Graphs of a function together with its first and second derivatives sit at the intersection of two assessment cultures: the IB Diploma's emphasis on structured mathematical communication and AP Calculus's emphasis on analytic justification. Most candidates who arrive at TestPrep Europe already know the vocabulary: increasing, decreasing, concave up, concave down, local maximum, inflection point. The gap is almost always in the justification: writing a sentence an examiner cannot dock a point from, on a paper where the question itself is already giving away the picture.
This article treats the three-graph problem as a single transferable skill rather than a chapter of two separate syllabuses. IB Mathematics: Analysis and Approaches HL and SL both assess the relationship between f, f' and f'' in Paper 1 and Paper 2 short-response items, and AP Calculus AB and BC test it in multiple-choice, free response and the BC-only 'analysis of graphs' style. The reasoning chain is identical; the mark scheme vocabulary is what shifts. A candidate who masters the chain can answer either exam on the same day without re-learning the mathematics.
What examiners are actually testing when they show three graphs
There is a single principle underneath every f, f' and f'' item, and once it is internalised the rest of the marks fall into place. The principle is this: the graph of f' is a graph of slopes, and the graph of f'' is a graph of how those slopes are changing. The question writer is not asking whether the candidate can label axes. They are asking whether the candidate can read behavioural information off one curve and translate it into language that the rubric accepts on a different curve.
Consider a typical Paper 2 short question at IB AA HL: a graph of f is given, and the candidate is asked to determine intervals on which f is increasing and concave down. The correct justification, in the language IB examiners prefer, is: 'On the interval (a, b), the graph of f rises while its tangent slopes decrease in magnitude then become negative; equivalently, the graph of f' crosses zero from above at x = a and the graph of f'' is below the x-axis on (a, b).' That is three references in one sentence, linking f, f' and f'' by behaviour. Candidates who only reference one curve leave two of the three available marks on the table.
AP Calculus, by contrast, will ask the same question with the answer 'f is increasing and concave down for a < x < b', then ask the candidate to 'justify your answer'. The AP reader awards credit for explicit reference to the sign of f' and the sign of f''. The mathematical content is identical; the rubric just wants a different surface form. I would recommend that IB students preparing for AP or vice versa deliberately write each answer twice, once in each rubric's voice, because the muscle memory is what carries you through a 90-minute paper under time pressure.
The three behaviours you must always verify
- Increasing or decreasing — read directly from the sign of f'.
- Concavity — read directly from the sign of f''.
- Local extrema — sign change of f' through zero, with f'' optionally confirming the nature.
If a candidate skips one of these three readings on a three-graph item, the resulting answer is almost always incomplete. The order of reading matters less than the order of writing. Many IB examiners want the answer in the order: behaviour of f, justification via f', confirmation or contrast via f''. AP readers accept that order but will also accept the reverse, so long as the chain is explicit.
The slope-reading checklist for f, f' and f''
Every three-graph question can be reduced to four reads. Memorising this checklist saves minutes per item, and minutes are the only currency that matters in Paper 2. The checklist is: zeros of f' (where is f flat), sign of f' (where is f rising or falling), zeros of f'' (where does concavity change), sign of f'' (which way is the curve bending). Two reads come from f', two from f''. The graph of f itself only provides the visual confirmation.
Zeros of f' are non-negotiable. They are the x-coordinates of local extrema on f, and they are also the points where the IB rubric wants the candidate to write the phrase 'f' changes sign from positive to negative'. That phrase, in that exact order, is worth one mark in HL Paper 2 Section B. A candidate who writes only 'maximum at x = a' loses that mark. The rubric is checking whether the candidate can read a sign change, not whether they know what a maximum is. For most candidates reading this, that distinction is the single largest source of avoidable point loss.
Sign of f' across an interval requires the candidate to look at the graph of f' and decide which side of the x-axis dominates between two consecutive zeros. The IB accepts the language 'f' > 0 on (a, b) so f is increasing on (a, b)'. AP accepts either order. The candidate who writes only 'f is increasing' leaves two marks on the table: one for the sign statement and one for the explicit reference to the derivative curve.
Zeros of f'' mark inflection points on f. This is the part most candidates miss in IB because the syllabus splits the language: at SL the term 'inflection' is sometimes avoided, while at HL it is required. For the avoidance of doubt, write 'point of inflection' on an HL paper and 'point where f changes concavity' on an SL paper. The mathematics is the same. AP uses the phrase 'inflection point' without hesitation in both AB and BC.
Sign of f'' is the cleanest of the four reads: where f'' is above the x-axis, f is concave up; where f'' is below, f is concave down. The subtlety is the inflection criterion: a sign change of f'' is necessary, and for AP the candidate must additionally verify that f' exists at that point. IB examiners are slightly more forgiving here, but the candidate who writes the criterion explicitly never loses a mark for being over-precise.
Translating behaviour between the three graphs in exam language
The single highest-leverage skill is translation. The examiner hands the candidate a graph of f and a graph of f' and asks which one is which. A surprising number of IB candidates identify them backwards on the first attempt, because they assume the 'higher' curve is f. The rule is: a horizontal tangent on f corresponds to a zero on f', a steep section on f corresponds to a large-magnitude value on f', and a flat section on f corresponds to f' near zero. Once this mapping is fluent, the question becomes self-checking.
Let me walk through a worked translation. Suppose a graph of f' is given: it crosses the x-axis at x = 1 and x = 5, is positive between 1 and 5, and negative outside. The candidate can immediately write, without ever seeing f: 'f is decreasing on (−∞, 1), increasing on (1, 5), decreasing on (5, ∞); f has a local minimum at x = 1 and a local maximum at x = 5.' That is a complete IB-style answer for the behaviour of f from the graph of f' alone, and it is also a complete AP-style answer if the rubric asks for the shape of f. The translation is symmetric: the same sentence structure works in reverse when the candidate is handed f and asked to sketch f'.
Adding f'' sharpens the picture. Suppose f'' is positive on (1, 3) and negative on (3, 5), with a zero at x = 3. The candidate can now add: 'on (1, 3), f is increasing and concave up; on (3, 5), f is increasing but concave down; x = 3 is an inflection point.' That is the full IB AA HL description in three clauses, and it is the full AP BC description in the same three clauses. The cost of writing it out is roughly 20 seconds; the reward is 4 to 6 marks depending on the rubric.
Common pitfalls and how to avoid them
- Confusing the sign of f' with the value of f. A negative f' means f is decreasing; it does not mean f is negative. Many IB candidates lose a mark by writing 'f is below the x-axis' when the rubric wants 'f is decreasing'.
- Forgetting to state the sign change at extrema. The phrase 'changes from positive to negative' is part of the IB definition of a local maximum. Without it, the answer is incomplete.
- Mixing up concavity and monotonicity. 'Concave up' is about bending; 'increasing' is about direction. They are independent. A curve can be increasing and concave down (think of the right half of a parabola opening downward).
- Skipping the existence check at inflection points. AP examiners want to know f is continuous and f' exists at the candidate's claimed inflection x. A one-line justification is enough.
- Reading zeros off the wrong curve. If the question gives f and asks for f' candidates must read zeros from the gradient of f, not from f's own zeros. This is the most common Year 1 mistake at IB AA SL.
IB Paper 2 short-response items: how the marks are distributed
At IB AA HL, three-graph items appear in two locations. In Section A they tend to be one or two marks, focused on a single behaviour such as 'state the x-coordinate of the local maximum' or 'state the interval of concave down behaviour'. In Section B they appear as 4 to 6 mark items, often worth a full structured-response rubric with one mark per behaviour, one for justification, and one for a written conclusion. The candidate's job is to know which item is which, because the answer length should scale with the marks available.
The one-mark item in Section A is a trap. The rubric awards the mark only for the explicit phrase, not for the underlying reasoning. Writing a paragraph when the rubric wants 'x = 3' is a waste of 90 seconds. The candidate should answer in the smallest form that earns the mark and move on. In contrast, the four-mark Section B item is a generosity question: the rubric is essentially giving the candidate the four behaviours for free, and the marks are awarded for stating each one explicitly. The candidate who writes one sentence covers one mark; the candidate who writes four sentences covers all four.