5 graph relationships that decide an AP-style f, f' and…

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.

For AP, the equivalent item in the multiple-choice section awards a single point for the right behaviour plus, sometimes, a small justification in the stem. In the free-response section, a three-graph item is typically a 3 to 4 point question worth approximately 6 to 8 minutes. The candidate who arrives with a pre-loaded answer template — 'behaviour of f, sign of f', sign of f'', conclusion' — finishes inside the time budget and has buffer for the harder items. In practice this template saves two to three minutes per item, which compounds across a six-item free-response paper.

Exam	Typical item weight	Rubric language	Time budget per item
IB AA SL Paper 1 Q7–9	4–6 marks	State, with reference to f' and f''	3–4 minutes
IB AA HL Paper 2 B	6–9 marks	Justify, identify intervals, name points	5–7 minutes
AP Calculus AB FRQ 2	3–4 points	Be specific, refer to the derivative	6–8 minutes
AP Calculus BC FRQ 2	4–6 points	Be specific, refer to the second derivative	7–9 minutes

Sketching f from f' and f'': a worked IB-style question

Take a concrete example. The graph of f' is a straight line crossing the x-axis at x = 2, with negative slope. The graph of f'' is a horizontal line below the x-axis for all x. The candidate is asked to sketch f for x in some closed interval, labelling any extrema and inflection points.

The reading: f' zero at x = 2, f' positive for x < 2, f' negative for x > 2. So f has a local maximum at x = 2. f'' is negative everywhere, so f is concave down on the whole interval. There is no sign change of f'', so there is no inflection point. The sketch is therefore a curve that rises, peaks at x = 2, and falls, with the entire shape bending downward. A clean IB answer would state each of these four facts in order, then describe the sketch in words before drawing it.

The same problem in AP language: the candidate writes 'f has a local maximum at x = 2, f is concave down on the interval shown, and there are no inflection points.' That sentence is worth full credit on the AP rubric, where the marks are split into 'maximum identified', 'concavity identified', 'no inflection identified'. Notice how the IB answer is longer but carries the same information; the AP answer is more compressed because the rubric tolerates 'concavity' as a single word where the IB rubric wants 'f'' < 0'.

From sketch to justification in BC-level items

For AP Calculus BC, the same three-graph problem is often embedded inside a multi-part item where the candidate is asked first to identify behaviours, then to compute an integral representing the area between f and the x-axis, and finally to interpret the result. The translation skill from the first part carries directly into the second: the sign of f tells the candidate whether the area should be added or subtracted, and the zeros of f identify the bounds. Candidates who can sketch f quickly are also the candidates who set up the integral without drawing a second picture, which is roughly a 30-second saving per item.

IB AA HL does not have a BC-style 'find the area' follow-up on every three-graph item, but the same principle applies whenever the rubric asks for a definite integral interpretation. A candidate who can read the graph of f and identify the interval on which f is positive and the interval on which f is negative will not need to re-draw the graph when the integral question appears. This is a quiet but real source of points: the candidate who skips the first part of the question carefully often gets the second part right by accident, and the candidate who skims the first part pays for it in the second.

Strategic preparation for f, f' and f'' items in the IB Diploma

Preparation for these items lives in three layers: recognition, justification and time. Recognition is the lowest cognitive load and the highest return; the candidate should be able to look at a graph of f' and immediately write the four-bullet description of f without consulting notes. Justification is the language layer, and the candidate should write out 20 to 30 complete answers from past IB and AP papers, marking each one against the official rubric to identify the verbs and phrases the examiner wants. Time is the disciplined layer: under timed conditions, the candidate should be able to do a Section A one-mark item in under 90 seconds and a Section B six-mark item in under 7 minutes.

Most candidates reading this should start with recognition. The fastest improvement I have seen in IB Diploma candidates is when they sit with a stack of f'-only graphs and write a behaviour description for each in 60 seconds flat. After 20 graphs the description becomes automatic; after 50 the candidate is finishing well inside the time budget. This is preparation strategy at its most efficient: the gain is in the response speed, not the underlying mathematics, because the mathematics was probably learned in Year 1 and just needs a refresh.

For the IB scoring system, these items matter disproportionately. The IB Diploma is awarded on a 1 to 7 scale, with band descriptors that explicitly reward 'justification' and 'communication' at the top of the scale. A candidate whose mathematics is correct but whose justification is sparse will plateau at band 4 or 5. The f, f' and f'' items are the easiest place in the IB AA HL paper to score full marks on justification, because the picture is already given to the candidate — there is no setup cost. The only barrier is the willingness to write the chain out in full.

Cross-format practice: IB and AP working together

The IB Diploma programme and AP Calculus are not the same exam, and no serious tutor would pretend they are. But for the specific sub-topic of three-graph interpretation they overlap so cleanly that practising them together is a net gain. The strategy I recommend for IB students considering AP, and AP students considering the IB, is to do two passes of the same set of past items: once in the IB rubric's voice, once in the AP rubric's voice. The exercise takes roughly 30% longer than a single-pass revision, but the reinforcement is asymmetric — the candidate ends up fluent in both rubrics and is no longer anchored to one set of magic phrases.

Exam format matters here. IB AA HL Paper 2 is a 2-hour written paper with a calculator restriction that does not affect three-graph items. AP Calculus AB and BC free response are 1 hour 30 minutes, with graphical and analytical sub-questions interleaved. The IB candidate should expect to defend a written answer; the AP candidate should expect to defend a written answer plus a sketch. The shared sub-skill is the same; the surface is different.

Question types also diverge at the margin. IB AA SL tends to test sign of f' and sign of f'' on closed intervals, with no follow-up. IB AA HL adds the existence criterion for inflection points and the link to integral interpretation. AP AB tests the same surface as IB SL plus the area-between-curves interpretation. AP BC adds the logistic differential-equation reading of f, f' and f'', which is a separate article altogether. For the purposes of this piece, the candidate should be aware that 'three graphs' in AP BC is sometimes 'three graphs and a function definition' and the rubric rewards the candidate who ties the differential equation back to the slope-reading on the graph.

What a band-7 IB answer looks like in practice

A band-7 IB AA HL answer to a six-mark three-graph item reads like a tight, three-sentence paragraph with a labelled sketch. The first sentence states the interval of monotonicity and references the sign of f'. The second sentence states the interval of concavity and references the sign of f''. The third sentence names the local extremum, names the inflection point if one exists, and ties them to the derivative curves in a single phrase such as 'f' changes sign from negative to positive at x = a, so f has a local minimum at x = a; f'' changes sign at x = b, so f has an inflection point at x = b'. The sketch that follows should label the extremum and the inflection point with their x-coordinates and indicate concavity with a small arrow or shading.

That paragraph is roughly 50 to 70 words. It takes an experienced candidate around 5 minutes to write, including the sketch. It is dense, and that density is the point: the IB examiner is reading 200 papers and awarding 6 marks per item. The candidate who writes a 200-word answer to a 6-mark item is, intentionally or not, asking the examiner to find 6 marks in 200 words. The candidate who writes 60 words to a 6-mark item is handing the examiner 6 marks on a plate. For most candidates reading this, the band-7 answer is not a more impressive answer; it is a more economical one.

Conclusion and next steps

The three-graph relationship between f, f' and f'' is one of the highest-leverage sub-topics in IB AA HL and AP Calculus, both because the marks are dense and because the underlying mathematics is the same across the two programmes. A candidate who can read slopes, concavity and sign changes off a derivative curve, and write the behaviour of the original function in language the rubric accepts, is in a strong position for the rest of the paper. The tactical advice above is, in summary: learn the four-bullet checklist, practise 20 to 30 timed justifications against official rubrics, and write band-7 answers in 50 to 70 words rather than 200.

TestPrep Europe's diagnostic assessment on f, f' and f'' graph interpretation is a natural starting point for IB Diploma and AP candidates building a sharper preparation plan around this specific question family.

Frequently asked questions

How are f, f' and f'' graph questions marked differently in IB and AP?

IB examiners typically require explicit reference to the sign of f' and f'' and the language of sign change at extrema, awarding one mark per stated behaviour. AP readers accept the same mathematics in a more compressed form, often rewarding a single combined statement. The mathematics is identical; the rubric vocabulary differs.

Do IB AA SL students need to use the term 'point of inflection'?

At SL, the rubric tolerates 'point where f changes concavity' as an alternative. At HL, the phrase 'point of inflection' is expected. Candidates preparing for both syllabuses should use the HL term because it never costs marks at SL and is required at HL.

What is the fastest way to improve at three-graph interpretation?

Practise 20 to 30 timed responses from past papers, writing the four-bullet description of f from a graph of f' alone. The repetition builds the response template so the candidate can finish a Section A one-mark item in under 90 seconds and a Section B six-mark item in under 7 minutes.

Is sketching f from f' a guaranteed mark in IB Paper 2?

A labelled sketch with the extremum and inflection points marked is worth at least one mark in any Section B item that asks for one, provided the labels match the candidate's written justification. A sketch that contradicts the written work loses marks, so the candidate should always write the justification before drawing the curve.

Does AP Calculus BC test f, f' and f'' differently from AB?

BC extends the same items with additional marks for integral interpretation and, occasionally, differential-equation readings. The core three-graph reasoning is unchanged; the BC candidate should be ready to translate the same behaviour description into an integral setup in the same item.

5 graph relationships that decide an AP-style f, f' and f'' analysis question