which transformations are needed to change the parent sine function to $y = \\frac{1}{4}\\sin\\left(4\\left(x…

which transformations are needed to change the parent sine function to $y = \\frac{1}{4}\\sin\\left(4\\left(x + \\frac{\\pi}{6}\\right)\\right)$?\nvertical stretch of $\\frac{1}{4}$, horizontal stretch to a period of $2\\pi$, phase shift of $\\frac{\\pi}{6}$ units to the left\nvertical compression of $\\frac{1}{4}$, horizontal compression to a period of $\\frac{\\pi}{2}$, phase shift of $\\frac{\\pi}{6}$ units to the left\nvertical stretch of 4, horizontal stretch to a period of $8\\pi$, phase shift of $\\frac{\\pi}{6}$ units to the right\nvertical compression of 4, horizontal compression to a period of $\\frac{\\pi}{4}$, phase shift of $\\frac{\\pi}{6}$ units to the right

which transformations are needed to change the parent sine function to $y = \\frac{1}{4}\\sin\\left(4\\left(x + \\frac{\\pi}{6}\\right)\\right)$?\nvertical stretch of $\\frac{1}{4}$, horizontal stretch to a period of $2\\pi$, phase shift of $\\frac{\\pi}{6}$ units to the left\nvertical compression of $\\frac{1}{4}$, horizontal compression to a period of $\\frac{\\pi}{2}$, phase shift of $\\frac{\\pi}{6}$ units to the left\nvertical stretch of 4, horizontal stretch to a period of $8\\pi$, phase shift of $\\frac{\\pi}{6}$ units to the right\nvertical compression of 4, horizontal compression to a period of $\\frac{\\pi}{4}$, phase shift of $\\frac{\\pi}{6}$ units to the right

Answer

Explanation:

Step1: Analyze vertical transformation

For the function $y = A\sin(B(x - C))$, the coefficient $A$ affects the vertical transformation. Here $A=\frac{1}{4}$. Since $0 < A< 1$, it is a vertical compression by a factor of $\frac{1}{4}$.

Step2: Analyze horizontal transformation for period

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

Step3: Analyze phase - shift

For the function $y=\sin(B(x - C))$, the phase - shift is given by $C$. Here $C=-\frac{\pi}{6}$, which means a phase - shift of $\frac{\pi}{6}$ units to the left.

Answer:

vertical compression of $\frac{1}{4}$, horizontal compression to a period of $\frac{\pi}{2}$, phase shift of $\frac{\pi}{6}$ units to the left