11-06: Sinusoidal Functions
Periodic Function: a function that repeats itself at regular intervals
Period: one cycle of the graph
Trough: the minimum point
Peak: the maximum point
Equation of the axis: the mean between the peak and trough (halfway)
Amplitude: the distance from the axis to the peak or trough
Sin Functions
Parent Function: y = sinx
| x | y |
|---|---|
| 0 | 0 |
| 90 | 1 |
| 180 | 0 |
| 270 | -1 |
| 360 | 0 |
^^axis → max → axis → min → axis^^
Cos Functions
Parent function: y = cosx
| x | y |
|---|---|
| 0 | 1 |
| 90 | 0 |
| 180 | -1 |
| 270 | 0 |
| 360 | 1 |
^^max → axis → min → axis → max^^
Transformations
Sin and Cos Functions Are Transformed Into…
Sin:
g(x) = asin(k(x-d)) + c
Cos:
g(x) = acos(k(x-d)) + c
Find:
- a: amplitude → a = (max - min)/2
- c: equation of the axis → c = (max + min)/2
- k: number of cycles (period) → k = 360/Period
- d: horizontal (phase) shift → if needed, it is the amount that has been shifted over
- Find max: c + |a| = max
- Find min: c - |a| = min
- Increments of the graph: P/4