The image contains a mathematical expression and a question related to it. Here is the transcription: Expression: \[ Y = -8 \cos \left(6x + \frac{2\pi}{3}\right) + 4 \] Question: "Is there a vertical expansion or compression? Pick one and explain why." --- ### Explanation The expression given is a transformation of the cosine function. - **Vertical Expansion/Compression**: - The coefficient of the cosine function, \(-8\), indicates a vertical transformation. Since \(|-8| > 1\), there is a vertical expansion by a factor of 8 and a reflection across the x-axis due to the negative sign. - **Phase Shift**: - The term \(\frac{2\pi}{3}\) inside the cosine function indicates a horizontal phase shift. However, this does not affect vertical expansion or compression. - **Vertical Shift**: - The "+4" outside the cosine function indicates a vertical translation up by 4 units. Overall, the function experiences a vertical expansion due to the amplitude being greater than 1, with an additional reflection and vertical shift.

Consider the given function $y=-8\cos \left(6x+\frac{2\mathrm{\pi }}{3}\right)+4$.

Note that, the general form of the cosine function is $y=A\cos \left(Bx\pm C\right)\pm D$.

Where, $\left|A\right|$ is amplitude, B - cycles from 0 to $2\mathrm{\pi }$. That implies, $\mathrm{period}=\frac{2\mathrm{\pi }}{\mathrm{B}}$.

C - horizontal shift (known as phase shift when B = 1), and D - vertical shift (displacement).

Compare the given function $y=-8\cos \left(6x+\frac{2\mathrm{\pi }}{3}\right)+4$ with $y=A\cos \left(Bx\pm C\right)\pm D$.

$A=-8, B=6, C=\frac{2\mathrm{\pi }}{3}, \mathrm{and} \mathrm{D}=4$.

That implies,

$\begin{array}{rcl}\mathrm{Amplitude}& =& \left|A\right|\\ & =& \left|-8\right|\\ & =& 8\end{array}$

$\begin{array}{rcl}\mathrm{Period}& =& \frac{2\mathrm{\pi }}{6}\\ & =& \frac{\mathrm{\pi }}{3}\end{array}$

$\begin{array}{rcl}\mathrm{horizontal} \mathrm{shift}& =& C\\ & =& \frac{2\mathrm{\pi }}{3}\end{array}$

$\begin{array}{rcl}\mathrm{vertical} \mathrm{shift}& =& D\\ & =& 4\end{array}$