Function study, part 6: the graph and three worked examples

The graph doesn’t appear at the end, all at once, like a prize you earn after doing all the calculations. It is built piece by piece, one at a time, while you work. The most useful advice I always give at this stage is a simple one, but it changes everything: sketch the Cartesian axes in a corner of the page as soon as you have the domain, and from then on mark each piece of information as soon as you find it: a dashed asymptote as soon as you compute it, a point as soon as you find it, a shaded zone as soon as you know the function can’t pass there. At the end you don’t “draw the graph”: you assemble it, because it is already almost all there.

This is the sixth and last part of the series on the complete guide to function study. In part 5: concavity and inflection points we closed the last piece of information, the second derivative. Here we put everything together: the order in which you draw, the check that saves you from the costliest mistakes, and three complete examples worked from start to finish.

The order in which you draw

Not all the information goes down in pencil at the same time. There is an order that reduces mistakes, because each new piece rests on those already drawn:

1. Dashed asymptotes. Vertical, horizontal, oblique: they are the “guides” of the drawing, and they come first because everything else has to respect them.
2. Regions forbidden by the sign, lightly shaded. Where you found $f(x)<0$ or $f(x)>0$, shade (even just mentally, or with light hatching) the zones of the plane where the graph cannot pass. It is a visual safety net.
3. Notable points. Intercepts with the axes, maxima, minima, inflection points, all with the coordinates written next to them, not just a dot.
4. Behaviour near the asymptotes, according to what the limits say: from above or from below, towards $+\infty$ or $-\infty$ from the right or from the left.
5. Join the pieces, respecting monotonicity (where it rises, where it falls, according to the sign of $f^{\prime }$) and concavity (where it bends upwards or downwards, according to the sign of $f^{\prime \prime }$).

Once you have done this, you have a drawing that is not an opinion: it is the direct consequence of your calculations.

The final consistency check

Before you consider the graph finished, stop for two minutes and go through this checklist. It is the step I see skipped most often, and the one that is worth the most: an error found now costs thirty seconds, the same error discovered after handing in costs points you never get back.

1. Does the graph respect the sign you computed? If you wrote $f(x)<0$ for $x\in (0,2)$ and the curve on that interval lies above the x-axis, you have an error somewhere, in the sign or in the drawing.
2. Does it approach the asymptotes as the limits say? If the limit as $x\to +\infty$ is $0^{+}$, the curve must approach $y=0$ from above, not from below.
3. Is the maximum you drew really higher than the nearby points? It sounds obvious, but it happens that you place a “maximum” lower than an adjacent point because the coordinates were copied wrongly.
4. Does the concavity match the sign of $f^{\prime \prime }$? Where $f^{\prime \prime }>0$ the curve must lie “above its tangents” (convex), where $f^{\prime \prime }<0$ below (concave). If the drawing does the opposite of what you wrote, one of the two is wrong.
5. Is the number of intersections with the x-axis the one you found? If the sign study gave you two zeros and the graph shows three, something doesn’t add up.
6. Does the graph cross a region the sign forbade? It is the quickest check and the most effective: just see whether the curve “trespasses” into a zone you shaded in step 2.
7. Is the stretch between two notable points consistent with the monotonicity? If between a maximum and a minimum the function has to be decreasing all the way, the curve can’t make a bump in the middle.

If something looks off, it isn’t the graph that is wrong: it is a calculation. Go back and recheck the sign, the limits, the first derivative or the second derivative until the drawing and the numbers tell the same story.

Example 1: a rational function

This is the function I use as the thread through the whole series: you have already met it in pieces in the previous chapters, and here you see it assembled from start to finish in one go.

1. Domain

The denominator vanishes at $x=1$, so the domain is

2. Symmetries

different from both $f(x)$ and $-f(x)$: neither even nor odd symmetry.

3. Intercepts with the axes

With the y-axis: $f(0)=0$. With the x-axis: $x^2=0\Rightarrow x=0$ (double zero). The only intercept is the origin, and the function touches the x-axis at that point without crossing it (zero of even multiplicity).

4. Sign

$x^2\ge 0$ always, so the sign of $f$ coincides with the sign of $x-1$: $f(x)>0$ for $x>1$, $f(x)<0$ for $x<1$ (excluding $x=0$ where $f=0$).

5. Limits and asymptotes

At $x=1$: from the left the numerator tends to $1$ and the denominator to $0^{-}$, so $f\to -\infty$; from the right $f\to +\infty$. Vertical asymptote $x=1$.

Dividing, $x^2=(x-1)(x+1)+1$, so

As $x\to \pm \infty$ the term $\frac{1}{x-1}\to 0$, so there is an oblique asymptote $y=x+1$: from above as $x\to +\infty$ (the remainder is positive), from below as $x\to -\infty$ (the remainder is negative).

6. First derivative

The denominator is always positive (where defined), so the sign of $f^{\prime }$ coincides with the sign of $x(x-2)$: positive for $x<0$, negative for $0<x<2$ (with $x\ne 1$), positive for $x>2$. Relative maximum at $(0,0)$, relative minimum at $(2,4)$.

7. Second derivative

Negative for $x<1$ (concave), positive for $x>1$ (convex). No inflection point, because the sign change happens at a point excluded from the domain.

8. Graph

What the graph looks like: on the left, the curve starts “resting” on the oblique asymptote $y=x+1$ from below, rises to the maximum at $(0,0)$, then falls (concave all the way) until it plunges towards $-\infty$ as $x\to 1^{-}$. On the other side of the vertical asymptote, the right branch starts again from $+\infty$ (as $x\to 1^{+}$), falls to the minimum at $(2,4)$, then rises again, always convex, approaching the oblique asymptote $y=x+1$ from above as $x\to +\infty$. The two branches never touch: they are separated by the vertical barrier at $x=1$.

Example 2: an exponential function

1. Domain

No restrictions: a product of functions defined everywhere.

2. Symmetries

different from both $f(x)$ and $-f(x)$: no symmetry.

3. Intercepts with the axes

$f(0)=0$: the only intercept, the origin.

4. Sign

$e^{-x}>0$ always, so the sign of $f$ is the sign of $x$: $f>0$ for $x>0$, $f<0$ for $x<0$.

5. Limits and asymptotes

As $x\to +\infty$: $x{e}^{-x}\to 0^{+}$ (the exponential always beats the polynomial), so horizontal asymptote $y=0$ on the right. Since $f(x)>0$ for large $x$, the curve rests on the asymptote from above.

As $x\to -\infty$: $x\to -\infty$ and $e^{-x}\to +\infty$ (because $-x\to +\infty$), so $f\to -\infty \cdot +\infty =-\infty$.

We need to check whether there is an oblique asymptote on the left:

The slope is not a finite number, so there is no oblique asymptote: on the left the function plunges faster than any line, as steep as you like.

6. First derivative

$e^{-x}>0$ always, so the sign of $f^{\prime }$ is the sign of $1-x$: positive for $x<1$, negative for $x>1$. Relative maximum at $\left(1,\frac{1}{e}\right)$.

7. Second derivative

Sign = sign of $x-2$: negative (concave) for $x<2$, positive (convex) for $x>2$. Inflection point at $\left(2,\frac{2}{e^2}\right)$.

8. Graph

What the graph looks like: on the left the curve comes in from $-\infty$ with an ever steeper slope (no asymptote slows it down), crosses the origin, and rises (concave) to the maximum at $\left(1,\frac{1}{e}\right)$. From there it falls, stays concave until $x=2$, then becomes convex at the inflection point $\left(2,\frac{2}{e^2}\right)$ and keeps falling, ever more slowly, flattening against the horizontal asymptote $y=0$ without ever touching it: for large $x$ the curve is pressed against the x-axis from above.

Example 3: a logarithmic function

1. Domain

The logarithm requires $x>0$, so

2. Symmetries

The domain is not symmetric about the origin (it doesn’t contain the negatives), so you can skip the study of symmetries altogether.

3. Intercepts with the axes

None with the y-axis (zero is not in the domain). With the x-axis: $\ln x=0\Rightarrow x=1$. A single zero at $(1,0)$.

4. Sign

The denominator $x$ is always positive in the domain, so the sign of $f$ is the sign of $\ln x$: negative for $0<x<1$, positive for $x>1$.

5. Limits and asymptotes

As $x\to 0^{+}$: $\ln x\to -\infty$ and $x\to 0^{+}$, so $f\to -\infty$. Vertical asymptote $x=0$.

As $x\to +\infty$: $\frac{\ln x}{x}\to 0^{+}$ (standard limit: the logarithm always loses against the power). Horizontal asymptote $y=0$, with the curve resting on it from above (since $f>0$ for $x>1$).

6. First derivative

The denominator is always positive, so the sign of $f^{\prime }$ is the sign of $1-\ln x$: positive for $x<e$, negative for $x>e$. Relative maximum at $\left(e,\frac{1}{e}\right)$.

7. Second derivative

The denominator $x^3$ is always positive in the domain, so the sign of $f^{\prime \prime }$ is the sign of $2\ln x-3$: negative (concave) for $x<e^{3/2}$, positive (convex) for $x>e^{3/2}$. Inflection point at $x=e^{3/2}$, with

8. Graph

What the graph looks like: the curve starts from $-\infty$ just above $x=0$ (vertical asymptote), rises steeply, crosses the x-axis at $(1,0)$, and keeps rising (always concave) to the maximum at $\left(e,\frac{1}{e}\right)$. From there it falls, stays concave until $x=e^{3/2}\approx 4.48$, becomes convex after the inflection point, and carries on in a very slow descent to the right, resting from above on the horizontal asymptote $y=0$ without ever reaching it.

Summary table

The pattern that jumps out: rational functions whose numerator has degree exactly one more than the denominator give an oblique asymptote (as in example 1, which happens to have no inflection point); exponentials of the type $x^n{e}^{-x}$ always have a horizontal asymptote on the right and unchecked growth on the left; and logarithmic functions of the type $\frac{\ln x}{x}$ live only to the right of zero, with a single maximum and a very slow descent towards the horizontal asymptote.

Exam variations

The area under the curve

If the question asks for the area of a region bounded by the graph, you usually first need to find the limits of integration (often exactly the intersections with the x-axis or with another curve that you already computed in step 3), then integrate $|f(x)|$ between those limits. Watch the absolute value if the function changes sign on the interval, otherwise the area comes out negative and wrong.

The tangent line at a point

You need $f^{\prime }(x_0)$, which you have almost always already computed in step 6. The tangent is

there is no other work to do, just substitute the numbers you already have in hand.

Discussion with a parameter

When a parameter appears (like $f(x)=\frac{x^2}{x-k}$), the method doesn’t change: you redo the same eight steps keeping $k$ as a letter, and at each step you note for which values of $k$ the conclusion changes (for example the domain always changes with $k$, but the number of intercepts might change only for certain values). The trick is to isolate straight away the conditions on $k$ that change the structure of the graph, rather than waiting until the end.

The graph of $|f(x)|$ and $f(|x|)$

For $|f(x)|$: keep the part of the graph where $f(x)\ge 0$ fixed, and flip above the x-axis the part where $f(x)<0$ (mirror in the x-axis). For $f(|x|)$: keep the part with $x\ge 0$ and reflect it to the left of the y-axis as well, erasing what you had drawn for $x<0$ (the graph becomes even-symmetric). They are geometric transformations and need no new calculations, but they need you to already have the graph of $f$ correct, otherwise you propagate the error.

Time management in the second paper

For a complete function study with all eight steps, a reasonable time in the exam is around 25-30 minutes. If you are behind, the order of cutting is precise:

1. Symmetries, if they are not obvious at a glance: you can mention “no obvious symmetry” and move on without the explicit calculation.
2. Second derivative, if time is really tight: a graph with the domain, sign, asymptotes and first derivative correct is still worth almost the whole mark, and you can draw a “plausible” concavity, stating that you haven’t checked it.

What you never cut: domain, asymptotes, first derivative. They are the three pieces of information that fix the shape and behaviour of the graph; without them the drawing is not a function study, it is a sketch.

The mistakes I see most often

1. The graph contradicts the calculations. You wrote $f(x)>0$ for $x>2$ but you draw the curve below the x-axis in that zone. It is the most serious mistake because it shows you didn’t do the final consistency check.
2. Absurd axis scales. If the maximum is $0.01$ and you use the same unit for an asymptote that reaches $50$, the graph becomes unreadable and the notable points pile on top of each other. Choose different scales on x and y if needed, and say so.
3. Asymptotes drawn with a solid line instead of a dashed one. It looks like an aesthetic detail, but it is a convention that signals whether you understood that this line is not part of the graph.
4. The curve touching the vertical asymptote. A vertical asymptote is never touched: the curve approaches it infinitely closely but always stays at a distance. If in your drawing the curve “lands” on the dashed line, it is a conceptual mistake, not just a graphical one.
5. Notable points without coordinates written next to them. A dot on the graph with no label doesn’t count as a “notable point identified”: always write the exact coordinates next to the point; that is what gets marked.

How we work on this in lessons

• Method out loud before the calculations. Before writing a single calculation, I have you describe in words what shape you expect from the graph, based on the type of function. If the prediction and the final result don’t match, we find the error straight away.
• Graphs to read backwards. I give you an already drawn graph and ask you to work out the domain, sign, sign of $f^{\prime }$ and of $f^{\prime \prime }$. It is the exercise that consolidates most, because it forces you to connect drawing and information in both directions.
• One family of functions at a time. Rational functions one week, exponentials the next, logarithms after that: each family has its specific traps (oblique asymptotes, growth with no oblique asymptote, restricted domain), and mixing them too early confuses more than it helps.
• Timed exam papers. In the last lessons before the maturità (Italy’s school-leaving exam), we simulate the function study in the real time of the second paper, to train time management too, not only the method.

On timing I am honest: if limits and derivatives are already solid (parts 3 and 4 of this series), 3-4 lessons are enough to assemble a complete function study without hitches. If instead we want to revise every step of the series properly, one at a time, the realistic path is 7-9 sessions, and a few more if the basics of limits and derivatives need recovering too. How it works and the hours are explained on the pricing page.

With this part the series on function study closes: domain and symmetries, intercepts and sign, limits and asymptotes, first derivative, second derivative, and now the final assembly into the graph. It is the chapter that on its own is often worth a quarter of the second maths paper, and among my students it remains one of those where you see the sharpest jump between those who treat it as a sequence of formulas and those who treat it as a piece of reasoning. Average grade 8.2/10, over 215 students followed, 97% of admission tests passed: these are numbers that come above all from here, from the “long” chapters done well.

If you have followed the whole series, the natural next step is to test yourself: the free maturità maths practice tests have several complete function studies with guided solutions, perfect for checking whether the method is really yours or whether you still need training on some specific step. To revisit the previous piece there is part 5: concavity and inflection points, and for the overall picture of the whole series the complete guide to function study remains valid. If instead you are preparing for an entrance test where function study appears in a more compressed form, the admission tests page explains how I treat it in that context; for the full final-year maths path you will find the general overview.

If this chapter is the one holding you back, or if you just want to understand where you stand before the exam, get in touch for a free 30-minute introductory call: we look together at a couple of functions that don’t work out for you, and I tell you honestly what it takes to arrive ready.

The other parts of the series, if you need them out of order: part 1: domain and symmetries, part 2: intercepts and sign study, part 3: limits and asymptotes, part 4: monotonicity, maxima and minima. If the derivative steps slow you down, derivatives explained properly is the targeted revision; for a timed check there is the Year 5 test on function study. Limits and asymptotes, where the graph starts, seen from the test’s point of view, are in limits, continuity and asymptotes: the Year 5 test.