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…

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/4,2), (3π/8,7), (π/2,2), (5π/8, - 3), and (3π/4,2). a. the period of this function is . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

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/4,2), (3π/8,7), (π/2,2), (5π/8, - 3), and (3π/4,2). a. the period of this function is . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Recall period - quarter - point relationship

The distance between two consecutive quarter - points of a sine function over one period is $\frac{T}{4}$, where $T$ is the period of the function. We can find the period by looking at the difference in $x$ - values of two non - overlapping quarter - points. Let's take the first and the last quarter - points $\left(\frac{\pi}{4},2\right)$ and $\left(\frac{3\pi}{4},2\right)$.

Step2: Calculate the period

The difference in the $x$ - values of these two points is $\Delta x=\frac{3\pi}{4}-\frac{\pi}{4}=\frac{3\pi - \pi}{4}=\frac{\pi}{2}$. Since the distance between two non - overlapping quarter - points separated by 4 quarter - points is $\frac{T}{4}$, and $\frac{T}{4}=\frac{\pi}{2}$. Then, we solve for $T$ by multiplying both sides of the equation by 4. So, $T = 2\pi$.

Answer:

$2\pi$