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

which transformations are needed to change the parent cosine function to $y = 0.35cosleft(8left(x-\frac{pi}{4}\right)\right)$?\nvertical stretch of 0.35, horizontal stretch to a period of $16pi$, phase shift of $\frac{pi}{4}$ units to the right\nvertical compression of 0.35, horizontal compression to a period of $4pi$, phase shift of $\frac{pi}{4}$ units to the left\nvertical compression of 0.35, horizontal compression to a period of $\frac{pi}{4}$, phase shift of $\frac{pi}{4}$ units to the right\nvertical stretch of 0.35, horizontal stretch to a period of $\frac{pi}{4}$, phase shift of $\frac{pi}{4}$ units to the right

which transformations are needed to change the parent cosine function to $y = 0.35cosleft(8left(x-\frac{pi}{4}\right)\right)$?\nvertical stretch of 0.35, horizontal stretch to a period of $16pi$, phase shift of $\frac{pi}{4}$ units to the right\nvertical compression of 0.35, horizontal compression to a period of $4pi$, phase shift of $\frac{pi}{4}$ units to the left\nvertical compression of 0.35, horizontal compression to a period of $\frac{pi}{4}$, phase shift of $\frac{pi}{4}$ units to the right\nvertical stretch of 0.35, horizontal stretch to a period of $\frac{pi}{4}$, phase shift of $\frac{pi}{4}$ units to the right

Answer

Explanation:

Step1: Analyze vertical transformation

For the function $y = A\cos(B(x - C))$, the value of $A$ determines the vertical transformation. Given $y=0.35\cos\left(8\left(x-\frac{\pi}{4}\right)\right)$, since $|A| = 0.35<1$, it is a vertical compression by a factor of $0.35$.

Step2: Analyze horizontal - period transformation

The period of the parent cosine function $y = \cos(x)$ is $2\pi$. For the function $y=\cos(Bx)$, the period $T=\frac{2\pi}{B}$. Here $B = 8$, so $T=\frac{2\pi}{8}=\frac{\pi}{4}$, which means there is a horizontal compression to a period of $\frac{\pi}{4}$.

Step3: Analyze phase - shift transformation

For the function $y=\cos(x - C)$, the phase - shift is $C$ units. In $y = 0.35\cos\left(8\left(x-\frac{\pi}{4}\right)\right)$, $C=\frac{\pi}{4}$, so there is a phase - shift of $\frac{\pi}{4}$ 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