3 · IB Maths AA
Transformations of trigonometric graphs
Change a, b, c and d in y = a sin(b(x − c)) + d and watch the amplitude, period, phase shift and principal axis change against the base graph. Drag the gold point to shift the graph.
transformed graphbase graphprincipal axis / one periodamplitude, phase shift
Key equation (x in radians):
y = a sin(b(x − c)) + d
amplitude = |a|, period = 2π / b
amplitude = |a|, period = 2π / b
y = sin(x)
Amplitude:
Period:
Phase shift:
Principal axis:
Range:
Expanded bracket:
Try this
- Set a = −2. Check that the amplitude is still 2 and the graph is reflected in its principal axis, not in the x-axis.
- Predict the period with 2π / b for b = 0.5, 2 and 3, then compare with the shaded cycle.
- Set b = 2 and c = π/4. Confirm a cycle starts at x = π/4, then read the expanded bracket to write the equation as sin(2x − k).
Common error: for y = sin(2x − π/3) the phase shift is π/6, not π/3 — always factor out b to get the form b(x − c) before reading off c.