General form
If we summarize the equations of stretching and shifting, the functions $y=\sin(x)$ and $y=\cos(x)$ have the following generalized equation:
$y=\color{red}{a}\cdot\sin(\color{blue}{b}\cdot(x+\color{green}{c}))+\color{grey}{d}$
$y=\color{red}{a}\cdot\cos(\color{blue}{b}\cdot(x+\color{green}{c}))+\color{grey}{d}$
$y=\color{red}{a}\cdot\cos(\color{blue}{b}\cdot(x+\color{green}{c}))+\color{grey}{d}$
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Remember
- $\color{red}{a}$ stretches / compresses the graph parallel to the y-axis (amplitude change).
- $\color{blue}{b}$ stretches / compresses the graph parallel to the x-axis (period change).
- $\color{green}{c}$ shifts the graph in the x direction.
- $\color{grey}{d}$ shifts the graph in the y direction.
Often you find the sine and cosine function also with the function equation:
$y=\color{red}{a}\cdot\sin(\color{blue}{b}x+\color{blue}{b}\color{green}{c})+\color{grey}{d}$
$y=\color{red}{a}\cdot\cos(\color{blue}{b}x+\color{blue}{b}\color{green}{c})+\color{grey}{d}$
$y=\color{red}{a}\cdot\cos(\color{blue}{b}x+\color{blue}{b}\color{green}{c})+\color{grey}{d}$
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Note
With the second function term, it is not always possible to read off the displacement in the x-direction. It is therefore advisable to exclude the factor $b$ in order to obtain the first spelling.