Ratios and Graphs of Trigonometric Functions · 6.2.2
Effect of a, b, c on trig graphs
Students investigate how the amplitude changes with a, how the period changes with b, and how the whole graph shifts vertically with c, then generalise these relationships: amplitude = a, period = 360° ÷ b, and the graph moves up c units. This applies across sine, cosine and tangent graphs.
The official learning standard (6.2.2)
“Investigate and make generalisations about the effects of changes in constants a, b and c on the graphs of trigonometric functions y = a sin bx + c, y = a cos bx + c and y = a tan bx + c, for a > 0, b > 0.”
What it means
Students investigate how the amplitude changes with a, how the period changes with b, and how the whole graph shifts vertically with c, then generalise these relationships: amplitude = a, period = 360° ÷ b, and the graph moves up c units. This applies across sine, cosine and tangent graphs.
How it is examined
Paper 1 commonly tests this as an objective item, reading off amplitude, period or vertical shift directly from an equation. Paper 2 may ask students to sketch a transformed graph, state its maximum and minimum values, or explain how changing a, b or c affects the graph's shape and position.
Worked example
State the amplitude, period, maximum value and minimum value of the graph y = 3 sin 2x + 1 for 0° ≤ x ≤ 360°.
- Compare with y = a sin bx + c: a = 3, b = 2, c = 1.
- Amplitude = a = 3.
- Period = 360° ÷ b = 360° ÷ 2 = 180°.
- Maximum value = a + c = 3 + 1 = 4; minimum value = -a + c = -3 + 1 = -2.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Does changing b affect the amplitude too?
No, b only affects the period (how many cycles fit into 360°), compressing the graph horizontally when b > 1. The amplitude is controlled entirely by a, which stretches or shrinks the graph vertically.
The two effects are independent of each other.
What exactly does the constant c do to the graph?
c shifts the entire graph vertically without changing its shape, amplitude or period, a positive c moves every point up by c units, and a negative c moves it down. The new maximum becomes a + c and the new minimum becomes -a + c.
How does this apply differently to a tangent graph?
For y = a tan bx + c, a still stretches the graph vertically and c still shifts it up or down, but tangent has no amplitude since it is unbounded. Changing b instead moves the asymptotes closer together or further apart, changing the period from 180° to 180° ÷ b.