the graph to the right is a function of the form y = a sin (bx - c)+d, b > 0. the five - quarter points of…

the graph to the right is a function of the form y = a sin (bx - c)+d, b > 0. the five - quarter points of one cycle of the graph, from left to right, are given below. these five - quarter points on the graph correspond to the five - quarter points on the graph of y = sin x over the interval 0,2π. determine the equation of the specific function that is represented by the given graph based on the association of the labeled quarter points and the quarter points of the graph of y = sin x over the interval 0,2π. the quarter points are (-π/6,-1), (0,-5), (π/6,-1), (π/3,3), and (π/2,-1). what is the function of the form y = a sin (bx - c)+d, where b > 0 and -π < c < π, that is represented by the given graph? (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)

the graph to the right is a function of the form y = a sin (bx - c)+d, b > 0. the five - quarter points of one cycle of the graph, from left to right, are given below. these five - quarter points on the graph correspond to the five - quarter points on the graph of y = sin x over the interval 0,2π. determine the equation of the specific function that is represented by the given graph based on the association of the labeled quarter points and the quarter points of the graph of y = sin x over the interval 0,2π. the quarter points are (-π/6,-1), (0,-5), (π/6,-1), (π/3,3), and (π/2,-1). what is the function of the form y = a sin (bx - c)+d, where b > 0 and -π < c < π, that is represented by the given graph? (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)

Answer

Explanation:

Step1: Find the amplitude $A$

The amplitude $A$ is half the vertical distance between the maximum and minimum values of the function. The maximum value is $3$ and the minimum value is $- 5$. So, $A=\frac{3 - (-5)}{2}=\frac{8}{2}=4$.

Step2: Find the vertical - shift $D$

The vertical - shift $D$ is the average of the maximum and minimum values. So, $D=\frac{3+( - 5)}{2}=\frac{3 - 5}{2}=-1$.

Step3: Find the period $P$

The period of the sine function $y = A\sin(Bx - C)+D$ is related to the horizontal distance between two corresponding quarter - points. The horizontal distance between two consecutive quarter - points of $y=\sin x$ is $\frac{\pi}{2}$. For the given function, if we consider two consecutive quarter - points, say $x_1 = 0$ and $x_2=\frac{\pi}{3}$, the change in $x$ is $\frac{\pi}{3}-0=\frac{\pi}{3}$. For $y = \sin x$, the change between two consecutive quarter - points is $\frac{\pi}{2}$. The period $P$ of the given function can be found using the proportion. The period of $y=\sin x$ is $2\pi$. Let the period of the given function be $P$. We know that the relationship between the horizontal displacements of quarter - points and the periods is the same. The period of the given function: The horizontal distance between two non - consecutive quarter - points (e.g., from $x =-\frac{\pi}{6}$ to $x=\frac{\pi}{2}$) is $\frac{\pi}{2}-(-\frac{\pi}{6})=\frac{3\pi + \pi}{6}=\frac{2\pi}{3}$. Since this represents half of the period, $P=\frac{4\pi}{3}$. And since $P=\frac{2\pi}{B}$, then $\frac{4\pi}{3}=\frac{2\pi}{B}$, solving for $B$ gives $B=\frac{3}{2}$.

Step4: Find the phase - shift $C$

The general form of the sine function is $y = A\sin(Bx - C)+D$. We know that for $y=\sin x$, the first quarter - point is at $x = 0$. For the function $y = A\sin(Bx - C)+D$, when $x = 0$, $y=-5$. Substituting $A = 4$, $B=\frac{3}{2}$, $D=-1$ into $y = A\sin(Bx - C)+D$, we get $-5=4\sin(-C)-1$. Then $- 4=4\sin(-C)$, so $\sin(-C)=-1$. Since $-\pi<C<\pi$, then $C=\frac{\pi}{2}$.

Answer:

$y = 4\sin(\frac{3}{2}x-\frac{\pi}{2})-1$