qualifies as a real answer to your question. $\endgroup$
    – Nick C
    Mar 16, 2023 at 16:40
  • $\begingroup$ "Explanatory"? "Delineating"? "Discriminating"? $\endgroup$ Mar 16, 2023 at 23:24
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    $\begingroup$ One time I couldn't think of a word for where the second derivative was zero, but not necessarily an inflection point. So I told the class they could make up their own name. It was for the rest of the semester called a "pizza" value. It went on the exams and everything. The point is your question is important! $\endgroup$
    – cheyne
    Mar 31, 2023 at 22:46

5 Answers 5


For function transformations I use "base point" ($y=a^x$ has base point $(0,1)$, $y = \frac{1}{x}$ has base point $(0,0)$, even though it isn't on the graph).

In calculus I use "point of interest" (for critical and hypercritical points), which can be useful language if you want to talk about multiple points moving as you transform your function.

  • $\begingroup$ I used the phrase "basic points" (mostly 1996-2004 I think), for example the two-three among (-1,1) & (0,0) & (1,1) that are relevant for $y = x^r$ when $r$ is an integer or $r = 1/2$ or $r = 1/3$ or $r = -2/3$ etc. Excerpt from instructions for a 24 Sep 1998 (handwritten) quiz of mine I just looked up: Draw a labeled sequence of graphs beginning with a basic graph (equation) and ending with what I give you. In each graph, give (and label) the points corresponding to those points in the basic graph whose $x$-coordinates I give you. The first (of two) questions (continued) $\endgroup$ Mar 16, 2023 at 21:59
  • $\begingroup$ has $y = -(x-2)^3 + 1$ along with "Basic points for $x=-1,\,0,\,1$" along with 4 hand-drawn unlabeled $xy$-axes (for students to sketch the various graphs in) each having below it the word "equation" with a long answer line after the word for students to write the equation for the graph they sketched. The second question is for $y = -\sqrt{x-3}$ and basic points $x=0,\,1$ provided with 3 unlabeled $xy$-axes and 3 blanks for equations. One of the things I checked for when grading was consistency between the graph and the equation they gave (continued) $\endgroup$ Mar 16, 2023 at 21:59
  • $\begingroup$ (thus, partial credit was possible even if wrong in other ways, unless they did something silly like using $y=x^2$ and its basic points for every graph), even if the graph and equation was not appropriate in the sense of arising from a single transformation (vertical, horizontal, reflection about an axis, etc.) applied to the previous graph/equation. (This was explained fully in class. Indeed, I did some on the board, and also passed out "sample quizzes" for class worksheets they worked on as I came around and helped as needed.) (continued) $\endgroup$ Mar 16, 2023 at 22:00
  • $\begingroup$ Obviously these were no-calculator quizzes -- an arithmetical calculator was not needed, and a graphing calculator would be inappropriate. In case I need to explain, being able to quickly sketch out things like this is very helpful in setting up definite integrals for areas and other applications of definite integrals, and in many other first year calculus tasks (e.g. applying intermediate value property to determine how many and approximate location of solutions to something like $e^x - x^2 + 4 = 0.)$ $\endgroup$ Mar 16, 2023 at 22:06
  • $\begingroup$ I say the same: ''points of interest". $\endgroup$
    – ryang
    Mar 29, 2023 at 6:22

When graphing a function (or just investigating it from some point of view), it is, indeed, useful to look at some special points where the behavior changes in some way. Most of them have well-established names like "root", "extremum", "critical point", or "inflection point" and one can just use those. Some are specific to the graphing task, like the easy to plot (say, integer) points where the function takes easy to compute (say, rational) values. What to look for is really task-dependent.

The main idea to convey is that once one has decided what features of the function are interesting in the context of some particular task, some points become "more equal than others" from that particular point of view, and you are absolutely right that understanding this fact is important. Unfortunately, the only way to communicate this general idea is through a variety of worked out examples with different tasks resulting in different sets and notions of "important points". There is no magic bullet that would convey the meaning by just coining a catchy name.

  • 1
    $\begingroup$ "There is no magic bullet that would convey the meaning by just coining a catchy name." Also using a catchy name may lead students to think it's a common term or that there's an exact definition – and make them very confused when their next teacher doesn't understand what they mean by that. $\endgroup$
    – JiK
    Mar 17, 2023 at 14:20

I've used the term "reference point" or "reference coordinate" when referring to the points on the original function, compared to their transformed locations. I've expected my students to know what happens to each of the "reference points" of the original graph. The most typical "reference coordinate" occurs where $x=0$, which in all three of the examples above is transformed to the coordinate $x=5$.

To add to this, I often use the original points on the graph at $x=1$ and $x=-1$ as the remaining reference coordinates, which works on functions like $f(x)=a^x$, $f(x)=x^n$ for integer $n$, etc. For inverse functions like $\log_b(x)$ it is occasionally better to use what occurs at original reference coordinates generated from $y=1$ and $y=-1$.


My advice would not be to strain for a special name like that. Just teach the practice of curve sketching instead (vice inventing a new term, not in the book). f and g are already "important" in the sense that they are the value for one of the step by step parts of the curve sketching practice. But I think you are better off teaching that as a step by step method, versus looking at "term roots" (that's the name I guess) to find which step it corresponds to (especially for weak/new students).


Also, I think it's arguable if the h function has an "important" (key) x point, like the other two do. It's not a value at one of the steps of the typical curve sketching method.

  • $\begingroup$ This is what I think, but I suppose if you (Chris) want a term, maybe something like "translation number". Or more accurately, but probably too long (and also scarier, which you want to avoid), "horizontal translation number". In my experience the key was to identify the underlying "standard graph" (a term I did use) and what was done to it, and for students I think identifying the standard graph and knowing its general shape was the primary task, and the $5$ was not something that had to be identified to avoid students using another number, e.g. preventing them from using $3$ instead of $5.$ $\endgroup$ Mar 16, 2023 at 17:46

Often questions about graph sketching will ask for a sketch showing all relevant points, or in some cases all relevant features. The actual points or features to be shown will vary depending on the specific graph and on what has been covered in the teaching unit. Students of the unit will understand from their studies and practice examples which features are relevant (intercepts, turning points etc) and which are not (points at every integer value of $x$ from $-4$ to $+4$).


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