# Given any sinusoidal equation, how do you identify the type of transformations that are made?

##### 1 Answer
Mar 12, 2018

Example: Describe the transformations to get $g \left(x\right) = 2 \sin \left(3 \left(x + \frac{\pi}{4}\right)\right) + 2$ from $f \left(x\right) = \sin x$

Here are what each of the parameters in the equation $y = a \sin \left(b \left(x - c\right)\right) + d$:

$a \to$ vertical stretch
$\frac{1}{b} \to$horizontal stretch
$c \to$ phase shift
$d \to$vertical transformation

So in the given equation, we have a vertical stretch by a factor of $2$, a horizontal stretch by a factor of $\frac{1}{3}$, a transformation $\frac{\pi}{4}$ units left and a transformation $2$ units up.

Hopefully this helps!