IB Maths AI HL Further Functions & Graphs Paper 2 Sketching, intercepts, asymptotes ~8 min read

Graphing Functions & Their Key Features

A useful sketch shows the shape of y = f(x) and labels its key features: intercepts, turning points and asymptotes. AI HL is a calculator paper for sketches — plot on the GDC, read off the key points, then copy the shape onto paper with everything labelled.

📘 What you need to know

Sketch = freehand, axes labelled, key points marked with coordinates. Draw = ruler, accurate to scale, points plotted from a table of values.
x-intercepts (zeros, roots): set y = 0 and solve. y-intercept: set x = 0 and evaluate.
Turning points: where the graph changes direction. A quadratic has one (the vertex); a cubic has 0 or 2.
Local max/min applies to a neighbourhood; global applies to the whole graph.
Asymptotes are lines the graph approaches but never crosses. Vertical from a zero in the denominator; horizontal from the long-run behaviour. Exponentials have a horizontal asymptote; reciprocals have both.
GDCs usually don’t draw asymptotes — spot them from the equation, e.g. 1/(x − 2) has a vertical asymptote at x = 2.

Sketching y = f(x)

The graph of y = f(x) pairs each x (input) with its output f(x) on the vertical axis. So f(a) = b puts (a, b) on the curve. A sketch just needs to capture the correct shape with axes labelled and key features marked — coordinates of intercepts and turning points, equations of any asymptotes. A draw question is stricter: pencil, ruler, accurate scale, points plotted from a table of values.

Always use the GDC: plot the function, identify the key points (intercept, vertex, max/min) using the menu, then sketch the shape on paper with those coordinates labelled. Check the viewing window so nothing is cut off.

The key features of a graph

Five features come up again and again: intercepts (where the graph crosses the axes), turning points (local max and min), symmetry (e.g. the vertical axis of a parabola), asymptotes (lines the curve hugs but doesn’t cross), and the overall shape (which depends on the function type). Each AI HL function family has a predictable set: quadratics have one vertex and at most two roots, exponentials have one horizontal asymptote, reciprocals have both a horizontal and a vertical asymptote.

Label every turning point, intercept and asymptote on your sketch — an unlabelled graph picks up few marks.

Key features to label on every sketch intercepts (y = 0 ⇒ x-intercepts; x = 0 ⇒ y-intercept) turning points (local max/min) · asymptotes (x = constant or y = constant)

Sketching with the GDC

The standard AI HL workflow: plot y = f(x) on the GDC, use the analyse graph or trace menu to read off intercepts, maxima and minima, then redraw the shape on paper with everything labelled. Asymptotes are usually invisible on the GDC screen — spot them by reading the equation: any factor of the form (x − a) in a denominator gives a vertical asymptote at x = a, and the long-run behaviour gives any horizontal asymptote.

🧭 Recipe — sketching a function

Identify the type (quadratic, cubic, exponential, rational, sinusoidal) so you know the general shape.
Find the y-intercept by evaluating f(0); find the x-intercepts by solving f(x) = 0.
Find any turning points using your GDC’s max/min function.
Find any asymptotes: vertical from a zero of the denominator; horizontal from the long-run behaviour.
Draw the axes, plot the labelled features, and sketch the curve through them.

Worked examples

WE 1

Key features of a factorised quadratic

For f(x) = (x − 3)(x + 1), find the x– and y-intercepts, the axis of symmetry, and the vertex.

x-intercepts: set f(x) = 0 (x − 3)(x + 1) = 0 ⇒ x = 3 or x = −1 x-intercepts: (−1, 0) and (3, 0) y-intercept: f(0) f(0) = (−3)(1) = −3 ⇒ (0, −3) axis of symmetry: midpoint of the roots x = (−1 + 3)/2 = 1 vertex: f(1) f(1) = (−2)(2) = −4 x-ints (−1, 0), (3, 0) · y-int (0, −3) · axis x = 1 · vertex (1, −4)

WE 2

Asymptotes of a translated reciprocal

State the equations of the asymptotes of g(x) = 3/(x − 2) − 1.

vertical: denominator = 0 x − 2 = 0 ⇒ x = 2 horizontal: long-run behaviour as x → ±∞, 3/(x − 2) → 0 so g(x) → 0 − 1 = −1 vertical x = 2 · horizontal y = −1 the −1 shifts the curve down by 1, taking the horizontal asymptote with it.

WE 3

Intercepts of an exponential

Find the x– and y-intercepts of h(x) = 2(3)^x − 6.

y-intercept: h(0) h(0) = 2(3)⁰ − 6 = 2 − 6 = −4 y-intercept: (0, −4) x-intercept: h(x) = 0 2(3)ⁿ − 6 = 0 ⇒ 3ⁿ = 3 ⇒ x = 1 x-intercept (1, 0) · y-intercept (0, −4) an exponential has at most one x-intercept — sometimes none.

WE 4

Full feature list of a quadratic

For f(x) = x² − 6x + 5, find the axis of symmetry, vertex, x-intercepts and y-intercept.

axis of symmetry: x = −b/(2a) x = 6/2 = 3 vertex: substitute x = 3 f(3) = 9 − 18 + 5 = −4 ⇒ (3, −4) x-intercepts: factor x² − 6x + 5 = (x − 1)(x − 5) = 0 x = 1 or x = 5 y-intercept: f(0) f(0) = 5 ⇒ (0, 5) axis x = 3 · vertex (3, −4) · x-ints (1, 0), (5, 0) · y-int (0, 5)

WE 5

Quadratic with no real roots

List the features needed to sketch f(x) = x² − 2x + 5.

check the discriminant b² − 4ac = 4 − 20 = −16 < 0 no real roots ⇒ no x-intercepts vertex x = 2/2 = 1; f(1) = 1 − 2 + 5 = 4 vertex (1, 4) y-intercept f(0) = 5 ⇒ (0, 5) opens upward · vertex (1, 4) · y-int (0, 5) · no x-intercepts when there are no roots, label the vertex and y-intercept — that’s enough to sketch the parabola.

WE 6

Full feature list of a translated reciprocal

For g(x) = 1/(x + 3) + 2, find the asymptotes and both intercepts.

vertical asymptote: x + 3 = 0 x = −3 horizontal asymptote: long-run as x → ±∞, 1/(x + 3) → 0 ⇒ g → 2 y = 2 y-intercept g(0) = 1/3 + 2 = 7/3 ⇒ (0, 7/3) x-intercept: g(x) = 0 1/(x + 3) = −2 ⇒ x + 3 = −1/2 ⇒ x = −7/2 x-int: (−7/2, 0) v.a. x = −3 · h.a. y = 2 · (0, 7/3) · (−7/2, 0)

💡 Top tips

Label every feature with coordinates on your sketch — an unlabelled curve picks up almost no marks.
Asymptotes go as equations (x = −3, y = 2), not as points; draw them dashed.
Read the equation for asymptotes — the GDC usually won’t draw them.
For a quadratic with two roots, the axis of symmetry is the midpoint of the roots.
Always check your viewing window; turning points outside the default range are easy to miss.

⚠ Common mistakes

Drawing the curve through the asymptote — by definition the curve approaches but never crosses it.
Forgetting asymptotes exist because the GDC doesn’t show them.
Reading vertex coordinates as both x-values rather than (x, y).
Confusing local with global max/min — a cubic has both a local max and a local min, but neither is global.
Sketching freehand when “draw” is required — “draw” needs a ruler and an accurate plot.

Next up: Intersecting Graphs — using your GDC to find where two curves meet and to solve equations of the form f(x) = g(x).

Need help with AI HL Graphing?

Get 1-on-1 help from an IB examiner who knows exactly what Paper 2 is looking for.

Book Free Session →