select the correct answer. function g is a transformation of the parent sine function such that g(x) =…
select the correct answer. function g is a transformation of the parent sine function such that g(x) = sin(-3x) + 2. which graph represents function g?
Answer
Explanation:
Step1: Analyze the transformation of the sine - function
The general form of a sine - function transformation is $y = A\sin(Bx - C)+D$. For the function $g(x)=\sin(-3x)+2$, we know that $\sin(-\alpha)=-\sin(\alpha)$, so $g(x)=-\sin(3x)+2$. The amplitude $A = 1$, the period $T=\frac{2\pi}{|B|}=\frac{2\pi}{3}$, and the vertical shift $D = 2$.
Step2: Consider the key - points
The parent function $y = \sin(x)$ has key - points at $(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2}, - 1),(2\pi,0)$. For $y =-\sin(3x)+2$, when $x = 0$, $y=-\sin(0)+2=2$. When $3x=\frac{\pi}{2}$ (i.e., $x=\frac{\pi}{6}$), $y=-\sin(\frac{\pi}{2})+2=-1 + 2=1$. When $3x=\pi$ (i.e., $x=\frac{\pi}{3}$), $y=-\sin(\pi)+2=2$.
Step3: Eliminate wrong options
The function $g(x)=-\sin(3x)+2$ has a period of $\frac{2\pi}{3}\approx2.09$. Option A has a period of approximately $2$ and is shifted up by $2$ units and is a negative - sine function. Option B has a different period and shape.
Answer:
A.