A sketch shows features, not every point: A sketch isn't a precise plot — it shows the right shape with the key features labelled.
Mark each of: axis intercepts, turning points (max/min), asymptotes, and the correct end-behaviour.
The sketch checklist: Every sketch marks the same four things:
• Intercepts — where it crosses each axis (label coordinates). • Turning points — any maximum or minimum (label coordinates). • Asymptotes — dashed guide lines the curve approaches (label equations). • Shape & ends — does it open up / down, rise / fall, level off?
Label everything you find: Marks come from labelled features.
A correctly-shaped curve with no values gets few marks — write the coordinates and asymptote equations on the sketch.
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Start where it crosses the axes: Find the y-intercept by setting x = 0, and the x-intercepts (the zeros/roots) by setting y = 0 and solving.
IB-style question — intercepts of a parabola
Find the intercepts of y = (x − 1)(x + 3) for a sketch.
Step by step
- x-intercepts: set y = 0 — each bracket gives a root.
- y-intercept: set x = 0.
Final answer
Crosses the x-axis at (1, 0) and (−3, 0), the y-axis at (0, −3).
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Factored form gives the roots free: If a quadratic is written as (x − a)(x − b), the x-intercepts are simply x = a and x = b — no solving needed.
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Mark the turn, fix the direction: Mark any maximum or minimum (for a quadratic, the vertex).
The leading term sets the shape: a > 0 opens up into a U — a smile 😊 — so the vertex is the lowest point (a minimum); a < 0 opens down into a ∩ — a frown — so the vertex is the highest point (a maximum).
Picture the smile or frown first, then you know min vs max.
IB-style question — a downward parabola
Sketch information for y = −x² + 4x: opening direction, vertex and intercepts.
Step by step
- Leading coefficient is −1 < 0, so it opens downward (a maximum).
- Axis of symmetry x = −b/(2a) = −4/(−2) = 2; vertex y = −(2)² + 4(2) = 4.
- x-intercepts: −x² + 4x = 0 ⇒ x(4 − x) = 0.
Final answer
Downward parabola, maximum at (2, 4), through (0, 0) and (4, 0).
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End-behaviour from the leading term: For any polynomial, the highest-power term controls the ends: e.g. a positive x³ falls on the left and rises on the right.
Draw asymptotes as dashed guides: For rational and exponential graphs, draw each asymptote as a dashed line the curve approaches but doesn't cross.
A vertical asymptote is where a denominator = 0; a horizontal asymptote is the value the curve levels off to.
IB-style question — a reciprocal-type graph
State the asymptotes for a sketch of y = 1/(x − 2) + 1.
Step by step
- Vertical asymptote: denominator zero.
- Horizontal asymptote: the +1 is the level it approaches.
Final answer
Dashed lines x = 2 and y = 1; the curve has two branches approaching them.
Animated graph
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Exponentials have one asymptote: y = aˣ approaches the x-axis (y = 0) on one side but never touches it — draw that as the dashed horizontal asymptote.
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Graph it, then transfer it: On Paper 2 a function may be too awkward to sketch by hand.
Graph it on the GDC, then transfer the picture to paper: keep the shape, and label the same intercepts, turning points and asymptotes — reading their values off the GDC.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
A sketch still needs labels: Copying the GDC's curve shape isn't enough — the marks are for the labelled features.
Always transfer the numbers, not just the shape.