In y = a sin bx + c, the three numbers each change one thing you can see. The value a sets the height, b sets how often the wave repeats, and c lifts or lowers the whole curve.
This lesson builds on reading sine, cosine and tangent graphs and belongs to the trigonometric ratios and graphs chapter.
What does each number do?
Compare each equation with the plain graph y = sin x, which has amplitude 1, period 360° and middle line y = 0.
| Number | Name | Effect on the graph | Reading it |
|---|---|---|---|
| a | Amplitude | Stretches up and down | Highest value above the middle line |
| b | Frequency factor | Squeezes or stretches sideways | Period = 360° ÷ b |
| c | Vertical shift | Moves the whole curve | Middle line is y = c |
Maximum value = c + |a|. Minimum value = c − |a|.
Worked example: y = 2 cos(x ÷ 2) + 1
Here a = 2, b = 1/2 and c = 1. Read each one in turn.
- Amplitude = 2.
- Period = 360° ÷ (1/2) = 720°, so one full wave takes two turns around the circle.
- Middle line is y = 1.
- Maximum = 1 + 2 = 3 and minimum = 1 − 2 = −1.
Since cosine starts at its highest point, the graph starts at (0°, 3). It falls to 1 at 180° and reaches −1 at 360°.
The mistake that costs marks
The common slip is to multiply instead of divide. For y = sin 3x, a student writes the period as 360° × 3 = 1 080°.
| Step | Wrong | Right |
|---|---|---|
| Period rule | 360° × b | 360° ÷ b |
| Period of sin 3x | 1 080° | 120° |
| Waves in 0° to 360° | less than one | 3 complete waves |
A sense check helps. The number 3 makes the graph repeat three times faster, so the period must get shorter, not longer.
Check yourself
For y = 4 sin 3x − 1, state the amplitude, the period, and the maximum and minimum values.
Answer
Amplitude = 4, from the number in front of sine.
Period = 360° ÷ 3 = 120°.
The middle line is y = −1. Maximum = −1 + 4 = 3 and minimum = −1 − 4 = −5.
What to study next
Put everything together in the trigonometry practice set, which mixes angles, graphs and transformations. If a slip comes from the algebra, such as dividing by a fraction, the algebra step repair trainer drills that step.
For a teacher to work through transformed graphs with you, see online one-to-one Mathematics tuition.