Sketching Sine and Cosine Graphs

Step-by-step interactive tutorial for sketching transformed sine and cosine graphs. Select parameters: function (sin or cos), amplitude a (range −3 to 3), frequency b (0.5, 1, 2, or 3), vertical shift d (−2 to 2). Seven guided steps: (1) read off amplitude, period T = 2π/b, midline y = d, and reflection status; (2) draw the midline as a dashed guide; (3) mark maximum and minimum bounds (midline ± amplitude); (4) divide one period into quarters and mark T/4 intervals with π-fraction labels; (5) plot five key points following the pattern (start/end on midline for sin, max/min for cos; max/min/max for sin; min/max/min for cos); (6) sketch a smooth wave through points (flat at extrema, steepest at midline crossing); (7) extend the periodic pattern across multiple periods. Random function generator, keyboard navigation (arrow keys, space, N for new), and real-time graph display. Teaches systematic method for hand-sketching trigonometric functions without memorizing the full curve.