Quick answer
Plot midline D, amplitude A, period 2π/B, and start the cycle at phase shift C/B for f(x) = A sin(Bx - C) + D.
Introduction
Graph visualization catches errors that algebra hides. A correct C/B with wrong period still produces a useless picture.
Graphing phase shift functions is a core skill in trigonometry and precalculus because it connects symbolic parameters to shapes.
Work through phase shift examples numerically first if you want reference values to plot.
After sketching, open the Phase Shift Calculator on our homepage and confirm phase shift and period rows match your axis marks.
What a good graph shows
A complete graph displays midline, amplitude rails, one full period, and labeled phase anchor.
Sine and cosine differ in starting feature, but the plotting steps are the same after you account for shape.
Technology graphers help, but hand sketches on tests require the manual sequence below.
Trigonometry graph movements explain why the same phase value moves every feature together rather than just the first peak.
Plotting anchors
Quarter-period points divide one cycle into readable segments.
Label axes with π multiples when B involves π to keep fractions exact.
Reflect across midline when A is negative before you shift horizontally.
Graphing steps
- Draw midline. Horizontal line y = D across the plotting window.
- Mark amplitude. Points at D + A and D - A.
- Set period width. Interval length 2π/B from the phase anchor.
- Place phase anchor. Start reference at x = C/B.
- Plot quarter points. Use sine or cosine shape rules within one period.
- Verify. Compare axis labels with the Phase Shift Calculator.
Full sketch walkthrough
For f(x) = sin(x - π/2), midline y = 0, amplitude 1, period 2π, phase anchor π/2.
Mark π/2, then add quarter periods π/2 apart to locate max, midline, min, and return.
The result matches a cosine-like placement, consistent with identity relationships.