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

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 (-π/2,-5),(-π/4,-2),(0,1),(π/4,-2), and (π/2,-5). (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) g. what is the function of the form y = a cos (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 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 (-π/2,-5),(-π/4,-2),(0,1),(π/4,-2), and (π/2,-5). (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) g. what is the function of the form y = a cos (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 is half the difference between the maximum and minimum values of the function. The maximum value is 1 and the minimum value is - 5. So, $A=\frac{1 - (-5)}{2}=\frac{6}{2}=3$.

Step2: Find the vertical - shift D

The vertical - shift D is the average of the maximum and minimum values. So, $D=\frac{1+( - 5)}{2}=\frac{-4}{2}=-2$.

Step3: Find the period P

The distance between two consecutive quarter - points with the same y - value (e.g., $(-\frac{\pi}{2},-5)$ and $(\frac{\pi}{2},-5)$) is half of the period. The distance between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ is $\pi$. So, the period $P = 2\pi$. Since $P=\frac{2\pi}{B}$ and $P = 2\pi$, then $B = 1$.

Step4: Find the phase - shift C

The general form of the cosine function is $y = A\cos(Bx - C)+D$. We know that $y = 3\cos(x - C)-2$. When $x = 0$, $y = 1$. Substitute these values into the equation: $1=3\cos(0 - C)-2$. First, add 2 to both sides: $3 = 3\cos(-C)$. Then, divide both sides by 3: $\cos(-C)=1$. Since $\cos(-C)=\cos(C)$ and $\cos(0)=1$, then $C = 0$.

Answer:

$y = 3\cos(x)-2$