10. Transformed Sine Waves

Ah, sinusoidal functions! Plus Bach and Desmos. Come for the stretching, stay for the shifting. This is the tenth of a sequence of supplementary videos for a trigonometry class taught from my book 𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡. See my website https://bravernewmath.com for more information on my various books: 𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠, 𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎, 𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡, 𝐿𝑜𝑏𝑎𝑐ℎ𝑒𝑣𝑠𝑘𝑖 𝐼𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑. The first three are available for sale as paperbacks at Amazon, and as pdfs at Lulu. (The Lobachevski book is available at Amazon and the American Mathematical Society) The brief excerpt of Bach's Musical Offering I used in this video was made available under a Creative Commons License at MusOpen. (Though alas, no performers were listed on the file.) The audio editing program on which I displayed its waveform is OcenAudio. The graphing program is Desmos. 0:00 Intro - Looking Forward, Looking Back 2:11 ...and looking Bach! 3:44 Desmos Demo - summing transformed sine waves 8:10 Vertical Stretches 15:01 Vertical Shifts 17:52 Setting the (Transformation) Table 19:41 Horizontal Shifts 25:35 Horizontal Stretches 28:41 Hidden Symmetry 30:27 Example: Graphing y = 3cos(4x) - 2 42:04 Example: Graphing y = -2sin(2x - π) + 1 48:04 Alternate solution to the previous problem 52:18 Given graph, find function