IB Maths AI SL Topic 2 — Further Functions & Graphs Paper 1 & 2 GDC-essential ~7 min read

Graphing Functions & Their Key Features

Most graphing questions in AI SL come down to the same workflow: plot it on your GDC, identify the key features (intercepts, turning points, asymptotes), then translate the screen into a clean labelled sketch. Marks are won by labelling every important point with coordinates, not by drawing a beautiful curve. This note covers what counts as a “key feature”, the sketch-vs-draw distinction, and how to extract everything you need from the GDC.

📘 What you need to know

Sketch: labelled axes, general shape (freehand OK), key points labelled with coordinates. Draw: pencil/ruler, accurate, to scale, often needs a table of values.
Intercepts: y-intercept where x = 0 (substitute); x-intercepts (zeros/roots) where y = 0 (GDC solver).
Local max/min (turning points): where the curve changes direction. A graph can have several — a local max isn’t necessarily the global maximum.
Asymptotes: lines the curve gets ever closer to but never touches. Vertical from forbidden x-values (denominator = 0); horizontal from the tail behaviour as x → ±∞.
GDC essential: use the trace/analyse menus to get exact coordinates for intercepts and turning points. Don’t eyeball values off the screen.
Asymptotes don’t auto-plot: most GDCs draw the curve only. Read the equation to spot asymptotes, then add the dashed lines yourself.

Sketch vs draw — what examiners want

Read the command word carefully. Sketch means a quick, freehand diagram showing the right shape with all key points labelled with coordinates — no graph paper needed. Draw means an accurate diagram on a scaled grid, usually with a table of values, drawn with pencil and ruler.

Both still need: labelled x– and y-axes, all intercepts marked with coordinates, any turning points marked with coordinates, and asymptotes shown as dashed lines with their equations. Missing one of these typically costs an “AG” mark (axes/graph-features) even if the curve itself is correct.

The key features to find & label

Finding intercepts y-intercept: put x = 0 into the equation
x-intercepts (roots/zeros): solve f(x) = 0

Left: quadratic with two roots, vertex (lowest point), y-intercept, and axis of symmetry. Right: exponential with horizontal asymptote (dashed) the curve approaches but never reaches.

Turning points (local max/min)

A point on the curve where it switches direction. On your GDC, use the maximum and minimum commands — they give exact x and y coordinates. A quadratic has exactly one (the vertex); a cubic has 0 or 2; higher-order curves can have more. “Local” means it’s a max/min of its immediate neighbourhood; the “global” max/min looks across the whole domain.

Asymptotes — the GDC blind spot

Vertical asymptotes happen where the function is undefined — typically where a denominator hits zero. So for y = 1x − 3, there’s a vertical asymptote at x = 3. Horizontal asymptotes are about long-term behaviour: as x → ±∞, does y level off at some value? For exponential models like y = ke^rx + c, the line y = c is the horizontal asymptote.

Most GDCs don’t draw asymptotes — only the curve itself. Spot them from the equation, then add dashed lines to your sketch with their equations labelled.

🧭 Recipe — sketching any function on a GDC

Type the function into your GDC and adjust the viewing window until you see the full shape — including any turning points and the long-term tails. Don’t sketch from a half-visible curve.
Find the intercepts: use the zero/root tool for x-intercepts; for the y-intercept just substitute x = 0 (or use trace).
Find turning points: use the maximum and minimum tools. Note all of them — one curve can have several local extrema.
Spot asymptotes from the equation: vertical where denominator = 0; horizontal from the long-term limit. Draw them as dashed lines.
Translate to paper: draw the axes, mark and label all intercepts, turning points, and asymptotes with coordinates and equations. Sketch the curve to match the shape on the GDC.

Worked examples

WE 1

Quadratic — label every key feature

Sketch the graph of y = x² − 2x − 8, labelling the x– and y-intercepts and the vertex.

Step 1 — shape: a > 0, so ∪ with a minimum opens upward, has a vertex (lowest point) Step 2 — y-intercept (set x = 0) y = 0 − 0 − 8 = −8 → (0, −8) Step 3 — x-intercepts (solve x² − 2x − 8 = 0 on GDC) x = −2 or x = 4 → (−2, 0) and (4, 0) Step 4 — vertex from GDC minimum tool x = 1, y = (1)² − 2(1) − 8 = −9 vertex (1, −9) · roots (−2, 0), (4, 0) · y-int (0, −8) cross-check using symmetry: axis of symmetry x = (−2 + 4)/2 = 1, matches the vertex’s x-coord ✓.

WE 2

Rational function — find the asymptotes

The function f(x) = 3 − 2x − 4 is to be sketched. Find the equations of both asymptotes and the coordinates of the y-intercept.

Step 1 — vertical asymptote: denominator = 0 x − 4 = 0 ⇒ x = 4 Step 2 — horizontal asymptote: as x → ±∞ 2/(x − 4) → 0, so f(x) → 3 asymptote: y = 3 Step 3 — y-intercept: set x = 0 f(0) = 3 − 2/(−4) = 3 + 0.5 = 3.5 asymptotes: x = 4, y = 3 · y-int (0, 3.5) on a sketch, draw both asymptotes as DASHED lines and label each with its equation. The curve sits in two separate branches on either side of x = 4.

WE 3

Exponential — intercepts and asymptote

For the function y = 2e^0.5x − 4, find the y-intercept, the x-intercept (to 3 s.f.) and the equation of the horizontal asymptote.

Step 1 — y-intercept (x = 0) y = 2e⁰ − 4 = 2(1) − 4 = −2 (0, −2) Step 2 — horizontal asymptote: long-term limit as x → −∞, e⁹.⁵ˣ → 0, so y → −4 asymptote: y = −4 Step 3 — x-intercept (use GDC solver: y = 0) solve 2e⁹.⁵ˣ − 4 = 0 x ≈ 1.39 (to 3 s.f.) y-int (0, −2) · x-int (1.39, 0) · asymptote y = −4 for “keˣ + c” type curves, the horizontal asymptote is ALWAYS y = c — read it straight from the equation. Saves a calculation.

WE 4

Cubic — local max & min

The cubic y = x³ − 3x² − 9x + 5 has two turning points. Use your GDC to find them and state which is the local maximum.

Step 1 — plot on GDC, set window so both turning points are visible curve rises → falls → rises Step 2 — use GDC maximum and minimum tools maximum tool: x = −1, y = 10 minimum tool: x = 3, y = −22 Step 3 — sanity-check by substituting y(−1) = −1 − 3 + 9 + 5 = 10 ✓ y(3) = 27 − 27 − 27 + 5 = −22 ✓ local max (−1, 10) · local min (3, −22) for a cubic with positive leading coefficient, the LEFT turning point is always the local max and the RIGHT one is the local min. Quick shape check.

WE 5

Real-world model — cooling curve

A cup of coffee cools according to T(t) = 20 + 60e^−0.1t, where T is in °C and t is in minutes after pouring. State T(0), the long-term temperature, and describe the shape of the graph.

Step 1 — starting temperature (t = 0) T(0) = 20 + 60e⁰ = 20 + 60 = 80 y-intercept (0, 80) Step 2 — long-term: as t → ∞, e⁻.¹ˣ → 0 T → 20 + 0 = 20 horizontal asymptote: T = 20 Step 3 — shape (negative exponent ⇒ decreasing) curve starts at 80, decays towards 20, never touches T(0) = 80°C · asymptote T = 20°C · decaying exponential interpretation: 20°C is the room temperature — the coffee cools towards it but never gets cooler than the surroundings. A classic exam scenario for asymptotes.

WE 6

Difference of two functions — find intercepts

The functions f and g are given by f(x) = x² − 4 and g(x) = x + 2. Sketch h(x) = f(x) − g(x) and find both intercepts.

Step 1 — form h(x) h(x) = (x² − 4) − (x + 2) = x² − x − 6 Step 2 — y-intercept (x = 0) h(0) = −6 → (0, −6) Step 3 — x-intercepts (solve on GDC) x² − x − 6 = 0 (x − 3)(x + 2) = 0 x = 3 or x = −2 x-ints (−2, 0), (3, 0) · y-int (0, −6) your GDC will happily plot y = f(x) − g(x) directly — you don’t have to simplify first. But simplifying makes the intercepts faster to find by hand.

💡 Top tips

Always sketch the GDC screen as part of your working: even when not asked. Examiners can give method marks for a sensible sketch alongside a wrong final answer.
Window settings matter: if you can’t see all the turning points or both ends of the curve, zoom out. A “missed” feature is the most common GDC slip.
Asymptotes from the equation, not the screen: vertical from where the denominator = 0; horizontal from “y = c” in y = ke^rx + c.
Label EVERY key point with its coordinates: examiners read labels first. An unlabelled vertex or root is worth zero, even if correctly placed.
Symmetry shortcut: a quadratic’s axis of symmetry sits halfway between its two roots. Useful to find the vertex quickly without calculus.

⚠ Common mistakes

Missing asymptotes: the GDC won’t draw them. Spot them from the equation; if you skip them, both the sketch and any “range” question downstream go wrong.
Eye-balling coordinates off the GDC screen: always use the trace/zero/maximum/minimum tools for exact values. “About 3.5” loses accuracy marks.
Labelling features without coordinates: a circle on the graph saying “vertex” is incomplete — it needs to be labelled (x, y).
Mistaking local for global extremes: a cubic’s local max is NOT the highest value the function ever reaches (the curve keeps going up at the right end). Don’t claim it’s the maximum overall.
Confusing “sketch” with “draw”: a sketch doesn’t need accurate scaling, but a “draw” question does — and that needs a table of values plotted carefully.

Up next: Intersecting Graphs. Once you can plot two functions cleanly, the natural next step is finding where they cross. Intersection points are solutions to equations: f(x) = g(x) becomes “where do the two curves meet?”. Your GDC’s intersect tool does the heavy lifting; the skill is setting the problem up so the right curves are being compared.

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