IB Maths AA HL Topic 2 — Functions Paper 1 & 2 ~7 min read

Graphing Functions & Their Key Features

When the IB asks you to “sketch” a graph, they want a clear picture with a small set of labelled features — intercepts, turning points, and asymptotes. They don’t want a beautiful work of art and they don’t want a freehand mess. The skill here is knowing exactly what to mark, using your GDC efficiently to find the values, and not getting caught out by the difference between “sketch” and “draw”.

📘 What you need to know

“Sketch” = freehand, accurate-enough shape, key points labelled with coordinates. Quick and clean.
“Draw” = ruler, pencil, plotted accurately to scale with axes labelled. Slower, only when asked.
Always include: labelled axes, the general shape, and key features with coordinates.
Key features: x-intercepts (where y = 0), y-intercept (where x = 0), turning points (local max/min), asymptotes (vertical and horizontal), symmetry.
Zeros / roots: x-intercepts of y = f(x) are the same as solutions to f(x) = 0.
GDC: use it to plot, then read off values for intercepts, turning points, and intersections. Most GDCs don’t draw asymptotes — find those from the equation.
Asymptote rules: vertical asymptotes occur where the denominator is zero (and the numerator isn’t); horizontal asymptotes occur where the function levels off as x → ±∞.

“Sketch” vs “Draw” — different jobs

Sketch (most common)

freehand, key features labelled

no graph paper or ruler needed. Show shape + intercepts + turning points + asymptotes

Draw (less common)

accurate, to scale, ruler & pencil

plot points from a table of values. Used mostly when graph paper is provided

If the question says “sketch”, don’t waste time being neat — focus on labelling the right features. Examiners care about the labels much more than the artistic quality.

The key features to label

y-intercept

set x = 0

find f(0)

x-intercepts

solve f(x) = 0

also called zeros / roots

Turning points

local max / min

where the curve changes direction

Asymptotes

lines the curve approaches

vertical or horizontal

All the features at a glance

Spotting asymptotes from the equation

Rational

y = 1x − a + b

vertical: x = a
horizontal: y = b

Exponential

y = a^x + c

horizontal: y = c
(no vertical asymptote)

Logarithmic

y = ln(x − k)

vertical: x = k
(no horizontal asymptote)

GDC limitation: most GDCs don’t actually draw the asymptote line. They just leave a “gap” or weird curve behaviour near it. Always work asymptotes out from the equation — don’t rely on the calculator’s screen to spot them.

GDC workflow for sketching

🧭 Recipe — sketching with the GDC

Plot the graph. Adjust the window so all important features are visible.
Find the y-intercept (substitute x = 0).
Find the x-intercepts using the zero/root function on the GDC.
Find any turning points using the max/min function.
Determine asymptotes from the equation (not from the GDC screen).
Sketch on paper: axes labelled, general shape, all features marked with coordinates.

Worked examples

WE 1

Identify all key features of a quadratic

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

Step 1: x-intercepts — solve f(x) = 0 x² − 6x + 5 = 0 → (x − 1)(x − 5) = 0 x = 1 or x = 5 Step 2: y-intercept — substitute x = 0 f(0) = 5 Step 3: Vertex — x = −b/(2a) x = 6/2 = 3 f(3) = 9 − 18 + 5 = −4 x-intercepts: (1, 0), (5, 0); y-intercept: (0, 5); vertex: (3, −4) a parabola is fully determined by these three pieces of info — opens upward since a = 1 > 0

WE 2

Find the key features of a cubic

For f(x) = x³ − 3x, find the x-intercepts, y-intercept, and the coordinates of any local maximum or minimum points.

Step 1: x-intercepts x³ − 3x = 0 → x(x² − 3) = 0 x = 0, x = ±√3 Step 2: y-intercept f(0) = 0 Step 3: Turning points — use GDC max/min function local max at x = −1 → f(−1) = −1 + 3 = 2 local min at x = 1 → f(1) = 1 − 3 = −2 x-intercepts: (−√3, 0), (0, 0), (√3, 0); local max: (−1, 2); local min: (1, −2) odd-power cubic: rotational symmetry about origin — local max and local min are mirror images

WE 3

Sketch a rational function — find both asymptotes

For g(x) = 1x − 4 + 3, find the vertical asymptote, the horizontal asymptote, the y-intercept, and the x-intercept.

Step 1: Vertical asymptote — denominator = 0 x − 4 = 0 → x = 4 Step 2: Horizontal asymptote — as x → ±∞, 1/(x−4) → 0 g(x) → 0 + 3 = 3 → y = 3 Step 3: y-intercept — set x = 0 g(0) = 1/(−4) + 3 = −0.25 + 3 = 11/4 Step 4: x-intercept — set g(x) = 0 1/(x − 4) + 3 = 0 → 1/(x − 4) = −3 x − 4 = −1/3 → x = 11/3 vertical asymptote: x = 4; horizontal: y = 3; y-int: (0, 11/4); x-int: (11/3, 0) always work asymptotes out from the equation — the GDC won’t draw them as lines

WE 4

Identify features of an exponential function

For f(x) = 3 · 2^x − 6, find the y-intercept, the x-intercept, and the equation of the horizontal asymptote.

Step 1: y-intercept f(0) = 3 · 1 − 6 = −3 Step 2: x-intercept — solve f(x) = 0 3 · 2ˣ − 6 = 0 → 2ˣ = 2 x = 1 Step 3: Horizontal asymptote — as x → −∞, 2ˣ → 0 f(x) → 0 − 6 = −6 → y = −6 y-int: (0, −3); x-int: (1, 0); horizontal asymptote: y = −6 exponentials of form a·bˣ + c have ONE horizontal asymptote (y = c) and NO vertical asymptote

WE 5

Find features of a sum of two functions using the GDC

Let h(x) = x² + 4x. Use a GDC to find the local minimum to 3 sf, and state the equation of the vertical asymptote.

Step 1: Vertical asymptote — denominator of 4/x is 0 when x = 0 x = 0 makes the function undefined → vertical asymptote at x = 0 Step 2: Plot h(x) on the GDC and use min function local minimum at approximately x ≈ 1.26, y ≈ 4.76 Step 3: Verify by setting h'(x) = 0 h'(x) = 2x − 4/x² = 0 → 2x³ = 4 → x = ³√2 ≈ 1.26 h(³√2) = (³√2)² + 4/³√2 ≈ 1.587 + 3.175 ≈ 4.76 local min ≈ (1.26, 4.76); vertical asymptote: x = 0 GDC is the workhorse here — finding the exact minimum by hand requires calculus, which is overkill for a sketch

WE 6

Identify symmetry from key features

The function f(x) = x² − 2x − 8 has x-intercepts at (−2, 0) and (4, 0). State the equation of the axis of symmetry and the coordinates of the vertex.

Step 1: Axis of symmetry — midway between the roots x = (−2 + 4) / 2 = 1 Step 2: Vertex — substitute x = 1 into f f(1) = 1 − 2 − 8 = −9 axis of symmetry: x = 1; vertex: (1, −9) for a quadratic with two real roots, the axis of symmetry is always the average of the roots — quick shortcut

💡 Top tips

“Sketch” wants labelled features, not artistic perfection. Identify the right things to mark; the rest is just a smooth curve.
Always label coordinates at every key point — examiners look for these as the “marks”.
Use your GDC for intercepts and turning points. Use algebra for asymptotes.
For sums/products of functions, plot on the GDC — algebraic features can be hard to spot from the formula alone.
Vertical asymptotes: look for divide-by-zero. Horizontal asymptotes: look at what happens as x → ±∞.
Axis of symmetry of a quadratic = average of the two roots, or use x = −b/(2a).
Label asymptotes with their equations (e.g. “y = 2″, “x = −1″) and draw them as dashed lines.

⚠ Common mistakes

Forgetting to label features. A perfectly-shaped curve with nothing marked usually scores zero.
Missing the asymptotes because the GDC didn’t draw them. Work them out from the equation.
Confusing “sketch” with “draw”. Don’t waste time being accurate when freehand is enough.
Reading turning points off the GDC by eye instead of using the max/min function. Always use the function — eyeballing leads to rounding errors.
Forgetting the y-intercept. It’s the easiest feature (just substitute x = 0) but easy to miss in a busy question.
Drawing asymptotes as solid lines. They should always be dashed to show the curve doesn’t actually touch them.
Using the wrong window on the GDC and missing a feature off-screen. Always zoom out to confirm the full shape before marking features.

A clear sketch with the right labels can earn more marks than a long algebraic derivation. The next note covers intersecting graphs — using sketches and the GDC’s intersect function to solve equations like f(x) = g(x), which is one of the most useful problem-solving tools across the whole syllabus.

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