Function study step by step

Studying a function (what is often called curve sketching) is the most dreaded exercise of the final year of high school, and at the same time the most predictable. That sounds like a contradiction, but it isn’t: it is dreaded because it is long and pulls together half of the calculus syllabus; it is predictable because the method is always the same, from the first function to the last one you will ever study. There is no flash of insight to have and no trick to guess. There is a list of steps, and you go through it in order.

The problem I see in lessons is almost always the same: the student knows the individual pieces (can take a limit, can differentiate) but doesn’t have the procedure that connects them in their head. So they jump from one step to another at random, get lost halfway, and end up drawing a graph that has nothing to do with the calculations they have just done. In this article I give you the complete method, in the right order, with the reason why each step sits where it does, and then I apply it to a whole function to show you how everything fits together.

The guide in six parts

This article is the map: you get the full method, one example worked all the way through and the most common mistakes. The first two chapters group two steps each (domain with symmetries, intercepts with sign), while the other four cover one step each. Each has a dedicated deep dive with all the special cases and more worked examples.

Domain and symmetries: conditions of existence, systems, even and odd functions, periodicity.
Intercepts and sign study: zeros, multiplicity, sign table.
Limits and asymptotes: vertical, horizontal, oblique, and the cases where there are none.
Monotonicity, maxima and minima: sign of $f^{'}$ , stationary points, cusps and corner points.
Concavity and inflection points: sign of $f^{''}$ , classifying inflection points, the inflectional tangent.
The graph and three worked examples: final assembly, consistency checklist, managing your time in the exam.

If you are short of time, read only this page; if you are preparing for the maturità (Italy’s school-leaving exam), do the full tour.

What function study is really for

Before the method, something almost nobody says: function study is not a calculation exercise. It is an exercise in gathering information. You don’t know the graph, and nobody can draw it for you. What you do is ask the function a series of questions (where does it exist? where is it positive? what does it do at infinity? where does it rise and where does it fall?) and every answer rules out possibilities, until practically only one graph is compatible with all the information.

This is the right mindset, and it changes how you approach the exercise. Each step is one more constraint on the final drawing. That is why the order matters: some questions only make sense once you have answered the earlier ones (asking “where are the vertical asymptotes?” before you have found the domain is hot air).

And above all, for this reason, the graph is drawn at the end, but the pencil is used from the start. The most useful advice I always give: sketch the axes in a corner of the page as soon as you have the domain, and every time you find a piece of information, mark it there. The graph doesn’t appear at the end from a stroke of genius: it is built piece by piece while you do the calculations.

The complete method, in the right order

Here are the steps. Learn this list by heart; it is the only thing to memorise in the whole topic.

Domain: where the function exists.
Symmetries and possible periodicity: even, odd, periodic.
Intercepts with the axes: where the graph crosses $x$ and $y$ .
Sign of the function: where it lies above and below the $x$ -axis.
Limits at the ends of the domain and asymptotes: vertical, horizontal, oblique.
First derivative: increasing, decreasing, maxima and minima.
Second derivative: concave, convex, inflection points.
Graph: putting it all together.

Let’s go through them one by one, with what you really need to know.

1. Domain

This is the step everything else depends on: if you get the domain wrong, every later calculation is wrong. The conditions to check are always the same three:

denominators different from zero;
argument of even-index roots greater than or equal to zero;
argument of logarithms strictly greater than zero.

To these you add the less frequent cases: the tangent (where the cosine vanishes), arcsine and arccosine (argument between $- 1$ and $1$ ), and functions with a non-integer exponent.

When there are several conditions together, they go in a system, not as alternatives: they must all hold at the same time. It is a trivial mistake but a very frequent one under test conditions.

2. Symmetries

Thirty seconds that can halve your work. Compute $f (- x)$ :

if $f (- x) = f (x)$ the function is even → graph symmetric about the $y$ -axis;
if $f (- x) = - f (x)$ it is odd → symmetric about the origin;
if it is neither, move on without another thought.

If the function is even or odd you can study it only for $x > 0$ and mirror the graph. At the maturità, with the clock ticking, that is a gift.

3. Intercepts with the axes

The intercept with the $y$ -axis is simply $f (0)$ , if $0$ belongs to the domain. The one with the $x$ -axis requires solving $f (x) = 0$ .

A practical warning: sometimes the equation $f (x) = 0$ can’t be solved with the tools you have. It isn’t a disaster and it doesn’t mean you did something wrong. You write “the equation cannot be solved by elementary methods” and carry on: you will work out the position of the zeros from the sign and the graph anyway.

4. Sign

Solve the inequality $f (x) > 0$ . The result tells you on which intervals the graph lies above the $x$ -axis and on which below.

The mistake I see most often here is one of method: many students treat the sign as an exercise in its own right and then don’t use it. Yet it is pure gold, because it closes off whole regions of the plane. If you know that between $1$ and $3$ the function is negative, it simply can’t pass above the axis on that stretch. Mark it straight away on the axes you drew: I add light shading on the forbidden zones, so the final graph almost draws itself.

5. Limits and asymptotes

You compute the limits at the ends of the domain: where the domain is open or excluded, and at $\pm \infty$ if the function reaches it. The asymptotes come from these limits, and there are three kinds.

Vertical asymptote at $x_{0}$ : you have one when

x \to x_{0} lim f (x) = \pm \infty

The right-hand and left-hand limits must always be computed separately, because the function can diverge to $+ \infty$ on one side and to $- \infty$ on the other.

Horizontal asymptote: if the limit at infinity is finite,

x \to \pm \infty lim f (x) = L

the line $y = L$ is a horizontal asymptote.

Oblique asymptote: you look for it only if the limit at infinity is infinite and there is no horizontal asymptote on that side. The line is $y = m x + q$ , with

m = x \to \pm \infty lim \frac{f ( x )}{x}

and, if $m$ is finite and non-zero,

q = x \to \pm \infty lim [f (x) - m x]

If you need a refresher on the actual computation of limits, I wrote about it in the post on limits explained properly; here I take them as known.

6. First derivative

Differentiate the function and study the sign of $f^{'} (x)$ , not its value. Where $f^{'} (x) > 0$ the function increases, where $f^{'} (x) < 0$ it decreases. The points where the derivative vanishes or doesn’t exist are the candidates for maximum and minimum: candidates, not yet winners. You confirm by looking at how the sign of $f^{'}$ changes around the point: from $+$ to $-$ is a maximum, from $-$ to $+$ a minimum, and if the sign doesn’t change it is an inflection point with a horizontal tangent.

Watch out too for points where $f^{'}$ doesn’t exist but the function does: cusps and corner points. These are what separate a 7 from a 9 in the maturità. If you are still unsure about the differentiation rules, the post on derivatives explained properly covers everything you need.

7. Second derivative

Same logic one floor up: you study the sign of $f^{''} (x)$ . Where it is positive the function is convex (concave up), where it is negative it is concave (concave down). The points where the concavity changes are the inflection points.

This is the step everyone cuts when time is short, and sometimes that is the right call, but know what you are losing: without the second derivative the graph has the right shape but not the right curvature, and in the marking schemes of the second paper (the maths paper of the maturità) this costs points.

8. The graph

If you have marked the information on the axes as you went along, it is now almost a tracing job. Draw the dashed asymptotes first, then mark the zeros, maxima, minima and inflection points, and finally join them up respecting the increasing/decreasing behaviour and the concavity you found.

Then do a consistency check, always. Does the graph respect the sign you computed? Does it approach the asymptotes as the limits say? Is the maximum you drew higher than the nearby points? If something looks off, there is an error in the calculations, and finding it now is worth far more than noticing it once the paper has been handed in.

A complete example

Take

f (x) = \frac{x ^{2}}{x - 1}

the classic that turns up on half the tests in Italy.

Domain: $x - 1 \neq = 0$ , so $D = R ∖ {1}$ .

Symmetries:

f (- x) = \frac{x ^{2}}{- x - 1}

which is neither $f (x)$ nor $- f (x)$ . No symmetry.

Intercepts: $f (0) = 0$ , so it passes through the origin. And $f (x) = 0$ requires $x^{2} = 0$ , that is $x = 0$ : the origin is the only intercept, and it is a double zero.

Sign: the numerator $x^{2}$ is always $\geq 0$ , so the denominator decides the sign. $f (x) > 0$ for $x > 1$ ; for $x < 1$ (excluding $x = 0$ where it vanishes) the function is negative. We already know that to the left of the asymptote the whole graph lies below the $x$ -axis.

Limits and asymptotes: as $x \to 1^{-}$ the denominator tends to $0^{-}$ and the numerator to $1$ , so $f \to - \infty$ ; as $x \to 1^{+}$ we have $f \to + \infty$ . Vertical asymptote $x = 1$ . As $x \to \pm \infty$ the function diverges, so there is no horizontal asymptote: we look for the oblique one.

m = lim \frac{x ^{2}}{x ( x - 1 )} = 1

q = lim [\frac{x ^{2}}{x - 1} - x] = lim \frac{x}{x - 1} = 1

Oblique asymptote: $y = x + 1$ .

First derivative:

f^{'} (x) = \frac{2 x ( x - 1 ) - x ^{2}}{( x - 1 ) ^{2}} = \frac{x ^{2} - 2 x}{( x - 1 ) ^{2}} = \frac{x ( x - 2 )}{( x - 1 ) ^{2}}

The denominator is always positive, so the sign is that of $x (x - 2)$ : positive for $x < 0$ and $x > 2$ , negative between $0$ and $2$ . Hence a relative maximum at $x = 0$ with $f (0) = 0$ , and a relative minimum at $x = 2$ with $f (2) = 4$ . Note the counterintuitive fact that throws everyone: the maximum is $0$ and the minimum is $4$ , so the “minimum” sits higher than the “maximum”. That is perfectly normal: they are relative extrema, belonging to two separate branches divided by the asymptote.

Second derivative: differentiating again gives

f^{''} (x) = \frac{2}{( x - 1 ) ^{3}}

which is positive for $x > 1$ and negative for $x < 1$ . Concave up on the right branch, concave down on the left one, no inflection points (at $x = 1$ the function doesn’t exist).

Graph: the left branch lies entirely below the $x$ -axis except for the origin, which it touches at a maximum, and then goes down towards $- \infty$ as it approaches $x = 1$ ; the right branch comes down from $+ \infty$ , touches the minimum $(2, 4)$ and then climbs back up, resting on the oblique asymptote. Everything is consistent.

The five mistakes I see most often

A sloppy domain. One forgotten condition (typically the logarithm, or a root inside a fraction) and the whole exercise is compromised. Write the conditions as a system, always.
Looking for the oblique asymptote when there is a horizontal one. On the same side they are mutually exclusive. If the limit at infinity is finite, you are already done.
Studying the value of the derivative instead of its sign. What you need to know is where $f^{'}$ is positive, not what it equals. Make the sign table, as for any inequality.
Mixing up the sign of $f$ with that of $f^{'}$ . They are two different tables and must be kept apart on the page. A function can be negative and increasing with no contradiction at all.
Drawing a graph that contradicts the calculations. It happens when the graph is drawn “by feel” at the end. The final consistency check, thirty seconds, wins back a lot of points.

How we work on this topic in lessons

The path I use is always the same, because it works:

Method first, then calculations. In the first lesson we do function studies “out loud”, calculating almost nothing: we look at a function and say which questions to ask and in what order. It builds the automatic habit, which is the real content of the topic.
Graphs to read, not just to draw. I give you a graph and ask you to work out the domain, the sign and the sign of the derivative. It is the reverse route, and it demolishes the idea that calculus is a sequence of disconnected calculations.
One family of functions at a time. Rational functions first, then irrational ones, then exponentials and logarithms, then absolute values. Each family has its own two or three typical stumbling blocks, and tackling them all mixed up is the fastest way to get confused.
Timed exam papers. In the second paper, function study has an allotted time: practising without a clock gives false confidence. We work on real papers from previous years.

On timing I am honest, and I distinguish two different things. To make the method automatic (always knowing which question to ask and in what order), with limits and derivatives already solid, 3-4 lessons are enough. To go into the detail of every step, with all the special cases you find in the six chapters above, the full path is 7-9 lessons. If you also need to recover the computation of limits or the differentiation rules, add a few sessions. Either way we are talking weeks, not months: the topic is finite and well bounded.

For how the hours are organised and what a path like this costs, everything is on the pricing page, with no surprises.

Among my students, final-year maths is bread and butter: average grade 8.2/10, over 215 students followed and 97% of admission tests passed. Function study is typically the point where the calculus syllabus “closes”: whoever masters it reaches the maturità feeling in control, and it shows in the grade.

If you are preparing for the exam, practise with the free maturità maths practice tests: they have step-by-step guided solutions, including a complete function study. And if you want to see how it fits with the rest of the syllabus, on the maths page you will find all the topics I cover, while those aiming at university can have a look at the admission tests. To see limits and asymptotes from the point of view of the final-year test, with three worked exercises, there is limits, continuity and asymptotes: the Year 5 test.

If function study is your stumbling block, or if the final year has just begun and you want to reach it without panic, get in touch for a free 30-minute introductory call: we work out where you are and build the shortest path to where you want to be.

Want to test yourself? Take the function study revision quiz: 15 questions, solutions explained step by step.

Function study step by step: the method that always works