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 distance between the maximum and minimum values. The maximum value is 2.5 and the minimum is - 6.5. So $A=\frac{2.5-(-6.5)}{2}=\frac{2.5 + 6.5}{2}=\frac{9}{2}=4.5$.

Step2: Find the vertical - shift D

The vertical - shift D is the average of the maximum and minimum values. So $D=\frac{2.5+( - 6.5)}{2}=\frac{2.5-6.5}{2}=\frac{-4}{2}=-2$.

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 for $y=\cos x$, the first quarter - point is at $x = 0$. For our function, when $x = 0$, $y = 1$. Substituting $A = 4.5$, $B = 1$, $D=-2$ into $y = A\cos(Bx - C)+D$, we get $1=4.5\cos(-C)-2$. Then $4.5\cos(-C)=3$, so $\cos C=\frac{3}{4.5}=\frac{2}{3}$. Since $-\pi<C<\pi$, we can use the inverse - cosine function. Also, we can use the fact that the cosine function $y = A\cos(Bx - C)+D$ passes through the point $(0,1)$. Substituting into the equation $y = 4.5\cos(x - C)-2$ gives $1 = 4.5\cos(-C)-2$. Since $\cos(-C)=\cos C$, we have $\cos C=\frac{2}{3}$. And since the graph is a cosine function and we know the quarter - points, we can see that the phase - shift $C = 0$.

Answer:

$y = 4.5\cos(x)-2$