For y = a sin bx + c, the number a sets the amplitude, b sets the period (360° ÷ b), and c shifts the curve up or down. Read these three values first, then mark key points and join them smoothly.
This lesson is part of trigonometric functions. It uses the same shapes you met in solving equations over a stated interval.
What does each number do?
| Feature | Formula | Effect |
|---|---|---|
| Amplitude | |a| | Height from the middle line to a maximum |
| Period | 360° ÷ b | Length of one complete cycle |
| Middle line | y = c | The horizontal line the curve oscillates around |
| Maximum and minimum | c + |a| and c − |a| | Top and bottom of the curve |
A negative a flips the curve upside down.
Worked example 1: y = 2 sin 3x + 1 for 0° ≤ x ≤ 360°
Read the values: amplitude 2, period 360° ÷ 3 = 120°, middle line y = 1. The maximum is 1 + 2 = 3 and the minimum is 1 − 2 = −1.
One cycle spans 120°, so divide it into quarters of 30°.
| x | 0° | 30° | 60° | 90° | 120° |
|---|---|---|---|---|---|
| y | 1 | 3 | 1 | −1 | 1 |
Repeat the cycle three times to reach 360°. The curve starts at (0, 1), has maxima at x = 30°, 150°, 270°, and minima at x = 90°, 210°, 330°.
Worked example 2: a modulus graph
Sketch y = |sin 2x| for 0° ≤ x ≤ 180°.
The graph of y = sin 2x has period 180°, with a maximum of 1 at x = 45° and a minimum of −1 at x = 135°. The modulus reflects the negative part upward, so the curve stays between 0 and 1.
The result has zeros at x = 0°, 90° and 180°, with maxima of 1 at x = 45° and x = 135°. The two humps are identical, and the curve has sharp corners at the zeros.
The mistake that costs marks
The common slip is to multiply instead of divide when finding the period, writing 360° × 3 = 1080° for y = sin 3x.
| Step | Wrong | Right |
|---|---|---|
| Period of sin 3x | 360° × 3 = 1080° | 360° ÷ 3 = 120° |
| Cycles in 0° to 360° | Less than one | Three |
| Sketch | One stretched wave | Three compressed waves |
The larger b is, the more cycles fit. A quick check: y = sin 3x must repeat faster than y = sin x, never slower.
Check yourself
For y = 3 cos 2x − 2, state the amplitude, period, maximum and minimum values, and the number of maximum points for 0° ≤ x ≤ 360°.
Answer
The amplitude is 3. The period is 360° ÷ 2 = 180°. The middle line is y = −2, so the maximum is −2 + 3 = 1 and the minimum is −2 − 3 = −5.
Maxima occur where cos 2x = 1, so 2x = 0°, 360°, 720°, giving x = 0°, 180°, 360°. There are three maximum points, including the two end points.
What to study next
Test the whole chapter with the trigonometric functions practice set, which includes a sketch-features question. If graph slips repeat, note them in the mistake log.
If you want a teacher to check your sketches with you, see online one-to-one Additional Mathematics tuition.