A sketch shows features, not every point: A sketch isn't a precise plot โ 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:
โข Intercepts โ where it crosses each axis (label coordinates). โข Turning points โ any maximum or minimum (label coordinates). โข Asymptotes โ dashed guide lines the curve approaches (label equations). โข Shape & ends โ 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 โ 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 โ intercepts of a parabola
Find the intercepts of y = (x โ 1)(x + 3) for a sketch.
Step by step
- x-intercepts: set y = 0 โ each bracket gives a root.
- y-intercept: set x = 0.
Final answer
Crosses the x-axis at (1, 0) and (โ3, 0), the y-axis at (0, โ3).
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Factored form gives the roots free: If a quadratic is written as (x โ a)(x โ b), the x-intercepts are simply x = a and x = b โ 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 โ a smile ๐ โ so the vertex is the lowest point (a minimum); a < 0 opens down into a โฉ โ a frown โ so the vertex is the highest point (a maximum).
Picture the smile or frown first, then you know min vs max.
IB-style question โ a downward parabola
Sketch information for y = โxยฒ + 4x: opening direction, vertex and intercepts.
Step by step
- Leading coefficient is โ1 < 0, so it opens downward (a maximum).
- Axis of symmetry x = โb/(2a) = โ4/(โ2) = 2; vertex y = โ(2)ยฒ + 4(2) = 4.
- x-intercepts: โxยฒ + 4x = 0 โ x(4 โ 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 xยณ 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 โ a reciprocal-type graph
State the asymptotes for a sketch of y = 1/(x โ 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.
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Exponentials have one asymptote: y = aหฃ approaches the x-axis (y = 0) on one side but never touches it โ 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 โ 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 โ the marks are for the labelled features.
Always transfer the numbers, not just the shape.