33 lines
1.4 KiB
Markdown
33 lines
1.4 KiB
Markdown
|
|
|
|
Given the above graph:
|
|
- At the origin, $sin(x) = 0$ and $cos(x) = 1$
|
|
- A full wavelength takes $2\pi$
|
|
|
|
# Manipulation
|
|
| Formula | Movement |
|
|
| ---------------- | ---------------------------------- |
|
|
| $y = cos(x) - 1$ | Vertical shift down by 1 |
|
|
| $y = 2cos(x)$ | Vertical stretch by a factor of 2 |
|
|
| $y = -cos(x)$ | Flip over x axis |
|
|
| $y = cos(2x)$ | Horizontal shrink by a factor of 2 |
|
|
# Periodic Functions
|
|
A function is considered periodic if it repeats itself at even intervals, where each interval is a complete cycle, referred to as a *period*.
|
|
|
|
|
|
# Sinusoidal Functions
|
|
A function that has the same shape as a sine or cosine wave is known as a sinusoidal function.
|
|
|
|
There are 4 general functions:
|
|
|
|
| $$A * sin(B*x - C) + D$$ | $$ y = A * cos(B*x -c) + D$$ |
|
|
| ------------------------ | ---------------------------- |
|
|
|
|
|
|
| Variable | Meaning |
|
|
| -------- | --------------------------------------------------------------------------------------------- |
|
|
| $A$ | The amplitude of a function can be found by taking $\|A\|$. The sign flips it over the x axis |
|
|
| $B$ | Horizontal, or phase |
|
|
|
|
|
|
$$ y = A * \sin(B(x-\frac{C}{B})) $$ |