Compare the function to $y=a\cos b\left(x-c\right)+d$. The transformed function $y=2\cos \left(4x+4\mathit{\pi }\right)-1$ is not quite in the correct form yet. Factor the $4$ from inside the brackets.


Now, the values of the transformation parameters can be read properly from the function.


Recall the order of transformations matters. Stretches must be performed before translations.

$a=2$ means the amplitude is $2$. There is a vertical stretch by a factor of $2$, and all the $y$-values are multiplied by $2$.

The function $y=\cos x$ has a point at $\left(0\mathrm{,}1\right)$, which is a maximum. The point on the graph of $y=2\cos x$ will be $\left(0\mathrm{,}2\right)$.

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$b=4$ means a horizontal stretch by a factor of $\frac{1}{4}$.

$\begin{array}{rcl}\mathrm{Period}& =& \frac{2\mathit{\pi }}{\left|b\right|}\\ & =& \frac{2\mathit{\pi }}{4}\\ & =& \frac{\mathit{\pi }}{2}\end{array}$

The transformed function has a period of $\frac{\mathit{\pi }}{2}$, or one cycle is completed in $\frac{\mathit{\pi }}{2}$.

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$c=-\mathit{\pi }$ means the phase shift is $-\mathit{\pi }$, or the function has shifted to the left $\mathit{\pi }$ units.

Note that this phase shift does not change the look of this function because of its cyclic nature and the size of the period.

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The vertical displacement is $1$ unit down. All the transformations have been performed, and the graph below represents the function $y=2\cos \left(4x+4\mathit{\pi }\right)-1$.