TheIBAxolotl
Sketch or draw?
These command words ask for different things.
- Sketch: a diagram showing the general shape and the key features (intercepts, turning points, asymptotes, end behaviour). Scale is not exact, but positions should be roughly right.
- Draw: an accurate graph on given axes or a grid, plotted to scale, with points placed carefully and a smooth curve through them.
In both cases, label the axes and the important points with their coordinates.
Using technology well
A graphing calculator will draw almost anything, but the picture is only as good as the viewing window. Too large a window flattens interesting features; too small hides intercepts or asymptotes. A good habit:
- Look at the function to predict where things happen (roots, where a denominator is zero).
- Start with a standard window, then zoom or adjust until all key features are visible.
- If the question gives a domain, use exactly that domain.
Use the calculator's analysis tools (zero, minimum, maximum, intersect) for coordinates rather than tracing by eye.
Worked example
Sketch \(y = x^3 - 4x\) for \(-3 \le x \le 3\).
- Factorise: \(y = x(x - 2)(x + 2)\), so the curve crosses the \(x\)-axis at \(-2\), \(0\) and \(2\), and passes through the origin.
- The GDC gives a local maximum near \((-1.15, 3.08)\) and a local minimum near \((1.15, -3.08)\).
- End points: \(y(-3) = -15\) and \(y(3) = 15\).
- The curve has rotational symmetry about the origin, since replacing \(x\) by \(-x\) changes the sign of \(y\).
A good sketch shows the S-shape, the three roots, both turning points labelled, and the end points.
Sums and differences of functions
The graph of \(y = f(x) + g(x)\) is built by adding heights at each \(x\). Useful checks:
- where \(g(x) = 0\), the new graph meets \(f\);
- where \(f\) and \(g\) have equal and opposite heights, the sum is zero.
For example, \(y = x + \frac{1}{x}\) behaves like \(\frac{1}{x}\) near \(x = 0\) and like \(y = x\) for large \(|x|\).
Labelling checklist
- Axes named, with scale or key values.
- Intercepts with both coordinates.
- Turning points with coordinates (3 s.f. if not exact).
- Asymptotes as dashed lines with their equations.
- The curve stops at the ends of any given domain, with endpoints shown.
Common mistakes
- Copying the calculator screen with the wrong window and missing a turning point.
- Drawing curves that touch or cross an asymptote where they should not.
- Extending the graph beyond the stated domain.
- Unlabelled features: a shape alone rarely earns full marks.
Exam tips
- Before sketching, write down the features you need: it takes ten seconds and prevents lost marks.
- Make turning points look like smooth curves, not sharp corners.
- On non-calculator questions, use factorised forms and test values to place the curve.