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

the graph to the right is a function of the form y = a cos (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 = cos 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 = cos x over the interval 0,2π. the quarter points are (3π/4,1), (5π/4,0), (7π/4, - 1), (9π/4,0), and (11π/4,1). 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.) b. what is the value of b? b = . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. what is the value of the phase shift, using a value greater than - p/2 and at most p/2, where p is the period? the phase shift is . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) d. what is the value of c, - π<c<π? c = . simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) e. what is the value of a? a =

the graph to the right is a function of the form y = a cos (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 = cos 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 = cos x over the interval 0,2π. the quarter points are (3π/4,1), (5π/4,0), (7π/4, - 1), (9π/4,0), and (11π/4,1). 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.) b. what is the value of b? b = . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. what is the value of the phase shift, using a value greater than - p/2 and at most p/2, where p is the period? the phase shift is . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) d. what is the value of c, - π<c<π? c = . simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) e. what is the value of a? a =

Answer

Explanation:

Step1: Find the period

The quarter - points of one cycle are given as $\left(\frac{3\pi}{4},1\right),\left(\frac{5\pi}{4},0\right),\left(\frac{7\pi}{4}, - 1\right),\left(\frac{9\pi}{4},0\right),\left(\frac{11\pi}{4},1\right)$. The difference between the first and the last quarter - point is $\frac{11\pi}{4}-\frac{3\pi}{4}=2\pi$. Since the difference between the first and the last of five quarter - points of a cycle is one period, the period $P = 2\pi$.

Step2: Find the value of B

The formula for the period of the function $y = A\cos(Bx - C)+D$ is $P=\frac{2\pi}{B}$. Since $P = 2\pi$, we substitute $P$ into the formula: $2\pi=\frac{2\pi}{B}$. Solving for $B$ gives $B = 1$.

Step3: Find the phase - shift

The standard cosine function $y=\cos x$ has its first quarter - point at $x = 0$. For the function $y = A\cos(Bx - C)+D$, if we consider the first quarter - point $\left(\frac{3\pi}{4},1\right)$ and $B = 1$, the phase - shift is $\frac{3\pi}{4}$.

Step4: Find the value of C

The general form of a cosine function with phase - shift is $y=A\cos(Bx - C)+D$. The phase - shift is given by $\frac{C}{B}$. Since $B = 1$ and the phase - shift is $\frac{3\pi}{4}$, then $C=\frac{3\pi}{4}$.

Step5: Find the value of A

The amplitude $A$ is the distance from the mid - line to the maximum or minimum value of the function. The mid - line is $y = 0$ (since the maximum value is $y = 1$ and the minimum value is $y=-1$), and the maximum value is $y = 1$. The amplitude $A=1$.

Answer:

a. $2\pi$ b. $1$ c. $\frac{3\pi}{4}$ d. $\frac{3\pi}{4}$ e. $1$