A sketch question always states an interval such as 0 <= x <= pi. Draw only that part of the curve, and make the key features — maximum, minimum, midline and endpoints — clearly visible. - Read the midline y=c, the maximum c+a and the minimum c-a; rule them as dashed horizontal lines. - Work out the period 360 b or 2pib and divide one period into four equal quarters. - Find the value on the y-axis by substituting x=0; the sketch starts there. - Step through the quarters — the curve moves between maximum, midline and minimum once per quarter period. - Repeat the cycle until the interval is filled, stopping exactly at both endpoints. **Worked sketch: y=2cos 3x-1 for 0 <= x <= pi** **Feature** | **Working** Midline | y=-1 Amplitude | 2 Maximum and minimum | -1+2=1 -1-2=-3 Period | 2pi3 Start point | x=0: y=2cos 0-1=1 The interval 0 <= x <= pi holds one and a half periods: maxima at x=0 and x= 2pi3, minima at x= pi3 and x=pi, and the curve crosses the midline y=-1 between them. Label the axes, mark the maximum and minimum values, and stop the curve exactly at the stated endpoints — a sketch that runs past the interval loses marks.

Sketching over a Stated Interval

Secondary 4

TGM Original Questions

A sketch question always states an interval such as \(0\le x\le\pi\). Draw only that part of the curve, and make the key features — maximum, minimum, midline and endpoints — clearly visible.

Method

Read the midline \(y=c\), the maximum \(c+a\) and the minimum \(c-a\); rule them as dashed horizontal lines.
Work out the period \(\dfrac{360^\circ}{b}\) or \(\dfrac{2\pi}{b}\) and divide one period into four equal quarters.
Find the value on the \(y\)-axis by substituting \(x=0\); the sketch starts there.
Step through the quarters — the curve moves between maximum, midline and minimum once per quarter period.
Repeat the cycle until the interval is filled, stopping exactly at both endpoints.

Worked sketch: \(y=2\cos 3x-1\) for \(0\le x\le\pi\)

Feature	Working
Midline	\(y=-1\)
Amplitude	\(2\)
Maximum and minimum	\(-1+2=1\) \(-1-2=-3\)
Period	\(\dfrac{2\pi}{3}\)
Start point	\(x=0:\ y=2\cos 0-1=1\)

The interval \(0\le x\le\pi\) holds one and a half periods: maxima at \(x=0\) and \(x=\dfrac{2\pi}{3}\), minima at \(x=\dfrac{\pi}{3}\) and \(x=\pi\), and the curve crosses the midline \(y=-1\) between them.

Exam check

Label the axes, mark the maximum and minimum values, and stop the curve exactly at the stated endpoints — a sketch that runs past the interval loses marks.

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