which transformations are needed to change the parent cosine function to $y = 0.35cosleft(8left(x-\frac{pi}{4…

Explanation:

Step1: Analyze vertical transformation

For the general form of a cosine - function $y = A\cos(B(x - C))+D$, the coefficient $A$ affects the vertical transformation. Here $A = 0.35$. Since $0<|A| = 0.35<1$, it is a vertical compression by a factor of $0.35$.

Step2: Analyze horizontal transformation (period)

The period of the parent cosine function $y=\cos(x)$ is $T_0 = 2\pi$. For the function $y=\cos(Bx)$, the period $T=\frac{2\pi}{|B|}$. In the given function $y = 0.35\cos(8(x-\frac{\pi}{4}))$, $B = 8$. Then $T=\frac{2\pi}{8}=\frac{\pi}{4}$, which means a horizontal compression to a period of $\frac{\pi}{4}$ compared to the parent - function with period $2\pi$.

Step3: Analyze phase - shift

For the function $y = A\cos(B(x - C))+D$, the value of $C$ gives the phase - shift. Here $C=\frac{\pi}{4}$, and since $C>0$, the phase - shift is $\frac{\pi}{4}$ units to the right.

Answer:

vertical compression of 0.35, horizontal compression to a period of $\frac{\pi}{4}$, phase shift of $\frac{\pi}{4}$ units to the right