The four transformations we have applied to functions have special names for trig functions:[br][br]"Standard" Transformation: [math]a\cdot f\left(b\left(x-h\right)\right)+k[/math][br]Trig Transformation: [math]a\cdot f\left(\omega\left(x-h\right)\right)+k[/math] or [math]a\cdot f\left(\omega x-\phi\right)+k[/math][br][br]The amount of vertical stretch/compression is called Amplitude (A), and is always positive such that A = |a|;[br]The amount of horizontal stretch/compression is called Frequency ([math]\omega[/math], "omega") instead of "b";[br]The [i]absolute [/i]amount of horizontal shift is still called h, but the [i]relative [/i]amount of shift is called Phase Shift ([math]\phi[/math], "phi") and is found as [math]\phi=h\cdot\omega[/math]. The Phase Shift is the same as the Horizontal Shift when [math]\omega=1[/math].[br]The amount of vertical shift is still called k, though it is also sometimes referred to as the "midline".[br][br]This app demonstrates the effect of each transformation.

Each slider controls one of the transformations. Notice that the Period (P), the length of one cycle, is inversely proportional to the Frequency ([math]\omega[/math]) - the higher the frequency, the shorter the period. If the sine function is a sound wave, then the Amplitude (A) corresponds to the volume (loudness), while the frequency corresponds to the pitch. For visible light waves, red has a longer period (lower frequency) than blue (shorter period, higher frequency).[br][br]Phase Shift can be a bit confusing, but it measures the horizontal shift in terms of one wavelength (cycle) being [math]2\pi[/math] radians (or 360 degrees) long, regardless of frequency. So, a 1/4 wavelength phase shift is always [math]\frac{\pi}{2}[/math] radians (or 90 degrees), no matter what the period or frequency is.